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li*Further, K GZ;li(K) is dense in KI with respect to 1111,because K = k & K is dense in k Bpk K. Hence K, is exactly the completion of K. V
= (5) (z), for (p,Q> =
CR,Q> =
where ui (1 5 j < P) is a set of representatives of o/p. Hence P” = PP”, 1. If we normalize
p by putting P(O) = 1,
(7.2)
we have p” = P“.
Conversely, without the theory of Haar measure, it is easy to see that there is a unique invariant measure on k+ subject to (7.2). Everything so far in this section has depended not on the valuation 1 1 but only on its equivalence class. The above considerations now single out one valuation as particularly important.
GLOBAL
FIELDS
51
DEFINITION. Let k be a field with discrete valuation 1 1 and residue class fieId with P C 00 elements. We say that 11 is normalized if 17cl= Pl, where p = (7~). THEOREM. Suppose,further, that k is complete with respect to the normalized valuation I I. Then Aa+Po> = IPI where ,a is the Haar measureon kf normalized by ,u(o) = 1.
We can express the result of the theorem in a more suggestive way. Let /? E k, p # 0 and let p be a Haar measure on k+ (not necessarily normalized as in the theorem). Then we can define a new Haar measure pfl on k+ by putting CL,@) = p(PE) (E c k+). But Haar measure is unique up to a multiplicative constant and so &(E) = pL(PE) = fp(E) for all measurable sets E, where the factor f depends only on p. The theorem states that f is just I/II in the normalized valuation. [The theory of locally compact topological groups leads to the consideration of the dual (character) group of k+ . It turns out that it is isomorphic to k+. We do not need this fact for class field theory so do not prove it here. For a proof and applications see Tate’s thesis (Chapter XV of this book) or Lang: “Algebraic Numbers” (Addison Wesley), and, for generalizations; Weil: “Adeles and Algebraic Groups” (Princeton lecture notes) and Godement: Bourbaki seminars 171 and 176. The determination of the character group of k” is local classfield theory.]
The set of nonzero elements of k form a group k” under multiplication. Clearly multiplication and taking the reciprocal are continuous with respect to the topology induced in k” as a subset of k, so k” is a topological group with this topology.? We have k” DEEE~
where E is the group of units of k and where E, is the group of einseinheiten, i.e. the E E k with Is 1I < 1. Clearly E and E, are both open and closed ink”. Obviously k”/E is isomorphic to the additive group Zf of integers with the discrete topology, the map being YC’E+V
(v E Z).
Further, E/E, is isomorphic to the multiplicative group rcx of the nonzero elements of the residue class field, where the finite group JC’ has the discrete topo1ogy.S Further, E is compact, so k” is locally compact, Clearly the t We shall later have to consider the situation for topological rings R, where R” in general is given a different topology from the subset topology. $ xX is cyclic of order P  1. It can be shown that k always contains a primitive P  1th root of unity p and so the elements of k” are just the Yp%, EEEl, i.e. k” is the direct product of Z, Z/(P  l)Z and El. In fact let f(X) = Xpel  1 and let M:E o be such that 0:mod p generates K’. Then 1f(a) 1 < 1, 1f’(a)1 = 1. Then by Hensel’s Lemma (App. C) there is p E k such that f(p) = 0, p = a (mod p).
52
J. W.
additive a Haar obvious Finally LEMMA.
S. CASSELS
Haar measure on Ei is also invariant under multiplication so gives measure on E,: and this gives the Haar measure on k” in an way. we note the k+ and k” are totally disconnected (the only connected sets are
points). Beweis. Klar. [It is perhaps worth mentioning that k” and k+ are locally isomorphic teristic 0. We have the exponential map 01+expo:
if k has charac
= X1$
valid for all sut?kiently small a with its inverse ,og a = z (I+%
 1)” n
valid for all a sufficiently near to 1.1
8. Normed
Spaces
Let k be a field with valuation 1 1 and let V be a vector space over k. A realvaluedfunction 1111on V is called a norm tf
DEFINITION.
(1) llall > 0 for a czV, a # 0.
(2)lla+fJII 5 lbll+II% (3) /aall = Ial llall @Ek, DEFINITION.
Two
norms
a EW. II II 1,
11 II2
on the same space are equivalent tf
there exist constants cl, c2 such that
II4 224lQllz IMI s 44 This is clearly an equivalence relation. LEMMA. Suppose that k is complete with respect to I I and that V isfinitedimensional. Then any two norms on V are equivalent. Note. As we shall see, completeness is essential. Proof. Let a1, . . . , a, be any basis for V. We define a norm II IlO by
IICr”a”llo= my ILI. It is enough to show that any norm II 11is equivalent
to 1 I,,.
IICe.4122c Ia IMI s clIICtmanllo with cl = C ,pnl,.
Clearly
,
53
GLGBAL FIELDS
Suppose that there is no cz such thatt ho 5 4lQII~ Then for any E > 0 there exist Ti,. . ., & such that 0 < IIC Ca,ll 5 Emax It.1. By symmetry we may suppose that max IL1 = ltNl and then by homogeneity
that &=
Form=
1,2,...,wethushave&,(l ll~~~t...a.
1.
+ aNIl f 0
(m + a>;
SO
Nl The lemma being trivial for N = 1, we may suppose by induction that it is true for the (N I)dimensional space spanned by al,. . . , a, I and hence ~5n,d~“,ml+o V,rrf+%~) for 1 5 n I N 1. Since k is complete there are c.* E k with Ic,nt*p
(m,aJ).
Then
.
IIN~‘t.*an+aNll li; IIN~ltn,nan+%ll +NilIt?tn.mjllanll+O cm+ co) in contradiction
to (1).
9. Tensor Product We need only a special case. Let A, B be commutative rings containing a field k and suppose that B is of finite dimension N over k, say with basis 1 =@1,02 ,..., 0,. Then B is determined up to isomorphism by the multiplication table % %I = c c~nrn~n c,, E k. We can define a new ring C containing k whose elements are expressions of t When k is not merelycompletewith respectto 11but locally compact,whichwill be the caseof primary interest,onecan arguemore simplyas follows. By what has been shownalready,the function Ilollis continuousin the II/lotopology,andsoattainsits lower bound 6 on Ilull = 1. Then 6 > 0 by condition (i), and then llalloI Slllajl by homogeneityfor all a.
54
J. W. S. CASSELS
the type a,EA c amwm where the w,,, have the same multiplication rule
as the q,,.
QJcPm = x GAmPJn There are ring isomorphisms i:a+aw,
and j:~L*m+~L*II, of A and B respectively into C. It is clear that C is defined up to isomorphism by A and B and is independent of the particular choice of basis w,. We write C=A@,B since it is, in fact, a special case of the ring tensorproduct. [The reader will have no difficulty in checking that C together with the maps i, j possesses the defining Universal Mapping Property.]
Let us now suppose, further, that A is a topological ring, i.e. has a topology with respect to which addition and multiplication are continuous. The map Cash ‘(al,. . .,aJ is a l 1 correspondence between C and N copies of A (considered as sets). We give C the product topology. It is readily verified (i) that this topology is independent of the choice of basis q,. . . cc, and (ii) that multiplication and addition in C are continuous with respect to it; i.e. C is now a topological ring. We shall speak of this topology on C as the tensor product topology. Now let us drop our supposition that A has a topology but suppose that A, B are not merely rings but fields. LEMMA. Let A, B be jields containing the field k and suppose that B is a separable extension of degree [B : k] = N < co. Then C = A & B is the direct sum of a finite number of fields Kj, each containing an isomorphic image of A and an isomorphic image of B. Proof. By a wellknown theorem (appendix B) we have B = k(B) where fvlh;nO,l f; someNseyble f(X) E k[X] of degree N irreducible in k[X]. . . , /l  is a basis for B/k and so A I& B = Au] where bit. . . , b” are linearly independent over A and f(J) = 0. Although f(X) is irreducible in k[X] it need not be in A[X], say fwl where gj(X) E A[X]
is irreducible.
= l Jp,cx, The gj(X) are distinct because f(X)
is
55
GLOBALFIELDS
separable.
Let K, = A(Bj) where Sjclsi> = 0. Clearly the map AC&B%,
given by h(x) E AX
wb 11: W)
is a ring homomorphism. We thus have a ring homomorphism Pl@...@PJ ARABS
8 Kj. (9l) t5jrJ Let h(B), h(X) E A[X] be in the kernel. Then h(X) is divisible by every g,(X), so also byf(X), i.e. h(p) = 0. Thus (9.1) is an injection. Since both sides of (9.1) have the same dimension as vector spaces over A it must be an isomorphism, as required. It remains to show that the ring homomorphisms QB+A&B:K, are injections. If Aj2j(B)# 0 for any /3 E B then Aj(81) # 0 for all PI # 0 H ence all we have to show is that A, does because Aj@) = njcs,)nj~~‘). not map the whole of B onto 0: and this is trivial. COROLLARY. Let a E B and let F(X) E k[X], Gj(X) E A[X] (1 I i < J) be the characteristic polynomial of a over k and of the image of a under B+A&B+Kj over A respectively. Then F(X) =
n
ISjSJ
Gj(X)*
(9.2)
Proof: We show that both sides of (9.2) are the characteristic T(X) of the image of a in A @‘k B over A. That E’(X) = T(X) once by computing the characteristic polynomial in terms where ol,. . ., o, is a basis for B/k. That T(X) m,,** WN, follows similarly by using a base of A&B= QK, composed of bases of the individual KjIA. COROLLARY. For a E B we have Norm,,,a = 1jp%L4 Trace,,, a = i *paCeK,,A
polynomial follows at of a basis = IIGj(X)
a a.
ProoJ For the norm and trace are just the second and the last coefficient in the characteristic equation.
J. 56 10. Extension of Valuations
W.
S. CASSELS
Let k c K be fields and 1 I, )I b e valuations on k, K respectively. We say that 1111extends I I if lb1 = IJbII for all b E k. THEOREM. Let k be complete with respect to the valuation k and let K be an extension of k with [K: k] = N < 00. Then there is precisely one extension of 1 1 to K namely (10.1) llall = INorm,,l,crll’N. Proof. Uniqueness. K may be regarded as a vector space over k and then I\ 11is a norm in the sense defined earlier. Hence any two extensions I( ((r and 11II2 of I 1 are equivalent as norms and so induce the same topology in K. But as we have seen two valuations which induce the same topology are equivalent valuations, i.e. 11I! 1 = )I 11; for some c. Finally c = 1 because llblll = /lb/l2 for all b E k. Existence. For a proof of existence in the general case see e.g. E. Artin: “Theory of Algebraic Numbers” (Striker, Gottingen) and for a proof valid for separable nonarch. discrete valuations see Chapter I, 5 4, Prop. Here we give a proof (suggested by Dr. Geyer at the 1, Corollary. conference) valid when k is locally compact, the only case which will be used. In any case it is easy to see that the definition (10.1) satisfies the conditions (i) that llal\ 2 0 with equality only for a = 0 and (ii) 11aflll = [[a11l//?/l: the difficulty is to show that there is a constant C such that /all 5 1 implies 111+a\] I C. Let 11Ilo be any norm on K considered as a vector space over k. Then /alI defined by (10.1) is a continuous nonzero function on the compact set (Ial(O = 1, so A 2 llall 2 6 > 0 for some constants A, 6. Hence by homogeneity ArM
26>0.
II40
(alla#O).
Suppose, now, that IIan 5 1. Then IIan 5 6r and so
lP+4 s Al11+4J 22~Wllo+11410~ 5 A(~~~~)o+~‘) = C (say),
as required. Formula. Geyer’s existence proof also gives (10.1). But it is perhaps worth noting that in any case (10.1) is a consequence of unique existence, as follows. Let L 1 K be a finite normal extension of k. Then by the above there is a unique extension of I I to L which we shall denote also by 1111. If r~ is an automorphism of L/K then
114~ = lbll
57
GLOBALFIEZDS
is also an extension of 11 to L, so 1 us = (I 11,i.e. lluall = llall (all c1E L). But now Norm,,,a = ulao2a.. . ona oN are automorphisms of L/k. foraoK,whereol,...,
Hence
INormK,k aI = IINo~Ic,G~~
= llQllN~ as required. COROLLARY. tit
O1,...,
an be a basis for KJk.
Then there are constants
ct, c2 such that
c * ICbJ@A< c ’
max lb.1
’
for b l,...,bNEk(notaNO). Proof. For 12 b, o “1 an d max lb,,1 are two norms on K considered as a vector space over k. COROLLARY 2. A finite extension of a completely valued$eld k is complete with respect to the extended valuation. For by the preceding corollary it has the topology of a finitedimension vector space over k. When k is no longer complete under I I the position is more complicated: THEOREM. Let K be a separable extension of k of degree [K : k] = N < CO. Then there are at most N extensions of a valuation I 1 of k to K, say II 11, (1 I j s J). Let I%, K, be the completion of k req. K with respect to I I, resp. 1111,. Then (10.2) k@kK= @ Kj 1SjS.l algebraically and topologically, where the R.H.S. is given the product topology. Proof. We know already that R @I K is of the shape (10.2) where the Kj are finite extensions of R. Hence there is a unique extension I 17 of I 1 to the K, and the K, are complete with respect to the extended valuation. Further, by a previous proof, the ring homomorphisms are injections.
+K&&K+K, Hence we get an extension II 11,of I I to K by putting
IlPllj = Inj
58
J. W. S. CASSELS
It remains to show that the 11Ilj are distinct and that they are the only extensions of I I to K. Let 11]I be any valuation of K extending I I. Then II 11extends by continuity to a realvalued function of R Ok K, a function also to be denoted by 11II. By continuity we have
Ib+~II5 maxII4 Ml c1pEk(g)K ll@ll= lbllIPII > ’ * We consider the restriction of II I/ to one of the Kj. If lltlll # 0 for some CIE Kj the ll~lll = ll~ll IIuP‘II for every P + 0 in Kj SO l/~/l # 0. Hence either I] II is identically 0 on Kj or it induces a valuation on Kj. Further, II 11cannot induce a valuation on two of the KP For (a,@O@. . . @O).(O@a,OO.. .@O) = (000. . .@O) and so II%II ll%II = 0
%EKlY
a2 E&*
Hence II II induces a valuation in precisely one of the Kj and it clearly extends the given valuation I I of li. Hence II II = II IIj for precisely one j. It remains only to show that (10.2) is also a topological homomorphism. For ul,. . .,pJ~K~@...@K,put 11& . * *,PJ>llO =l~~~JllPjllj* Clearly, II Ijo is a norm on the R.H.S. of (10.2), considered as a vector space over k and it induces the product topology. On the other hand, any two norms are equivalent, since R is complete, and so II II0 induces the tensor product topology on the lefthand side of (10.2). COROLLARY. Let K = k(P) and let f(x) E k[X] be the irreducible equation for j7. Suppose that
in k[X],
where the gj are irreducible.
Then Kj = &?j) where gj
11. Extensions of Normalized Valuations Let k be a field with valuation 1 I. We consider (1) 1 1 is d iscrete nonarch. and the residue (2(i)) The completion of k with respect to (2(ii)) The completion of k with respect to II
the three cases: class field is finite. is R. is C.
[In virtue of the remarks in 3 7, these cases can be subsumed in one: the completion is locally compact.]
k
In case (1) we have already defined a normalized valuation (9 7). In case (2(i)) we say I I is normalized if it is the ordinary absolute value and in
GLOBAL
59
FELDS
case (2(ii)) if it is the square of the absolute value. map a: <+a<
<EL+
Thus in every case the
(aEE)
of the additive group R+ of the completion of k multiplies on li’ by la[: and this characterizes the normalized equivalent ones.
the Haar measure valuation among
LEMMA. Let k be complete with respect to the normalized valuation 11 and let K be an extension of k of degree [K: k] = N < co. Then the normalized valuation 11II of K wh’ICh is equivalent to the unique extension of I I to K is given by the formula
llall = INor+ka(
(a E K).
Proof. By the preceding section we have
(11.1) llall = INormK,kalc (a E K> for some real c > 0 and all we have to do is to prove that c = 1. This is trivial in case 2 and follows from the structure theorems of Chapter I in case 1. Alternatively one can argue in a unified way as follows. Let ml,. . *, ON be a basis for K/k. Then the map ~=~bwG,...,5~) (tf1,.,LEk) gives an isomorphism between the additive group K+ and the direct sum ONkf of N copies of kf, and this is a homomorphism if the R.H.S. is given the product topology. In particular, the Haar measures on Kt and CBNkf are the same up to a multiplicative constant. Let b E k. Then the map b:E+bS
of Kt is the same as the map of eNkt Hence
(t 1,...,5N),(b~l,...,br~) and so multiplies the Haar measure by IbIN, since I I is normalized.
llbll= IbIN* But NormKltb = bN and so c = 1 in (11.1). In the incomplete case we have THEOREM. Let I I be a normalized valuation of a field k and let K be a finite extension of k. Then
where the (1IIj are the normalized valuations equivalent to the extensions of to K.
II
60
J. W.
Proof.
S. CASSELS
Let
E@kK=
@ Kj,
15lZZ.I
where li is the completion of K. Then (I 9) Norm,,, a = n (Norm,,,x a). lSj<J The theorem now follows from the preceding lemma and the results of 5 10. 12. Global
Fields By a global field k we shall mean either a finite extension of the rational field Q or a finite separable? extension of F(t), where F is a finite field and t is transcendental over F. We shall focus attention in the exposition on the extensions of Q (algebraic number case) leaving the extension of F(t) (function field case) to the reader. LEMMA. Let a # 0 be in the global field k. Then there are only finitely many unequivalent valuations 1 1 of k for which [aI > 1. Proof. We know this already for Q and F(t). Let k be a finite extension of Q, so a”+a,a”‘+. . . +a, = 0 forsomenanda,,...,a,,. If[ 1 is a nonarch. valuation of k we have Iaj”=jaala”l...aa,i 5 max(l,Iar‘)max(jarj,.
. .,\a$
and so Ial 5 maxU,lall,. . .,la$. Since every valuation of Q has finitely many extensions to k and since there are only finitely many arch. valuations altogether, the theorem for k follows from that for Q. All the valuations of a global field k are of the type described in $ 11, since this’is true of Q and F(f). Hence it makes sense to talk of normalized valuations. THEOREM. Let a E k, where k is a globalfieId and a # 0. Let I IV run through all the normalized valuations of k. Then IaIV = 1 for all except finitely many v and
Note. We shall later give a less computational
proof of this.
t Thii condition is not redly necessary. If k is any Ikite extension of F(t) there is a “separating element” s, i.e. an s E k such that k is a finite separable extension of F(S).
GLOBAL
Proof. By the lemma Ial0 < 1 for many). Similarly la‘lV I 1 for almost Let V run through all the normalized VI V to mean that the restriction of u to
by the preceding But if now
section.
61
FIELDS
almost all 1) (i.e. all all u, so lalD = 1 for valuations of Q [or Q is equivalent to K
This reduces the theorem
except finitely all 0. F(t)] and write Then almost
to the case k = Q.
b=kn~+Q, P
where p runs through all the primes and BP E Z, we have lblp = p+p for the padic valuation
1 IP and lblo3 =,pz’* P
for the absolute value I Im. Q.E.D. Let K be a finite separable extension of the global field k. Then for every valuation v of k we have an isomorphism
where k, is the completion of k with respect to u and K,, . . . , K,, are the completions of K with respect to the extensions V,, . . . , V, of u to K (5 lo), the number J = J(u) depending on u. We shall later need the LEMMA. Let cut,. . ., a+,?be a basis for K/k. Then for almost all normalized u we have ~,o~~~o~...~~No=~l~...~~J (12.1) where N = [K: k], o = o, is the ring of integers of k for I I,, and 0, is the ring of integers for I IV, (1 5 j 5 J). Here we have identified with its canonical image in k, @ K. Proof. The L.H.S. of (12.1) is included in the R.H.S. provided lo.&, 5 1 (1 I n 5 N, 1 5 j < J). Since IaIr 5 1 for almost all follows that L.H.S. c R.H.S. for almost all u. To get an inclusion the other way we use the discriminant NYl,.
* .,YN> =
det,,
dtraCeK/k
where yr,. . ., yN E k, Bk K. If y. E R.H.S. traceKlk
YntYn
=
c l&jsiJ
Y~Y,),
(1 < n I N) we have (5 9)
traceKj/Eymyn
and so %b.
c K, aEK
. * >YN) E 0,.
E o =
Ov
that Y it
J. W. S. CASSELS
62
Now suppose that a E R.H.S. and that ,!I=$b,,w,ER.H.S.
1 Then for any m, 1 I m I N we have
I.@, ,..., w,,~,P,w,,+~
(12.2)
(b,Ek,).
,..., d=b:W’,
,..., R>,
and so dbi E o,
(1 I m 5 N)
where d=D(o,,...,w,)Ek.
But (Appendix B) we have d # 0, and so ldlv = 1 for almost all u. For almost all u the condition (12.2) thus implies b, E o, (15 m 5 N), i.e. R.H.S. c L.H.S. This proves the lemma. [COROLLARY.
Almost all v are unramijied in the extension K/k.
For by the resultsof Chapter I a necessary and sutlicientconditionfor v to be unramifiedis that there are yl,. . ., yN E R.H.S. with ID(y,,. . ., yN)lv= 1. And for almost all v wecanput yn= ~f”~.]
13. Restricted Topological Product We describe here a topological tool which will be needed later: DEFINITION.Let sZA(A EA) be a family of topological spacesand for almost alIt L let On c Q, be an open subsetof a,. Consider the spacen whosepoints are sets u = h&h where aA E a, for every A and cr, E Onfor almost all 1. We give Q a topology by taking as a basis of open sets the sets lm where rl c G12,is open for all rZ and rA = O1 for almost all 1. With this topology R is the restricted topologicalproduct of the R, with respect to the O1. COROLLARY. Let S be a finite subset of A and let Rs be the set of a E fi with aAE O1 (2 q!S), i.e.
(13.1) Then G!, is open in Sz and the topology induced in as as a subset of R is the sameas the product topology. Beweis. Klar.
The restricted topological product but not on the individual 0,: t i.e. all exceptpossiblyfinitely many.
depends on the totality
of the 0,
GLOBAL
FIELDS
63
LEMMA. Let 0; c Jz, be open sets defined for almost all II and suppose that OA = 0: for almost all 2. Then the restricted product of the f2, with respect to the 0; is the same ast the restrictedproduct with respect to the 0,. Beweis. Klar. LEMMA. Suppose that the RA are locally compact and that the 0, are compact. Then R is locally compact. Proof. The as are locally compact by (13.1) since S is finite. Since n = u C& and the S& are open in 0, the result follows. DEFINITION. Suppose that measures ,uI are defined on the RA with ~~(0,) = 1 when OA is defined. We define the product measure p on R to be that for which a basis of measurable sets is the
where MA c Sz, has finite PAmeasure and MA = On for aImost all 1 and where
COROLLARY.
The restriction of ,u to Q, is just the ordinary product measure.
14. Adele Ring (or Ring of Valuation Vectors) Let k be a global field. For each normalized valuation 1 1” of k denote by k, the completion of k. If 1 1” is nonarchimedean denote by D, the ring of integers of k,. The adele ring V, of k is the topological ring whose underlying topological space is the restricted product of the k, with respect to the D, and where addition and multiplication are defined componentwise: (14.1) (a+/% = av+/L a,BE 6. W% = aA It is readily verified (i) that this definition makes sense, i.e. if a, fi E V, then an, a+ fl whose components are given by (14.1) are also in V, and (ii) that addition and multiplication are continuous in the V,topology, so V, is a topological ring, as asserted. V, is locally compact because the k, are locally compact and the o, are compact (5 7). There is a natural mapping of k into V’ which maps ccE k into the adele every one of whose components is a: this is an adele because a E o, for almost all O. The map is an injection, because the map of k into any k, is an injection. The image of k under this injection is the ring of principal adeles. It will cause no trouble to identify k with the principal adeles, so we shall speak of k as a subring of Yk. LEMMA. Let K be afinite (separable) extension of the globaljield k. Then (14.2) &c&K = v, t A purist wouldsay “canonicallyisomorphicto”.
64
I.
W.
S. CASSELS
aIgebraicaIIy and topologically. In this correspondence k &K = K c V’ Blk K, where k c V,, is mapped identically on to K c V,. Proof. We first established an isomorphism of the two sides of (14.2) as topological spaces. Let ml,. . . , oN be a basis for K/k and let u run through the normalized valuations of k. It is easy to see that the L.H.S. of (14.2), with the tensor product topology, is just the restricted product of the k,C&K=k,o,@...@kvcoN (14.3) with respect to the o,,t& @ . . . @ o,wN. (14.4) But now (cf. $lO), (14.3) is just (14.5) G, CB. . .c?3Kv,,
GLOBAL
FIELDS
65
structure to find a neighbourhood U of 0 which contains no other elements of k+. We take for U the set of a = (GC,}E Vo! with Iaalg, < 1 Ia& 5 1 W ~1, where 1 IP, 1 I are respectively the padic and the absolute values on Q. If b E Q n U then in the first place b E Z (because lblp 5 1 for all p) and then b = 0 because lblm < 1. Now let W c V$ consists of the a = {a”} with la,la S 3, l~plp 5 1 We show that every adele /I is of the shape beQ,
j = b+a,
For each p we can find an rP = zpp such that
611 10.
(Z,EZ,
(14.7)
aEW.
XpEZ,
x, 2 0)
I&4 5 1 and since a is an adele we may take rP = 0 (almost all p). Hence r = c rp is well defined and Nowchoos:soZsuchthatlhr”l
(allp)S
Ifi,
rs1
S $.
Then b = r +s, fl = a  b do what is required. Hence the continuous map W + V,‘/Q’ induced by the quotient map V$ + V,‘/Q’ is surjective. But W is compact (topological product of la,la 2 3 and the D& and hence so is V,‘/Q’. As already remarked, Vz is a locally compact group and so it has an invariant (Haar) measure. It is easy to see that in fact this Haar measure is the product of the Haar measures on the k, in the sense described in the previous section. COROLLARY 1. There is a subset W of Vk defined by inequalities of the type I<& 5 a,, where 6, = 1 for almost all v, such that every cpE V, can be put in the form
yak Proof. For the W constructed in the proof is clearly contained in some W of the type described above. (P = e+y,
OEW,
COROLLARY 2. V,‘/k’ hasfinite measure in the quotient measure induced by the Haar measureon V,’ .
66
J. W. S. CASSELS
Note. This statement is, of course, independent of the particular choice of the multiplicative constant in the Haar measure on Vk+. We do not here go into the question of finding the measure of V,+/k+ in terms of our explicitly given Haar measure. (See Tate’s thesis, Chapter XV of this book.) Proof. This can be reduced similarly to the case of Q or F(t), which is almost immediate: thus W defined above has measure 1 for our Haar measure. Alternatively finite measure follows from compactness. For cover Vz /k’ with the translates of F, where F is an open set of finite measure. The existence of a finite subcover implies finite measure. [We give an alternative proof of the product formula lI l<jv = 1 for 6 E k, 6 # 0. We have seen that if b,, E kv then multiplication by /3. magnifies the Haar measure in kc by the factor I& Hence if B = {jV} E Vk, multiplication by B magnifies Haar measure in I’,,? by I2 I& In particular multiplication by the principal adele c magnifies Haar measure by l7 I&. But now multiplication by < takes k+ c V,+ into k+ and so gives a welldefined 1  1 map of V,+/k+ onto V,+/k+ which magnifies the measure by the factor I7 I<[.. Hence Ii’ I& = 1 by the Corollary.]
In the next section we shall need the LEMMA. There is a constant C > 0 depending only on the global field k with the following property: Let dl = {cI”> c V, be such that
lJ IU”l” ’ c
(14.8)
Then there is a principal adele B Ek c V,, /I # 0 such that lPlu 5 ‘l~l,
(all 9.
Proof. This is modelled on Blichfeldt’s proof of Minkowski’s Theorem in the Geometry of Numbers and works in quite general circumstances. Note that (14.8) implies [a,l” = 1 for almost all u because l~,l, 5 1 for almost all v. Let co be the Haar measure of V,+/k+ and let c1 be that of the set of y = (yJ c Vc with IY”l” s ik if v is arch. lY”l” 2 1 if u is if u is nonarch.
Then 0 < co < co and 0 < ci < co because the number finite. We show that c = CO/Cl will do. The set T of z = (tU) c V> with Iz,I, I &$x,1, if u is arch. if u is nonarch. j~“l” 5 I@“\”
of arch. u’s is
GLOBAL
FIELDS
67
has measure
Hence in the quotient map F’k’ + V,+/k+ there must be a pair of distinct points of T which have the same image in Vz/k+, say z’ = (2;) E T,
Z” = {TV”}E T
and z’r”
= p (say) E k’.
Then IPI” = I+cl”
2 I%I”
for all u, as required. COROLLARY. Let v0 be a normalized valuation and let 6, > 0 be given for all v # I,+, with 6, = 1 for almost all v. Then there is a /? E k, p # 0 with
I& 5 6,
(all u # ud.
Proof. This is just a degenerate case. Choose a, E k, with 0 < aolv I 6, and ]a,/, = 1 if 6, = 1. We can then choose avOE kUOso that all “K ”0 IavIu ’ =. Then the lemma does what is required. [The character group of the locally compact group V,+ is isomorphic to V,+ and k+ plays a special role. See Chapter XV (Tate’s thesis), Lang: “Algebraic Numbers” (AddisonWesley), Weil : “Adeles and Algebraic Groups” (Princeton lecture notes) and Godement : Bourbaki seminars 171 and 176. This duality lies behind the functional equation of 5 and Lfunctions. Iwasawa has shown (Annuls of Math., 57 (1953), 331356) that the rings of adeles are characterized by certain general topologicoalgebraic properties. ]
15. Strong Approximation Theorem The results of the previous section, in particular the discreteness of k in V, depend critically on the fact that all normalized valuations are used in the definition of V,: THEOREM. (Strong approximation theorem.) Let v,, be any valuation of the global field k. Define Y to be the restricted topological product of the k, with respect to the Q~,where v runs through all normalized v # vO. Then k is everywhere densein Y. Pro0f.t It is easy to see that the theorem is equivalent to the following
statement. Suppose we are given (i) a finite set S of valuations u # ue, (ii) elements a, E k, for all u E S and (iii) E > 0. Then there is a /I E k such that Ipa”l “< E for all u E S and l/31u 5 1 for all v 4 S, v # ve. By Corollary 1 to the Theorem of $14 there is a W c V, defined by inequalities of the type Ir,lu 5 6, (6, = 1 for almost all U) such that every t Suggested by Prof. Kneser at the Conference.
68
J. W.
S. CASSELS
q E V, is of the form OEW, yEk. (15.1) cp=e+y, By the corollary to the last lemma of § 14, there is a rZE k, A # 0 such that I$ < 43 (u E 8, (15.2) I& s 6,’ (1)4 s,u # uo). Hence, on putting cp = Ala in (15.1) and multiplying by L we see that every a E V, is of the shape (15.3) a=$+P, $EAW, BEk, where AW is the set of AC, C E IV. If now we let a have components the given a, at u e S and (say) 0 elsewhere, it is easy to see that fi has the properties required. [The proof clearly gives a quantitative form of the theorem (i.e. with a bound for I&,). For an alternative approach, see K. Mahler: Inequalitiesfor ideal basea,J. Australian Math. Sot. 4 (1964),425A48.1
16. Idele Group The set of invertible elements of any commutative topological ring a group R” under multiplication. In general, R” is not a topological if it is endowed with the subset topology because inversion need continuous. It is usual therefore to give R” the following topology. is an injection x+,x1)
R form group not be There (16.0)
of R” into the topological product Rx R. We give to R” the corresponding subset topology. Clearly R” with this topology is a topological group and the inclusion map RX + R is continuous. DEFINITION. The idele group Jk of k is the group V,” of invertible elements of the adele ring V, with the topology just defined. We shall usually speak of Jk as a subset of V, and will have to distinguish between the Jk and V’topo1ogies.t We have seen that k is naturally embedded in V, and so k” is naturally embedded in Jk. We shall call kx considered as a subgroup of Jk the principal ideles. LEMMA. k” is a discrete subgroup of Jk. ProojI For k is discrete in V, and so k” is injected into V, x V, by (16.0) as a discrete subset. LEMMA. Jk is just the restricted topoIogica1 product of the k,” with respect to the units U,, c k. (with the restricted product topology). Beweis. Klar. t Let a@)for a rationalprime4 bethe elementof JQ with components a?) = q, a:” = 1 (u # q). Then a@)+ 1 (q + 00) in the Y,topology, but not in the J,topology.
GLGBAL
DEFINITION.
FIELDS
69
For a = (Q~> c Jk we define c(a) = fl IQ& lo be the conall ”
tent of a. LEMMA. The map a + c(a) is a continuous homomorphism of the topological group Jk into the multiplicative group of the (strictly) positive real numbers. Beweis. Klar. [Lemma. Let a E .&. Then the map 5 + a6 of V,+ onto itself multiplies Haar measure on V,+ by a factor c(a). Beweis. Klar. Note also that the &topology is that appropriate to a group of operators on V,+ : a basis of open sets is the S(C, 0) where C, 0 C V,+ are respectively Kcompact and Vkopen and S consists of the a l Jk such that (1  a)C c 0, (1  al)C c 0.1
Let Ji be the kernel of the map a + c(a) with the topology as a subset of Jk. We shall need the LEMMA. Jl considered as a subset of V, is closed and the Vksubset topology on Jj coincides with the J,topology. We must find a Vkneighbourhood W of a Proof. Let aE Vk, a#Ji. which does not meet Ji. 1st Case. n [a& < 1 (possibly = 0). Then there is a finite set S of v such that (i) S contains all the v with lcc,l, > 1 and (ii) n Ic(& < 1. Then the set W can be defined by ves ls”~“l” <& VES IC”lS 1 v#S for sufficiently small E. 2nd Case. n Ia& = C (say) > 1. Then there is a finite set S of v such that (i) S con:ains all the v with Ia&, > 1 and (ii) if v 4 S an inequality It,]” < 1 implies? l&l, < *C. We can choose E so small that I&a&, < E (v E S) implies 1 < ,?[, l&,1 < 2C. Then W may be defined by ILslo
< 8 (0 Es) jr”1 5 1 (0 4 s). We must now show that the Jk and &topologies on Ji are the same. If a E Ji we must show that every J,neighbourhood of a contains a I’,neighbourhood and viceversa. Let: W c J,,! be a V,neighbourhood of a. Then it contains a V,neight If k =) Q and u is a normalized extension of the padic valuation then the value group of II consists of (some of the) powers of p. Hence it is enough for (ii) to include in S all the arch. u and all the extensions of padic valuations with p 5 2C. Similarly if k = F(r). $ This half of the proof of the equality of the topologies makes no use of the special properties of ideles. It is only an expression of the fact noted above that the inclusion R” + R is continuous for any topological ring R.
70 bourhood
J. W. S. CASSELS
of the type
ILa,l, < 8 CUE s) (16.1) [rvlv 5 1 (v # s) > where S is a finite set of v. This contains the J,neighbourhood in which 5 in (16.1) is replaced by = . Now let H c Ji be a J,neighbourhood. Then it contains a J,neighbourhood of the type 15UC1,1,<~ CS) (16.2) Ir”l” = 1 (u # 9 > where the finite set S contains at least all arch. Y and all v with ICI,], # 1. Since n Ia& = 1 we may also suppose that E is so small that (16.2) implies rJ Ir”l” < 2. Then the intersection of (16.2) with Jk is the same? as that of (16.1) with Ji, i.e. (16.2) defines a V,neighbourhood. By the product formula we have k” c Jkr . The following result is of vital importance in classfield theory. THEOREM. JiJkx with the quotient topology is compact. Proof. After the preceding lemma it is enough to find a V,compact set W c V, such that the map W n J: + J,‘/kX is surjective. We take for W the set of c = {&,} with It& 5 Ia& where a = (a”} is any idele of content greater than the C of the last lemma of § 14. Let fl = {B,} E J,f. Then by the lemma just quoted there is a g E k” such that I$ I IP;‘a,l, (all 0). Then qfi E W, as required. [&l/c” is totally disconnected in the function field case. For the structure of its connected component in the number theory case see papers of Artin and Weil in the “Proceedings of the Tokyo Symposium on Algebraic Number Theory, 1955” (Science Council of Japan) or ArtinTate: “Class Field Theory”, 1951/Z (Harvard, 1960(?)). The determination of the character group of Jk/kx is global class field theory.]
17. Ideals and Divisors Suppose that k is a finite extension of Q. We define the ideal group Ik of k to be the free abelian group on a set of symbols in 1  1 correspondence t See previous footnote.
GLOBAL
with the nonarch. valuations
FIELDS
v of k, i.e. formal sums n,.v c
71 (17.1)
“nonarch.
where n, E Z and n, = 0 for almost all v, addition being defined componentwise. We call (17.1) an ideal and call it integral if n, 2 0 for all v. This language is justified by the existence of a ll correspondence between integral ideals and the ideals (in the ordinary sense) in the Dedekind ring o= n 0,: nonarch.
cf. Chapter I, $2, Prop. 2. There is a natural continuous
map
Jk+& of the idele group on to the ideal group? given by a = (a,> f C (ord, a). v. The image of k” c Jk is the group of principal ideals. THEOREM. The group of ideal classes, i.e. Ik module principal ideals, is jinite. Proof. For the map Ji + I, is surjective and so the group of ideal classes is the continuous image of the compact group Jl/k” and hence compact. But a compact discrete group is finite. When k is a finite separable extension of F(t) we define the divisor group Dk of k to be the free group on all the v. For each v the number of elements in the residue class field of v is a power, say qdy of the number q of elements in F. We call d, the degree of v and similarly define C n,d, to be the degree of c n, . v. The divisors of degree 0 form a group Dz. One defines the principal divisors similarly to principal ideals and then one has the THEOREM. 0,” module principal divisors is a jinite group. For the quotient group is the continuous image of the compact group J,‘/k”. 18. Units
In this section we deduce the structure theorem for units from our results about idele classes. Let S be any finite nonempty set of normalized valuations and suppose that 5’ contains all the archimedean valuations. The set of q E k with (18.1) [?I” = 1 (v#s) are a group under multiplication, the group Hs of Sunits. When k 3 Q and S is just the archimedean valuations, then Hs is the group of units tout court. t Zk being
given
the discrete
topology.
72
J. W.
S. CASSELS
LEMMA 1. Let 0 < c c C c co. Then the set of Sunits 4 with (18.2) c~[lrlltsC CUES) is finite. Proof. The set W of ideles at = (CL,} with (18.3) c 2 lct”l” s c (u E S) I~“10 = 1 (u6s)s is compact (product of compact sets with the product topology). The required set of units is just the intersection of W with the discrete subset k of Jk and so is both discrete and compact, hence finite. LEMMA 2. There are only finitely many E E k such that l&Iv = 1 for every v. They are precisely the roots of unity in k. Proof. Ifs is a root of unity it is clear that IsI0 = 1 for every v. Conversely, by the previous lemma (with any S and c = C = 1) there are only finitely many EE k with Islo = 1 for all u. They form a group under multiplication and so are all roots of 1. THEOREM. (Unit theorem.) Hs is the direct sum of a finite cyclic group and a free abelian group of rank s 1. Proof. To avoid petty notational troubles we treat only the case when Q c k and S is the set of arch. valuations. Let J, consist of the ideals at = {a”} with lcl&, = 1 (t, 4 S) and put Ji = J, n Ji. Clearly Jl is open in Jk and so Jfi/H, = Ji/(Ji n k”) (18.4) is open in Jl/k” . Since it is a subgroup, it is also closed, and so compact (§ 16). Consider the map L:J,+R+@R+&?&R+, \ , s times
where R+ is the additive group of reals, given by a,(logI~,ll,logla,l,,. . .Joglcl,l,), where 1,2,. . ., s are the valuations in S. Clearly A is both continuous and surjective. The kernel of L restricted to Hs consists just of the E with Is],, = 1 for every t), so is a tite cyclic group by Lemma 2. By Lemma 1 there are only Iinitely many q E H, with (18.5) 341~1.~2 VES. Hence the group A (say) = A(H,) is discrete. Further, T = A(J,‘) is just the set of (x,, . . . , x,) with x,+x2+. . . +x, = 0,
GLOBAL
73
FIELDS
i.e. an S 1 dimensional real vector space. Finally, T/A is compact, being the continuous image of the compact set (18.4). Hence A is free on S 1 generators, as asserted. Of course this structuretheorem (Dirichlet) and the finiteness of the classnumber (Minkowski) are older than ideles. It is more usual to deduce the compactness of J,‘/k” from these theorems instead of vita versa. 19. Inclusion and Norm Maps for Adeles, Ideles and Ideals Let K be a finite extension of the global field k. We have already seen ($ 14, Lemma) that there is a natural isomorphism (19.1) Kc&K = VK algebraically and topologically. Hence V, = V, Bk k can naturally be regarded as a subring of V, which is closed in the topology of I’,. This injection of V, into I’, is called the injection map or the conorm map and is written con : a + con a = con,,, a E V, (a o V,). Explicitly if A = con a, then the components satisfy Ay=uvEk,c
K,
(19.2)
where V runs through the normalized valuations of K and v is the normalized valuation of k which extends to V. If k c L c K it follows that conKlk a = conLlk (conKIL a). (19.3) Finally, for principal adeles the conorm map is just the usual injection of k into K. It is customary, and usually leads to no confusion, to identify conKlk a with a. One can also define norm and trace maps from V, to vk by imitating the usual procedure (cf. Appendix A). Let or,. . . , w, be a basis for K/k. Then by (19.1) every A E I’, is uniquely of the shape A=Cajmj ait? vk (19.4) and the map A + aj of V, into vk is continuous by the very definition of the tensor product topology (5 9). Hence if we define aij = aij(A) E Vk (19.5) the n x n matrices (Q) give a a continuous over vk. In particular, the Sxc,kA = 1 aii NK,kA = det (aij)
representation
of the ring VR (19.6) (19.7)
74
J. W.
S. CASSELS
are continuous functions of A and have the usual formal properties SK,,& + AZ) = SK,& SKIkconK,ka = na
+ SK,,&
(19.8)
(19.9) (19.10) NK,PCWJ = NK,A NK,L& NKIR con,,, a = a”. (19.11) Further, the norm and trace operations are compatible with the embedding of k, V in V,, V, respectively, i.e. if A E K c V, we get the same answer whether we compute NKIkA, SKIKA in K or in V,, so there is no ambiguity in the notation. Finally if K 2 L 2 k we have V’ c V, c V, (on regarding conorm as an identification), and so the usual relations (cf. Appendix A) (19.12) S,,,(SK,A = S,,,A and (19.13) NL,~NKILA = = N,,,AWe can express the maps (19.6), (19.7) componentwise if we like. Let VI,. . *, V, be the extensions of any given valuation u of k to K. Then (§ 9) K, (say) = 0 Ki = k, Blk K = 0 kvq (19.14) 1sijS.J 15is;n where k,, Ki are the completions of k, K with respect to v, Vj respectively. Any A E V, can be regarded as having components A,,O...OA,,=A, (19.15) in the K, and then the components in the matrix representation (19.5) of A are just the representations of the A,. In particular (19.16) SK,IM = {SK,,~, A,) and (19.17) NKIJ = {NK,,v%)Finally, making use of the final remarks of $9, we deduce that (19.18) SK/IA = c SK~,,JAV) { Vlv 1” and (19.19) NK,& = NK.p,Av 9 f/VU >” where VIv means “V is a continuation of v”. We now consider the consequences for ideles. If et is an idele, it is clear from the definition (19.2) that con,,, a is an idele, so we have an injection conKlk : Jk ) JR whichis clearly a homomorphism of Jk with a closed subset of V,. Further,
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if A EJR c V,, so A is invertible, it follows from (19.9) that NKIk A is invertible, i.e. is an element of Jn. Hence we have a map N K/k JK+Jk which is continuous by the definition of the idele topology ($16) and which clearly satisfies (19.10), (19.11), (19.13) and (19.19). On the other hand, the definition of trace does not go over to ideles. FinalIy, we consider the conorm and norm maps for ideals, where k is a finite extension of Q. The kernel of the map ($17) Jk
+
Ik
of the idele group into the ideal group is just the group u,, (say) of ideles V. If K is a finite extension of k, it is clear that a = a, which have ICI& = 1 for every nonarchimedean uk
ConKlk
=
UK
and from the Lemma of $11 and (19.17) we have NK,kUK
=
uk*
Hence on passing to the quotient from Jk we have the induced maps . I,
ConKlk
N
K/k
* I,
)
I,
,
Ik
with the usual properties (19.10), (19.11) and (19.13); and these maps are compatible with the norm and conorm maps for elements of K and k on taking principal ideals. By definition (19.2) we have (19.20) conKlkv = c ev v Vlv where the positive integers ey are defined by b”I” = pblevv, rr, and lT, being prime elements of k,, Kr, respectively. from (19.19) that NK,k v = fv v,
Similarly,
(19.21) it follows (19.22)
where fV is the degree of the residue class field of V over that of v. We note in passing that (19.11), (19.20) and (19.22) imply that
zvevfv = h as it should since
+fv = [Kv: bl. Similarly, when k is a finite extension of F(t) one defines conorm and norm of divisors, with the appropriate properties.
76
I. W. S. CASSELS
A
APPENDIX
Norms and Traces Let R be a commutative ring with 1. By a vector space V over R of dimension n we shall mean a free Rmodule on n generators, say q,. . . , o, (a basis). If o;, . . ., c$, is another basis, there are z+,, uu E R such that 01 =zy utp;,
0; = c %j@f
(A.1)
and 7
%J vjl
=
C
"lj"jZ
=
(A2)
6il
(Kronecker 8). The set of all Rlinear endomorphisms of V is a ring, which we denote by End, V. The ring R is injected into End, V if we identify b E R with the module action of b on V, and we shall do this. The ring End, V is isomorphic but not canonically, to the ring of all n x n matrices with elements in R. The isomorphism becomes canonical if Y is endowed with a fixed choice of basis. In fact if /I E End, (V) and PA = T hf./w,,
(bij E R)
(A3)
the I 1 correspondence between /I and the transposed matrix (b,J is a ring isomorphism. For p E End, we denote by FB(x) = det (x6,1  b,) (A4) the characteristic polynomial of R. On using (A.2) it is easy to see that I;B(x) is independent of the choice of bases of V. The CayleyHamilton theorem? states that F&3) = 0. (A5) We define further the truce &pdPl=
WI
= C b,
=  coefhient
of 9r
in FP(x)
64.6)
and the norm &I~(B)
= NP) = det (hi)
= ()”
(A7)
constant term in F@(x),
t Proof. Write (A.3) in the form T (4,8  wwi
= 0.
Working in the commutative ring R[Bl multiply the equations (*) by the cofactors of the and one obtains “coefficients” of wl, and add. Then war. . . , w, are “eliminated” F&?)w~ = 0. Similarly F$./!QD, = 0 (2 5 j I n) and so F&9) = 0.
77
GLOBAL FIELDS
of the choice of basis because Q(x) is. Clearly
which are independent
WI
+BJ
= WI)
S(b) = nb
+ WJ
(A.9
(b E R)
(A.9
(A.10) N(P, PJ = NmwM (A.ll) N(b) = b” (b E R), because the correspondence (A.3) between /? E End, (V) and the matrix bji is a ring isomorphism. LEMMA A.1. Let t be transcendental over R. Then (A.12) N(tp) = F,(t). Pedantically, what is meant is, of course, that we consider a vector space V with basis ol, . . ., w, defined over R[t] and a p given by (A.3). Proof. We have (tP)oi
= C (tsijbij)wj j
and so = det(t$b,J
N(tp)
= F,(t) by (A.4) and (A.7). COROLLARY. (A.12) holds for any t E R. LEMMA A.2. Let /II,. . . , j3, E End, V and let t be transcendental
over R.
Then N(t’+j&
t''+
. . . +/3,) = t”‘+g1
Pl+
. . . +g”l
(A.13)
where gl,. . . , gnl E R and in particular 91 = WI);
9.1 = NW
(A.14)
Similar to that of Lemma A.1 and left to the reader. Now let R and P c R be commutative rings with 1 and suppose that R regarded as a Pmodule is free on a finitenumber, say, m of generators a,,.  *, 0, (i.e. an mdimensional Pvector space). Let V be an ndimensional Rvector space with basis or,. . . , 0,. Then V can also be regarded as an mndimensional Pvector space with basis (l
n,Oj
Then &,I4
= &4f&J9,
NvpB
= &,P(&,R
P>
(A.16) (A.17)
78
J. W.
S. CASSELS
Further w3
(A.18)
= NR,PWh
where a,(x) E P[x], F(x) E R[x] are the characteristic polynomials Endp (V) and End, (V) respectively. Proof. If j? is given by (A.3) let y E End, (V) be given by Y%=%
jFl
of p in
bijwj
ywi = b,lo,
(A.19)
(i > 1).
Then for cc= yp we have ami = bllw, awi = Bill + C (bll bijbil = bilw,
blj)wj
(say).
+ji’aijw,
(A.20)
j>l
Hence (A.21) Nv,Ra = bllNwlRa* where W is the n 1 dimensional Rvector space spanned by 02,. . . , w, and a* is the Rlinear map COi +jFIU*jWj (i > 1). Consequently (A.22) NRIPWvIR a> = NRIp 4 I . NRIP(NWIRa*). We now use induction on the dimension n, since the Theorem is trivial for n = 1. Since W has dimension n  1 we have by the induction hypothesis
%,PWV,R a*)
= N,,,a*.
(A.23)
On the other hand, it follows directly from (A.20) that hIpa=
NR~~blN~,~a*
and so NV/G
= NRIPNvIpa.
(A.24)
Further, clearly Nv,,Y = NR,,Nv,PY = WR,PW’~Since a = j?y and both Nv,r and NRIPNYIR are multiplicative it follows from (A.24) that W~,dd”%,J = (NR,Pbll)nlNR,p(Nv,RP). If NRIPbll were invertible, this would give (A.17) at once. however, this is not the case and we must use a common trick.
(by (A. lo)), (A.25) In general,
GLOBAL
19
FIELDS
Let t be a transcendental over R and let /?, be the transformation obtained from /? by replacing b,, by bil +t but leaving the remaining b,j unchanged. Then (A.25) applied to /?, gives WW (&p&l + 0)“%,~Bt = NR,Y@II + O”lN~,~Wv,dr)~ All the norms occurring in (A.26) are polynomials in t. On comparing coefficients of powers of t in (A.26), starting at the top, we deduce that
NvpPt
(A.27)
= NR,PWY,R PJ
because the coefficient of the highest power of i in iVRIP(bl 1 + t) is 1. Then (A.17) follows on putting t = 0. We now prove (A.18). By Lemma 1 we have W4 = NV,& PI, F(x) = NV,& B) and so (A.18) is just (A.17) with xp for p. Finally, (A.16) follows from (A.18) on using (A.6) and the first half of (A.14). When R = k is a field there is some simplification, since every finitelygenerated module V over k is free, i.e. is a vector space. Further each /? E End, (V) has a minimum polynomial, i.e. a nonzero polynomial f(x) of lowest degree, with highest coefficient 1, such that f(p) = 0. Then g(p) = 0 for g(x) E k[x] if and only iff(x) divides g(x) in k[x]. In particular the CayleyHamilton theorem (A.5) now states thatf(x) divides the characteristic polynomial FP(x). Finally we have THEOREM A.2. Let K be a field of finite degree n over the field k and let p E K. Then the degree m (say) of the minimum polynomial f(x) of p over k divides n and J64 = (fW>“‘“, where
F(x)
is the characteristic &,k(P) &,&V
polynomial
of p.
In particular
= ;GB,+...+P,h = (PI 9IL  . 9 AJ”‘m,
where PI,. . . , p,,, are the roots of f(x) in any splitting field. Proof. Suppose first that K = k(B). Then the minimum
polynomial f(x) and the characteristic polynomial F(x) of p have the same degree and highest coefficient, so F(x) = f(x) by the remarks preceding the enunciation of the Lemma. The general case now follows from Theorem A. 1, with V = K, R = k(P), P = k on using (A.ll).
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J. W. S. CASSELS APPENDIX R
Separability In this book we are primarily interested in separable algebraic field extensions. Here we recall their most important elementary properties. LEMMA B. 1. Let K, M be extensions of finite degree of the field k. Then there are at most [K: k] injections of K into M which leave k elementwise fixed. Proof. Trivial when K = k(a) for some a on considering the minimal polynomial for a. For general K we have a chain k=K,,CK,cK,... cKJ=K (B.1) where Kj = Kj_,(ai_,) and use induction on J. The finite field extension K/k is separable if there is some DEFINITION. finite extension M/k such that there are [K: k] distinct injections of K into k which leave k elementwise fixed. If K/k is not separable then it is said to be inseparable. COROLLARY 1. Let K 2 L I k. If K/k is separable then so are K/L and L/k. Proof. By Lemma 1 there are at most [L: k] distinct injections of L into M and by Lemma 1 again each of these can be extended in at most [K: L] ways into injections of K into M. By definition, there are [K:k]=[K:L][L:k] injections of K into M, and so there must be equality both times. COROLLARY 2. Let a E K where K/k is separable and let CQ,. . ., a,,, be the roots in M of the irreducible poIynomiaI f(x) for a over k. Then the oia (1 I i I n = [K: k]) are just the ul,. . . , LX,,,each taken n/m times, where cl,. . . , a,,, are the injections of K into some M. Proof. For put L = k(a) in the preceding argument. COROLLARY 3.
Proof.
Follows from Theorem A.2 and the preceding Corollary. extension and let o be an injection of k into some field M. Then there is a finite extension M, of M and an injection al of K into Ml which reduces to o on k. Proof. Trivial if K = k(a), and then follows for general K on using a chain (B. 1). THEOREM B.1. Let K/L and L/k be separable extensions. Then K/k is a separable extension. Proof. Let U/L be a finite extension and pi : K ~ U (1 I i I [K : L]) LEMMA B.2. Let K/k be aJinitejield
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be injections extending the identity on L and similarly let V/k be a finite extension and Cj:L+‘v (lljl[L:k]) extend the identity on k. By repeated application of Lemma 2 there is a finite field extension M/V and [L : k] injections oi:U+M
(lljl[L:k])
which extend the aj. Then the ojri give [K:L][L:k]=[K:k]
distinct injections of K into it4 extending the identity on k. COROLLARY. In characteristic zero livery finite Jield extension is separable. Proof. For a simple extension k(cr)/k clearly is, and then apply the theorem
to a tower of simple extensions. THEOREM B.2. Let K/k be a separable extension. Then it is simple, i.e. K = k(y) for Note. The Proof: If some u E K,
some y.
converse is, of course, false. k is a finite field then so is K and so indeed K = U(a) for where II is the prime field, by the structure theory of finite fields. Hence we need consider only the case when k has infinitely many elements. Suppose first that K = k(cr, /I) and let pi,. . . , gn where n = [K: k] be the distinct injections of K into M (say). If i # j, distinctness implies that either oia# aja or o,Q# oJp (or both). Hence we may find a, b E k to satisfy the finitely many inequalities a(oiorajcr)+
b(a;Poj~)
# O(i # j).
Put y = act+ b/3, SO
Oiy # CjY
The a,y are all roots of the irreducible
(i #j).
equation fcr y over k and so
[k(y) : k] 2 n.
But k(y) c K, so K = k(y). For the genera1 case when K = k(cr,, CQ,. . . , aJ) with J > 2 one uses induction on J. We have k(cc,, . . ., crJ) = k(B) for some /I and then W,, l9 = k(y). THEOREM B.3. Let K/k be a separableextension. Then 0, PI = &&dO is a nondegenerate symmetric bilinear form on K considered as a vector space over k.
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Proof. Only the nondegeneracy needs proof. Let oi, . . . , w, be a base of K/k. The statement of the Theorem is equivalent to
D (say) = det {S~/d~i”‘));;~~; Let or,. . ., a,, be distinct injections Corollary 3 we have
Z 0.
of K into some M.
By Lemma
B.l,
D = A2
where A = det(aiwj)lSiS.. l<j
By Theorem B.2 we have K = k(y) and so can take Oj = yj‘.
Then
as required.? We now consider when a simple extension k(z)/k is separable. Let f(x) be an irreducible polynomial in k[x] and let f’(x) be its derivative. If f’(x) # 0 it must be coprime tof(x), since it is of lower degree, and so there are a(x), b(x) E k[x] such that a(x)f(x> + b(x)f’(x) = 1. Hence f(p) = 0 for /I in any extension of k, implies that f’(/3) # 0, and so B is a simple root. Hence the number of roots off(x) in a splitting field is equal to the degree. On the other hand, iff’(x) = 0, every root off(x) is multiple, and so the total number of roots is less than the degree. In the first case we say thatf(x) is separable, in the second inseparable. The second case occurs if and only iff(x) = g(x”> for some g(x) E k[x], where p is the characteristic. LEMMA B.3. A necessary and suficient condition for k(or)/k to be separable is that the irreducible polynomial f(x) E k[x] for cc be separable. Proof. Clear. COROLLARY 1. Let K I k, and suppose that k(cr)/k is separable. Then K(or)/K is separable. Proof. For the irreducible polynomial F(x) E K[x] over K divides f(x). COROLLARY 2. A necessary and suficient condition that K/k be separable is that every element of K be separable /k. Proof. Suppose that every element of K is separable and that K is given
by a chain k=KOCK1c
. . . cK,=K
t Instead of using the fact that K = k(y) we could haveusedArtin’s theoremthat any set of injectionsof one field into another is linearly independent. SeeArtin: “Galois Theory” (Notre Dame)or Adamson:“Introduction to Field Theory” (Oliver andBoyd).
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where Kj = K~_,(Orj_ 1). Then KJIK, 1 is separable by the previous corollary and so K/k is separable by Theorem 1. The converse follows from Lemma 1 Corollary. In striking contrast to Theorem B.3 we have THEOREM B.4. Let K/k be inseparable. Then the trace S,,,(/?) vanishes for all /I E K. Proof. Suppose, first, that K = k(a) where up E k, cc$ k and p is the characteristic. Then o~~=l,o~=a,...,o~=cI~~ is a basis for K/k. If /3 = b,+b,a+ . . . +bpaP’ with bj E k, then PWi = C b,jOj bij E k where clearly bii = b, (1 < i I p). Hence SK,LB = C bii = pb, = 0. L Now let K/k be any inseparable extension. By the latest Corollary there is an inseparable CIE K. Put L = k(x), M = k(uP), so L/M is an extension of the kind just discussed. The general result now follows because of the transitivity of the trace: &,I4
= &&L,M(SX,LP)l.
APPENDIX C Hensel’s Lemma In the literature a variety of results go under this name. Their common feature is that the existence of an approximate solution of an equation or system of equations in a complete valued field implies the existence of an exact solution to which it is an approximation, subject to conditions to the general effect that the approximate solution is “good enough”. These results are essentially just examples of the process of solution by successive approximation, which goes back to Newton (at least). In this appendix we give a typical specimen. LEMMA. Let k be a field complete ,l,ith respect to the nonarchimedean valuation 1 1 and let
J(X)E4IXlY
cc.1)
where o c k is the ring of integers for 1 I. Let a0 E o be such that w,here f'(X)
is the (formal) .I@> = 0,
If( < I~‘
(C.2)
Then there is a solution of
5 lf<4l/lf’<41.
(C3)
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S. CASSELS
(Sketch.) Let f,(X) E o[X] be defined by the identity j(X+ Y) =f(X)+f,(X)Y + . . . +fj(x>Yj+ . . ., (C.4) where X, Y are independent variables, so fi(X) = f’(X). Define PO by Proof.
f&J + Pof1(4 = 0. Then by (C.4) and sincefj(a,) E D we have lfh
+/Ml
5 max
On using the analogue of (C.4) forfi(X), Ifi(ao+Po>fi(ao>l Thus on putting a1 = aO+po, we have
(C.5)
Ifj<%>P',I
it is easy to verify that < Jfi(ao)l.
lf
CHAPTER III
Cyclotomic
Fields and Kummer
Extensions
B. J. BIRCH 1. Cyclotomic Fields ............... 2. Kummer Extensions .............. Appendix. Kummer’s Theorem ............
I
85 89 92
1. Cyclotomic Fields Let K be any field of characteristic zero, and m > 1 be an integer. Then there is a minimal extension L/K such that x”  1 splits completely in L. The zeros of x”‘ 1 form a subgroup of the multiplicative group of L; this subgroup is cyclic (since every finite subgroup of the multiplicative group of a field is). The generators of this subgroup are called the primitive mth roots of unity. If [ is a primitive mth root of unity then every zero of (x”  1) is a power of i, and L = K(c). Clearly, L is a normal extension of K; we write L = K(ql). If u is an element of the Galois group G(L/K), then oi must be another primitive mth root of unity, so a( = ck for some integer k, (k, m) = 1. If [ is another primitive root of unity then a[” = pk; accordingly, (TH k is a canonical map of G(L/K) into the multiplicative group G(m) of residues modulo m prime to m. In particular, [L : K] I 4(m). If m = rs where (r, s) = 1 then there exist integers a, b with ar + bs = 1, I = (rywb, so K(l) = K(5’, r”) ; one obtains the extension K(c) by composing K(c) and K(r). So to some extent it is enough to consider K(mJ1) when m is a prime power. If p is odd then the group G(p”) is cyclic, so if m =pn, L = K(vl), then G(L/K) is cyclic; on the other hand G(2”) is generated by  1 and 5, so if we write q = [+c’ where c2” = 1 then K(c) = K(i, q) and G[K(q)/K] is cyclic. We are particularly interested in the extensions Q(r;l/l) and Q&‘/l); by Chapter I (Section 4 and start of Section 5) the study of the factorization of the prime p in the extension Q(vl)/Q is essentially the same as the study of the extension Q,(vl)/Q,. A s one of my jobs is to supply explicit examples for abstract theorems, I will prove things several times over by different routes. Good accounts of cyclotomic extensions are given by Weyl: “Algebraic Theory of Numbers” (Princeton U.P., Annals of Math. Studies 85
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No. l), pp. 129140, and by Weiss: “Algebraic Hill), Chapter 7.
Number
Theory”
LEMMA 1. Q(rC/l) is a normal extension of Q of degree 4(m); group is naturally isomorphic to G(m).
(McGraw its Galois
Proof (after van der Waerden). Let 5 be a primitive mth root of unity; after what has been said already, it is enough to show that an equation of minimal degree for [ over Q has degree d(m) so we need to show that if j”(x) E Z[x] and f(c) = 0 then f(p) = 0 whenever (a, m) = 1. For this, it is enough to show thatf(cP) = 0 whenever p is a prime not dividing m. Consider the field kp with p elements; if f(x) E Z[x], denote its natural image in k,[x] byf*(x). Let L*, a finite extension of kp, be a splitting field for (xl 1) ; (xm  1) is prime to its derivative mxmwl, so has distinct roots. Suppose that the factorization in Z[x] of (xm 1) isfi(x).fZ(x). . .L(x), so that the A(x) are irreducible. Then in k&l, (x” 1) = nfr(x), and all the zeros in L* of all the f *, being roots of x’” = 1, are distinct. Choose the numl&ing so that [ is a root of fi(x), and suppose that CPis a root of fJ(x). Then tiI(x) divides fJ(xp), so f T(x) divides f J(xp). Let [* be a zero off T(x); then fT([*p) = 0, and on the other hand f :(c*3 = ~~(~*)]p = 0. So f, is the same as fi. COROLLARY 2. If p ,/’ m there is a unique element op of the Galois group such that opu s or(p) for all integers a E Q(r;/l). In fact, zf[ is a primitive mth root of unity then op is given by o,(c air’) = xaicip for a, E Q. (op is the Frobenius automorphism.)
G[Q(vl)/Q]
Proof. The field basis 1, 5, . . . , [@(“‘)l has discriminant prime to p, so by Chapter I, Section 3, certainly every integer can be expressed as c ail ‘lb with al,. . ., ap2, b E Z and (b,p) = 1; so cp defined above does what is wanted. We need to show that it is unique. The effect of an element c of the Galois group is clearly determined by its effect on 5; and clearly every CJtakes c to another primitive root, p say, with (a,m) = 1. We need to show that r” I cb(p) only if r” = 5’; that is, 1  cb E O(p) only if lb = 1. First method. Suppose cb =j= 1. We know
x”1
= fi (xq), i=l
so mxmml = T i~(x,I), and
So (1 lb)
m1 m = ,I! (1 C)divides m, so p does not divrie (1  cb).
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Second method. Take the completion Q, of Q, and form the extension Q&‘/l). Let L* be the residue class field of Q,(r;/l). Then L* is a finite extension of k, and is a splitting field for x” 1. Q(‘;/l) c Q,(r;l/l), so the residue class map makes a root of xm 1 in L* correspond to each root of x” 1 in Q(vl). Conversely, the roots of x” = 1 in L* are distinct, and each of them is the residue class of an element of Q,(r;l/l), by Hensel’s Lemma (Chapter I, Section 7, Lemma 1, or Chapter II, Appendix C). So we have a oneone correspondence between the mth roots of unity in L* and in Q(ql), and everything follows. LEMMA 3. If q = pf is a prime power and c is a primitive qth root of unity, then the prime p is totally ramified and in fact (p) = (1  c)9(q).
First Proof. Write Iz = 1  [. Then cq = 1 but cqtp # 1, so I is a root of the polynomial F(x) = [(l +x)“ l]/[(l +x)~‘~ I]. f has leading coefficient 1 and constant term p, and it is readily verified that all the other coefficients are divisible by p. So F is an Eisenstein equation, and by Chapter I, Section 6, Theorem 1, Q,(ql) is a totally ramified extension of Q, of degree 4(q) and (p) = (A)4(q). Going back to Q(ql), we see that Q(ql) is an extension of Q of degree 4(q) (so we get another proof of Lemma 1 in this case) and p is totally ramified in this extension. Second Proof. We can see all this more explicitly. If (a,p) = (b,p) = I, then we can solve a I bs(q), so
(lr”)/(l[b)
= (l[b”)/(l56)
is an integer, and similarly so is (1 [‘)/l whenever (a,~) = (b,p) = 1. Also,
p=lim
xq1 =f.. x1 xq’p 1
(Jpi”)
= 1+[b+. tJ;
. . +rb(sl)
so (1  y)/(l lb)
= (li)“(q)
is a unit
n ls
O&
so (p) is simply the +(q)th power of (A) = (1  5). LEMMA 4. Let c be a primitive mth root of 1. If p is a prime not dividing m then it is unramifed in Q(r;/l) and its residue class degree f, (see Chapter I, Section 5) is the least integer f 2 1 such that p* E l(m).
Proof [after Serre: “Corps Locaux” (Hermann, Paris)]. Consider the extension Q,(c) of Q,; the residue class field k, has p elements. The polynomial x’“ 1 splits in k,, if and only if ml(pf 1). Take the least f with p/ = l(m), and construct the unramified extension L of Qp with residue class field k,, (see Chapter I, Section 7, Theorem 1). There is a multiplicative map k$ + Lx as in the second proof of Corollary 2, so xrn 1 splits in L, and clearly L is minimal with this property. So L = Q,(c).
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Going back to Q(c), we see that p is unramified in Q(c) and has residue class degree f, as described. (In fact, the lemma is an immediate consequence of the properties of the Frobenius automorphism obtained in Corollary 2.) COROLLARY. If p,j’ m, then p splits completely if and only ifp s l(m). LEMMA 5. If q = p’ is a prime power and [ is a primitive qth root of unity, the discriminant of the extension Q(c) of Q is q4(4)/pp’p; a basis for the Zmodule of integers @“Q(c) is 1, [, 5’, . . . , [9(q)1. Proof. Consider first the case t = 1. By Lemmas 3 and 4, the only prime ramified in Q(5) is p, and the ramification ofp is tame with e = p 1. Hence by Chapter I, Section 5, Theorem 2, the discriminant is ppm2. Hence by Chapter I, Section 4, Prop. 6, l,c, c2,. . . , cpm2 is a basis as asserted. In case t 2 2, the ramification is wild, so all that is obvious from Chapter I (until the final section) is that the discriminant is a power of p, at least p4(q). We give a direct proof that the powers of 5 form a basis for the integers, which works for all t. Consider the Zmodule Z[[]. By Chapter I, Section 4, Prop. 6, Z[[] has discriminant N,,I,,,[g’(c)], where g(x) =(kgcl(xIk)i after dull computation this turns out to be q”(q)/pq’p. We want to show that Z[l] is the whole ring of integers of Q(c); after Chapter I, Section 3, Prop. 4 it remains to check that
c
OSiS$(q)1
airi
with ai E Z is divisible by p if and only if all the ai are divisible by p; that is, that C b,(l5)’ OSij+(q)1
is divisible
by p if and only if all the bi are. To check this, suppose that PIE b,(l[‘); recollect that (p) = (1 #‘(q), and suppose that plbi for i= l,..., s 1. Then (l~)+‘q’lb,(l~)“+b,+,(l~)“+‘+. . ., so (1Ojb,, so plb,; so by induction we get what we want. The conclusions of the Lemma follow. LEMMA 6. If C is a primitive mth root of unity, then Q(C) is an extension of Q of degree 4(m); the discriminant of Q(c) over Q is m9(m) pmol(P1) . , /nPM a basis for the Zmodule of integers of Q(C) is 1, [, r2, . . , C4Cm) ’ ; p is ramified if and only ifplm.
Most of this has been proved already (Lemmas 1, 3 and 4); the extra assertions about the discriminant and the basis for integers are equivalent to
CYCLOTOMIC
each other, and may may also pick up the Q(q1) where q runs First, remark that
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89
in fact be proved directly (see Weiss, Section 7.5). One assertions about Q(r;/l) by fitting together the extensions through the prime power factors of m. if q,q’ are powers of different primes,
Q(ql> n Q($) = Q. Now work out the discriminant. Suppose phlIm. By Lemma 5 the discriminant of Q,(phJ1) over Q, isphph(h+l)ph‘. There is no further ramification of p in Q,(Ij/l), so by Chapter I, Section 4, Prop. 7, the discriminant of Q,(r;/l) Over Q, is pW(~)WO/(pl)~ By Chapter I, Section 4, Prop 6, the discriminant of Q(c) over Q is p~(W(p  1). mWO InPh By computation this is the discriminant of Z[(], so by Chapter I, Section 3, Prop. 4, 1, t;, c2,. . . , [0cm) l really is a basis for the Zmodule of integers. Remark. All assertions about factorization of primes in Q(y1) may be regained from Lemma 6, by using Kummer’s theorem (see for instance Weiss, Section 4.9; or the Appendix). “If Z[[] is the complete ring of integers of Q(c), and g(x) = 0 is the characteristic equation of [ over Q, then a prime p factorizes in Q(c) in the same way as g(x) factorizes modulo p, and in fact if g(x) E n gi(x) (mod P> then (PI = fl [P, sLO1.” It is, of course, perfectly possible to work out the ramification groups explicitly in this casefor this we refer to Serre : “Corps Locaux” (Hermann, Paris), pp. 8487. To conclude. We have shown that every cyclotomic extension, and so every subfield of a cyclotomic extension of Q has abelian Galois group. The converse is trueevery abelian extension of Q is a subfield of a cyclotomic extension (Kronecker’s Theorem); but that is Class Field Theory (Chapter VII, § 5.7.) 2. Kummer Extensions Throughout this lecture, K is a field of characteristic prime to n in which x” 1 splits; and [ will be a primitive nth root of unity. It will be shown that the cyclic extensions of K of exponent dividing n are the same as the socalled “Kummer” extensions K(ya). Let a be a nonzero element of K. If L is an extension of K such that x”=a has a root CI in L, then all the roots ct,[tL,[2a,...,~1a of x”=a are in L, and any automorph of L over K permutes them. We denote the minimal splitting field for x”  a by K(qa). If r~ is an element of the Galois group G[K(!$‘a)/K], then, once we have chosen a root of x” = a, d is determined completely by the image ba = cba. In particular, if a is of order n in the multiplicative group K “/(K “>“, then d is a nth power only if nlr, so
B. J. BIRCH
90
X”  a is irreducible; in this case, the map B I+ lb gives an isomorphism of the Galois group on to a cyclic group of order n. Let us state all this as a lemma. LEMMA 1. If a is a nonzero element of K, there is a welldefined normal extension K(va), the splitting field of x”  a. If a is a root of x” = a, there is a map of G[K(va)/K] into K” given by ok au/a; in particular, if a is of order n in K “/(K “>“, the Galois group is cyclic and can be generated by u with OCR= ccc. LEMMA 2. If L is a cyclic extension of K (i.e. G(L/K) for some b E K. (b must generate K “/(K”)“.)
is cyclic)
then
L = K(yb)
The proof is by direct construction, which is providentially possible. Let CJbe a generator of G(L/K); then L = K(y) for some y, since all separable extensions are simple, and we can choose y so that y, ay,. . . , rr”l y is a basis for L over K. Form the sum n1 P =,zo rse Then o/3 = C‘p, and p # 0 since y,. . . , a”‘~ are linearly independent over K; so p” E K and p’ # K for 0 < r < n. Thus p” = b is of order n in Kx /(Kx >“; by Lemma 1, K(lb) is a cyclic extension of degree n contained in L, so L = K(;lb). LEMMA 3. Two cyclic extensions K(qa), K(gb) of K of the same degree are the same if and only if a = b’ c” for some c E K and r E 2 with (r, n) = 1.
In fact, “if” is obvious; we have to prove the “only if” part. This is easy by elementary Galois theory. Suppose that K(a) = K(p) with ci’ = a, j3” = b. Let (T be a generator of the Galois group with aa = &x, then a/? = [‘/3 for some i. Suppose p =
nl T
cjc2,
then ab = 1 cjjicli, so we can only have /3 = cic2. COROLLARY. Let L be a jinite extension of K with abelian Galois group G of exponent dividing n. Then G is the direct product of cyclic subgroups of G, x . . . x Gi+l x . . . x G,; GI,. . ., Cr. For each i, let Li be the$xedfield then G(Li/K)
= Gt, Lt = K(ai) with UT= a, E K, and L = K(a,, . . . , a,).
We may approach the last couple of lemmas via Galois cohomology Serre, p. 163). We quote a generalization of Lemma 2.
(see
LEMMA 4. If L is a normal algebraic extension of K with Galois group G then H1(G, Lx) = 0.
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91
[A Icocycle is a “continuous” map G + Lx in which a +u, with u(aJ = a01 a,l; “continuous” means that the map may be factored through a finite quotient G’ of G. Form the sum /I = X a, O(Y); by the linear independence OEG’
of automorphisms [Artin: “Galois Theory”, (Notre Dame), Theorem 12 and Corollary and Theorem 211, we may choose y so that p # 0; and then a, = ,9/u fl.1 Apply Lemma 4 taking L as the (separable) algebraic closure of K. There is an exact sequence 0 + E,, .Y+L x , Lx ) 0 where v is the map XHX” and E,, is the nth roots of unity, Taking homology, we get H’(G,E,) + H’(G,L”) ” H’(G, Lx) + H’(G,E,) + H’(G, Lx) +. . . Since G acts trivially on E,, H’(G, E,) is Hom(G, E,), and of course H’(G, L “) is the part of L” fixed under G, so this is E,+K”&KX + Hom(G, EJ ) 0, i.e. K ‘/(K ’ >” ESHom(G, E,). One can make the isomorphism explicit. Recollect that if 0 + A + B + C + 0 then an element c E Cc gives a map G , A by taking b with b + c, and then crI+ blab is a map G + A. In our case, we start with c E Kx ; take y so that y” = c; and then 6~ y/oy is a map G + E.. Conversely, if 4 E Hom(G, E,), let G’ be the kernel of 4. Then G/G’ is the Galois group of a cyclic extension K’ of K of exponent dividing n, and if u E G is such that 4(o) = 5, there is an element y E K’ such that oy = [y; then y” E K ‘, so determines a class of K”IK”“. The elements of G whose exponent divides n are preserved in Hom(G, EJ. Let K, be the union of the abelian extensions of K whose Galois groups have exponent dividing n; then Hom[G(KJK), E,] E K “/(K ‘>,. In particular the finite abelian extensions of G whose exponents divide n correspond to finite subgroups of Kx /(Kx >“. Finally we want to look at the factorization of primes p of K in the extension K(J’/a); by Chapter I, as before, this is the same as the study of the extensions K+,(qu) of the local field KP. LEMMA 5. The discriminant of K(va) over K divides n” a”’ ; p is unram$ed if p J’na. If af is the least power of a such that af = x”(p) is soluble, then f is the residue class degree.
Suppose a” = a; then, if o is the ring of integers of K, o[tl] is a submodule of the ring of integers of K(a), and by Chapter I, Section 4, Prop 6, its discriminant is @a”‘)” = n” u” l; so the discriminant of K(U) over K divides .“a”I. In particular, by Chapter I, Section 5, p is unramified if p$na, and indeed by Kummer’s theorem (or by Chapter, II Section 16 and Hensel’s Lemma) the factorization of p then mimics that of x” a modulo p. Hence, if a/ is the least power of a such that x” = a/(p) is soluble,
92
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J.
BIRCH
then p factors as the product of n/fprime ideals in K(a), and the residue class degree isf. Alternatively, this last bit can be proved like Lemma 4 of the cyclotomic lecture, by forming the unramified extension of KP in which x”  a splits. LEMMA 6. Zf pla, p,j’n and p”,f a then p is tamely ramified pla and p2,fa then p is totally ramijied in K(qa).
in K(qa);
if
Let u be a root of x” = a. The second part of the lemma is easy; if pla but pz #a then, quite explicitly, p = (p, cc)“. The result can also be obtained from Chapter I, Section 6, Theorem 1, since in this case x” a is an Eisenstein polynomial. If p,t’n, so that p does not divide the degree of the extension then any ramification must certainly be tame (see Chapter I, Section 5). On the other hand, if pla but +~“,/‘a then p is certainly ramified, since (p, or)lpl(p, a)“, but a 4p. There remains the case where flu, p’+ ’ )(a, p ,j’n with 2 I r I n  1. If (r,n) = 1, we have k,m so that rk+nm = 1, and chooseqGK with p/q, p’,j’q. Then K[;i@q“‘“)I = K($’ a) , and we have got back to the case r = 1. If (r,n) = s > 1, the ramification is no longer total, and there may or may not be splitting as well. The ramification index is n/s : p is unramified in the extension K(la) of K, and then the factors of p are totally ramified in the extension K(;/u) of K(qa). APPENDIX
Kummer’s Theorem Throughout this appendix, K is an algebraic number field with ring of integers o, and L = K(O) is an extension of degree n; we suppose that 13is an integer and that f(x) 5 o[x] is the characteristic polynomial of 8. p is a prime ideal of o, v is the associated valuation, Kg is the completion of K at p, oP is the ring of integers of K,, and K,* is the residue class field. The prime ideal factorization of p in L is p = n q:J; the associated valuations, completions, rings of integers, and residue class fields are Vj, L,, oj, LT. If f, g are polynomials in o[x], o,[x] their images under the residue class map are denoted
by f *, g*. In Chapter II, Section 10, it was shown that, if the irreducible factorization off(x) in o[x] is Ax> = 1 %jgJ 9 jCx> then Lj = Kp(Oj) where gj(0j) = 0. There is an injection map pj : L + L,, with p, 0 = tlj; and a residue class map *j : oj + LT. We wish to relate the factorization of p in L to the factorization off;(x) in K~[x]. First we need a lemma.
CYCLOTOMIC FIELDS AND KUMMER EXTENSIONS
93
LEMMA. With the above notations, g;(x) is a power of an irreducible polynomial of K,*[x], say g;(x) = [GT(x)rJ, and $j 61jis a zero of G:(x).
This will be clear enough if we show that the modulo p reduction of every irreducible polynomial of op[x] is a power of an irreducible polynomial of K:[x]. Put another way, we wish to show that if h(x) E op[x] and h*(x) = h:(x). h;(x) with (h:(x), h;(x)) = 1, then we can find h,(x), h,(x) from op[x] such that h(x) = h,(x). h,(x). This isjust a version of Hensel’s Lemma (see Chapter II, Appendix C; Weiss 2.2.1.). KUMMER’STHEOREM. Supposethat p, 8 have the property that nl
T a,&K[B] is an integer only if v(ai) 2 0 for i = 0,. . . , n  1. Supposethat the irreducible factorization off*(x) in K,*[x] is f*(x) = n [G,*(x)]~‘, and for each j let Gj(x) E o[x] be a manic polynomial whose image by the residue class map is G;(X). Then the prime idealfactorization of p is p = fl q;l, with qj = (p, Gj(0)). [cf. Chapter II, 19.20; p = IT q;J is equivalent to saying that the valuations ofL extending v are V, for j = 1, .... J, the corresponding ramification indices are the e,, and the set of integers a of L with V,(a) > 0 is q,. In our notation, V,(a) > 0 if v,a = 0.1
We have to check that the ramification indices are correct and that q, = (p, Gj(0)); also, that different polynomials GT come from different polynomials gj. We know already that Lj E KP[x]/(gj(x)). By the hypothesis of the theorem, every integer of L is of form c a, 0’ with u(ai) 2 0, so every element of LT is of form c a*(tij8)‘; so Lj* z K,*[x]/(GT(x)), and ej = [Lj : K,]/[LT; K,*] is the correct ramification index. If a E (p, G,(e)) then clearly wjcr = 0, SO,CIE qj. Conversely, suppose aeqi; then n1 a = 1 a&3*, 0
and by the hypothesis of the theorem v(aJ 2 0 for i = 0,. . ., n 1. Write h(x) = 1 ai.xi so that h(x) E oP[x]. Since u E qj, tjj CI= 0 so h*(ll/j 0) = 0. We can express h(x) = Gj(x)qj(x)+ri(x), with qj, rj E o~[x] and
de&j) < deg(Gj) ; and then rj*(t,bjO) = 0, so r;(x) is identically zero, and h(B) E (p, Gj(@). Hence qj = (p, Gj(B)),‘as required. Finally, if GT = GTlwith i #.j, then qi = qj, so g,(x) would be the same as gj(x),Ia
contradiction.
CHAPTER IV
Cohomology
of Groups
M. F. ATIYAH and C. T. C. WALL 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
Definition of Cohomology . . . The Standard Complex . . . . Homology. . . . . . . . Change of Groups . . . . . The RestrictionInflation Sequence TheTate Groups. . . . . . Cupproducts . . . . . . . Cyclic Groups; Herbrand Quotient Cohomological Triviality . . . Tate’s Theorem . . . . . .
......... ......... ......... ......... ......... ......... ......... ......... ......... .........
94 96 97 g8 100 101 105 108 111 113
1. Definition of Cohomology Let G be a group, A = Z[G] its integral
group ring. A (left) Gmodule is the same thing as a (left) Amodule. By Gmodule we shall always mean left Gmodule. Note that if A is a left Gmodule we can define a right Gmodule structure on A by putting a. g = g  ’ . a. If A, B are Gmodules, the group of all abelian group homomorphisms A + B is denoted by Horn (A, B), and the group of all Gmodule homomorphisms by Horn, (A, B). Horn (A, B) has a Gmodule structure defined as follows: if cpE Horn (A, B), g. cp is the mapping a H g. cp(g‘a) (a E A). For any Gmodule A, the subset of elements of A invariant under the action of G is denoted by AG. AG is an abelian group which depends functorially on A. It is the largest submodule of A on which G acts trivially. If A, B are Gmodules then Horn, (A, B) = (Horn (A, B))G; u.0 in particular, Horn, (Z, A) = (Horn (Z, A))G r AG, regarding Z as a Gmodule on which G acts trivially. Since the functor Horn is leftexact, it follows that AG is a leftexact covariant fun&or of A, i.e. if O+A+B+C+O
is an exact sequence of Gmodules, then O+AG~BG+CG
is an exact sequence of abelian groups. 94
(1.2)
.
95
COHOMOLOGY OF GROUPS
If X is any abelian group we can form the Gmodule Horn (A, X). A Gmodule of this type is said to be coinduced. By a cohomological extension of the functor AG we mean a sequence of functors H4(G, A) (q = 0, 1,. . .), with H’(G, A) = A’, together with connecting (or boundary) homomorphisms 6 : Hq(G, C) ) H”+‘(G, A) defined functorially for exact sequences (1.2), such that (i) the sequence . . . f Hq(G, A) + Hq(G, B) + Hq(G, C): Hq+l(G, A) f . . . is exact; and (ii) Hq(G, A) = 0 for all q 2 1 if A is coinduced.
(1.3)
THEOREM 1. There exists one and, up to canonical equivalence, only one cohomological extension of the functor AG.
The groups Hq(G, A), uniquely determined cohomoIogy groups of the Gmodule
by Theorem
1, are called the
A.
The existence part of Theorem 1 is established by the following construction. Choose a resolution P of the Gmodule Z (G acting trivially on Z) by free Gmodules: . . . +P1+Po+Z+O and form the complex K = Horn, (P, A), i.e. O+Hom,(P,,A)+Hom,(P,,A)+.... Let Hq(K) (q > 0) denote the qth cohomology group of this complex. Then Hq(G, A) = Hq(K) satisfies the conditions for a cohomological extension of the functor AG. For by a basic theorem of homological algebra, the Hq(G, A) so defined satisfy the exactness property (1.3); also H’(G, A) = Ho(K) = Horn, (Z, A) = AG; finally, if A is coinduced, say A = Horn (A, X) where X is an abelian group, then for any Gmodule B we have Horn, (B, A) z Horn (B, X) (the isomorphism being as follows: if cp: B + A is a Ghomomorphism, then cp corresponds to the map B + X defined by b I+ e@)(l), where 1 is the identity element of G). Hence the complex K is now O,Hom(P,,X)+Hom(P,,X)+ .. . which is exact at every place after the first, because the Pi are free as abelian groups; and therefore Hq(G, A) = 0 for all q > 1. To prove the uniqueness of the cohomology groups we consider, for each Gmodule A, the Gmodule A* = Horn (A, A). There is a natural injection
96
M. F. ATIYAH
AND
C. T. C. WALL
A + A* which maps a E A to cp,,, where cp,,is defined by p,(g) = ga. Hence we have an exact sequence of Gmodules O+A+A*+A’,O
where A’ = A*/A;
(1.4)
since A* is coinduced, it follows from (1.3) that 6 : Hq(G, A’) + H4+ ‘(G, A)
is an isomorphism
(1.5)
for all q > 1, and that A) g Coker (H’(G, A*) + H’(G, A’)).
H’(G,
The Hq(G, A) can therefore be constructed inductively are unique up to canonical equivalence. This procedure as an inductive definition of the Hq. Remark. It follows from the uniqueness that the Hq(G, of the resolution P of Z used to construct them. So we venient choice of P.
(1.6)
from Ho, and so could also be used A) are independent may take any con
2. The Standard Complex As a particular choice for the resolution P we can take Pi = Z[G’+‘], i.e. Pi is the free Zmodule with basis G x . . . x G ((i+ 1) factors), G acting on each basis element as follows: s(go, 91,. * . , Si> The homomorphism Go,.
d: Pi + Pil
=
(sS09sSl,.
. *,sC7i)*
is given by the wellknown formula
. ~.83=j~o(1)j(80~~~~~9jl,Bj+l~~
. .sgi)7
(2.1)
and the mapping E: PO + Z is that which sends each generator (go) to 1 E Z. (To show that the resulting sequence d
d
. ..+P.+P,:z+o
(2.2)
is exact, choose an element s E G and define h : Pi + Pi+ 1 by the formula Mg 0,...,9i)=(s,B0,91,...,91). It is immediately checked that dh +hd = 1 and that dd = 0, from which exactness follows.) An element of K’ = Horn, (Pi, A) is then a function f: G’+ ’ + A such that mo,
?I13
* * *, W3
=
s*fh709f?l,+
* *,Si>*
Such a function is determined by its values at elements of G’+ ’ of the form t1,gl,glg,,...,glg,....g3:ifweput (Pkll,. ..,gj)=f(l,gl,g,Y,,...,Yl...Yi~ the boundary is given by the formula
COHOMOLOGY
tdv)(g,, . . ,9*+1) = 91.(Pt92,.
97
OF GROUPS
. ..Bi+l)+j~l(1Ym(gl....,8j8j+l..
+(iY+w,,. This shows that a Icocycle is a crossed homomorphism,
* *,Si+l)+
.4x
(2.3) i.e. a map G + A
satisfying and cp is a coboundary if there exists a E A such that cp(g) = gua. In particular, if G acts trivially on A then H’(G, A) = Horn (G, A). (2.4) From (2.3) we see also that a 2cocycle is a function 40: G x G + A such that 91~t92,93)4p(9192,93)+(Pt91~9293)(Pts1,92) = 0. Such functions (called factor systems) arise in the problem of group extensions, and H’(G, A) describes the possible extensions E of G by A, i.e. exact sequences 1 + A + E + G + 1, where A is an abelian normal subgroup of E, and G operates on A by inner automorphisms. If E is such an extension, choose a section o : G + E (a system of coset representatives). Then we have 4g1). h72) = (P(s19g2M3192) for some (p(gr, gJ E A. The function cp is a 2cocycle of G with values in A; if we change the section (r, we alter cp by a coboundary, so that the class of cp in H’(G, A) depends only on the extension. Conversely, every element of H2(G, A) arises from an extension of G by A in this way. For later use, we give an explicit description of the connecting homomorphism 6 : H’(G, C) + H’(G, A) in the exact sequence (1.3). Let c E H’(G, C) = CG, and lift c up to b E B. Then db is the function s H sb  b; the image of sb b in C is zero, hence sb b E A and therefore db is a Icocycle of G with values in A. If we change b by the addition of an element of A, we change db by a coboundary, hence the class of db in H’(G, A) depends only on c, and is the image of c under 6. 3. Homology If A, B are Gmodules, A @I B denotes their tensor product over Z, and A @o B their tensor product over A. A @ B has a natural Gmodule structure, defined by g(a 8 b) = (gu) @I (gb). Let I, be the kernel of the homomorphism A + Z which maps each s E G to 1 E Z. IG is an ideal of A, generated by all s 1 (s E G). From the
exact sequence o+I,+A+z+o
(3.1)
98
M. F. ATIYAH
AND
C. T. C. WALL
and the rightexactness of @I it follows that, for any Gmodule
A,
Z BG A z A/Z,A. The Gmodule A/&A is denoted by A,. It is the largest quotient module of A on which G acts trivially. Clearly A, is a rightexact functor of A. For any two Gmodules A, B we have A&B
z (A@&.
(3.2) A Gmodule of the form A @ X, where X is any abelian group, is said to be induced. By interchanging right and left, induced and coinduced, we define a homological extension of the functor A,. THEOREM 2. There exists a unique homological extension of the functor A o The homology group H&G, A) given by Theorem 2 may be constructed from the standard complex P of $2 by taking H&G, A) = f&U’ 63~ A). Uniqueness follows by using the exact sequence O+A’+A*+A+O
(3.3) where A, = A @ A. The details are exactly similar to those of the proof of Theorem 1. The connecting homomorphism 6 : H,(G, C) + &(G, A) may be described explicitly as follows. A lcycle of G with values in C is a functionf: G , C such thatf(s) = 0 for almost all s E‘ G and such that df = C (sl  l)f(s) =O. SEG
For each s E G lift f(s) to j(s) E B (if j(s) = 0, choose j(s) = 0). Then dj has zero image in C, hence is an element of A. The class of djin H,(G, A) is then the image under 6 of the class off. PROPOSITION 1. H,(G, Z) g G/G’, where G’ is the commutator subgroup of G. Proof. From the exact sequence (3.1) and the fact that A is an induced Gmodule, the connecting homomorphism 6 : H1(G, z) ) H,(G, zo) = I& is an isomorphism. On the other hand, the map s H s 1 induces an isomorphism of G/G’ onto &/I$ 4. Change of Groups Let G’ be a subgroup of G. If A’ is a G’module, we can form the Gmodule A = HomGP (A, A’): A is really a right Gmodule, but we turn it into a left Gmodule as described in $ 1 (if cpE A, then g . cp is the homomorphism g’ H (p(g’gI)). Then we have
COHOMOLOGY PROPOSITION
OF GROUPS
2 (Shapiro’s Lemma). Hq(G, A) = Hq(G’, A’)
99
for all q Z 0.
Proof. If P is a free Aresolution of Z it is also a free A’resolution, and Horn, (P, A) z Horn, (P, A’). The analogous result holds for homology, with Horn replaced by 63. Note that Prop. 2 may be regarded as a generalization of property (ii) of the cohomology groups (9 1) : if G’ = (l), then A’ = Z and A is a coinduced module, and the Hq(G’, A’) are zero for q 2 1. Iff: G’ + G is a homomorphism of groups, it induces a homomorphism P’ + P of the standard complexes, hence a homomorphism f * : Hq(G, A) + Hq(G’, A)
for any Gmodule A. (We regard A as a Gmodule via J) In particular, taking G’ = H to be a subgroup of G, and f to be the embedding H ) G, we have restriction homomorphisms Res : Hq(G, A) + Hq(H, A). If H is a normal subgroup of G we considerf: G + G/H. For any Gmodule A we have the G/Hmodule AH and hence a homomorphism Hq(G/H, AH) + Hq(G, AH). Composing this with the homomorphism induced by AH + A we obtain the inflation homomorphisms Inf : Hq(G/H, Ai + Hq(G, A). Similarly, for homology, a homomorphism f: G’ y G gives rise to a homomorphism f* : K#X A) + H,G A) ; in particular, taking G’ = H to be a subgroup of G, and f: H + G the embedding, we have the corestriction homomorphisms
Cor : H,(H, A) + H&G, A). Consider the inner automorphism s H tst ’ of G. This turns A into a new Gmodule, denoted by A’, and gives a homomorphism Hq(G, A) + Hq(G, A’).
Now a H t Ia defines an isomorphism
A’ + A and hence induces
Hq(G, A) + Hq(G, A). PROPOSITION
of H’(G,
3.
The composition
(4.1)
of (4.1) and (4.2) is the identity
(4.2) map
A).
The proof employs a standard technique, that of dimensionshifting: we verify the result for q = 0 and then proceed by induction on q, using (1 S) to shift the dimension downwards.
M. F. ATIYAH AND C. T. C. WALL
100
For q = 0, we have H’(G, A’) = (A’)G = t. AG, and (4.1) is just multiplication by t. Since (4.2) is multiplication by tr, the composition is the identity. Now assume that q > 0 and that the result is true for q  1. Corresponding to the exact sequence (1.4) we have an exact sequence 0 f A’ + (A*)’ + (A’)’ * 0.
Since (A*)’ is Gisomorphic to A*, it is a coinduced have functorial isomorphisms Hq(G, A’) z Hq‘(G,
(A’)‘)
module,
hence we
(q Z 2)
and H’(G,
A’) z Coker (H’(G,
(A*)‘)
t H’(G,
(A’)‘)).
Now apply the inductive hypothesis. 5. The RestrictionInflation
Sequence
PROPOSITION 4. Let H be a normal subgroup of G, and let A be a Gmodule. Then rhe sequence Inf RCS 0 ) H’(G/H,AH) + H’(G, A) + H’(H, A) is exact.
The proof is by direct verification on cocycles. (1) Exactness at H’(G/H, AH). Let f: G/H + AH be a 1cocycle, then f induces f : G + G/H f AH + A, which is a Icocycle, and the class off is the inflation of the class off. Hence if f is a coboundary, there exists But f is constant on the cosets of aEA such that f(s) = saa (sEG). H in G, hence saa = sta a for all t E H, i.e. ?a = a for all t E H. Hence a E AH and thereforef is a coboundary. (2) Res 0 Inf = 0. If rp : G + A is a Icocycle, then the class of cplH: H + A is the restriction of the class of cp. But if q = j, it is clear thatf/H is constant and equal to f(l) = 0. (3) Exactness at H’(G, A). Let cp : G + A be a 1cocycle whose restriction to H is a coboundary; then there exists a E A such that cp(t) = taa for all t E H. Subtracting from cp the coboundary s H sa a, we are reduced to the case where rp\H = 0. The formula then shows (taking t E H) that cp is constant on the cosets of H in G, and then (taking s E H, t E G) that the image of cp is contained in AH. Hence rp is the inflation of a 1cocycle G/H + AH, and the proof is complete.
COHOMOLOGY
101
OF GROUPS
PROPOXTION 5. Let q > 1, and suppose that H’(H, A) = Ofor 1 < i
1.
is exact.
This is another example
of dimensionshifting:
we reduce to the case
q = 1, which is Proposition 4. Suppose then that q > 1 and that the result is true for q 1. In the exact sequence (1.4), the Gmodule A* is coinduced as an Hmodule (since A = Z[G] is a free Z[H]module), hence
Hi(H,A’)zHi+‘(H,A)=O Also, since H’(H, A) = 0, the sequence
forl
0 + AH + (A*)H + (A’)H + 0
is exact, and (A*)a is coinduced as a G/Hmodule Horn (Z[G/H], A)). Hence in the diagram
(because (A*)H
E
0 + H4‘(G/H, (A’)H) + Hq‘(G, A’) + Hq ‘(H, A’) ld 16 16 0 + Hq(G/H, AH) + Hq(G, A) + Hq(H,A)
the three vertical arrows are isomorphisms, the diagram is commutative, and by the inductive assumption applied to A’, the top line is exact. Hence so is the bottom line. COROLLARY. Under the hypothesesof Prop. 5, H’(G/H, Aa) g H’(G, A),
1 < i < q  1.
6. The Tate Groups From now on we assume that G is finite, and we denote by N the element 1 s of A. For any Gmodule A, multiplication by N defines an endomor
S6G
phism N: A + A, and clearly I,A 6 Ker (N), Hence N induces a homomorphism
Im(N)
5 AG.
N* : H,(G, A) + H’(G, A)
and we define fi,(G, A) = Ker (N*),
fi’(G, A) = Coker (N*) = AGIN(
Since G is finite, we can define a mapping Horn (A, X) + A @ X (where X is any abelian group) by the rule
and it is immediately verified that this is a Gmodule isomorphism. Hence for a finite group the notions of induced and coinduced modules coincide.
102
M. F. ATIYAH AND C. T. C. WALL
PROPOSITION 6. If A is an induced Gmodule, then &(G,
A) = Z%‘(G, A) = 0. Proof. Let A = A @ X, X an abelian group. Since A is Zfree, every element of A is uniquely of the form c s @ x,. If this element is Ginvariant, SGG
then c gs @ x, = C s @ X, for all g E G, from which it follows that all the x,‘are equal. hence such an element is of the form N.(l 8 X) and therefore lies in N(A). Hence fi’(G, A) = 0. Similarly, if N.C s @ X, = 0, we find that c X, = 0, and therefore 1 s @I x, = C (s lj(l
@I x,) E Io A. Hence fj,(G, A) = 0. Now we define the Tate cohomology groups I?*(G, A) for all integers q by forq 2 1 fiq(G, A) = Hq(G, A) 8 ‘(G, A) = fl,(G, A) PQ(G, A) = Hq ,(G, A) for 4 2 2. THEOREM 3. For every exact sequence of Gmodules O+A*B+C+O we have an exact sequence . . . + Aq(G, A) + A~(G, B) + A~(G, c) L f++ ‘(G, A) .+ . . . Proof. We have to splice together the homology and cohomology sequences. Consider the diagram . . . + H,(G, C) : H,(G, A) ) H,(G, B) + H,(G, C) + 0 &G 1% 1 PE. d 1 0 + H’(G, A) + H’(G, B) + H’(G, C) + H’(G, A) + . . . where N: is the homomorphism N* relative to A, and so on. It is clear that the inner two squares are commutative, and for the outer two squares commutativity follows immediately from the explicit descriptions of the connecting homomorphism 6 given in $I 2 and 3. We define 8 : fj,(G, C) + I?‘(G, A) as follows. If c E fio(G, C) = Ker (NE), lift c to b E H,(G, B), then N;(b) E H’(G, B) has zero image in H’(G, C) and therefore comes from an element a E H’(G, A), whose image in A’(G, A) is independent of the choice of b; this element of fi’(G, A) we define to be (c). The definitions of the other maps in the sequence H,(G, C> + fio(G, 4 ) fio(G, B> ) fio(G C) : AO(G, A) + fi’(G, B) + BO(G, C) t H’(G, A) are the obvious ones, and the verification that the whole sequence is exact is a straightforward piece of diagramchasing. The Tate groups can be considered as the cohomology groups of a
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complex constructed out of a complete resolution of G. Let P denote a Gresolution of Z by finitelygenerated free Gmodules (for example, the standard resolution of $2), and let P * = Horn (P, Z) be its dual, so that we have exact sequences . ..tP.tP,:Z+O z* O+Z+P;r+PT+
.. .
(the dual sequence is exact because each P, is Zfree). Putting P, = Pzdl and splicing the two sequences together we get a doublyinfinite exact sequence L:
. . . +P1+Po+P1+P2+
.. . .
The Tate groups are then the cohomology groups Hq(Hom, any Gmodule A. This assertion is clear if q > 1. If q < 2 following fact: if C is a finitely generated free Gmodule, let C* = be its dual; then the mapping B: C 8 A + Horn (C*, A) follows : o(c 8 a) mapsfe C* tof(c).a
is a Gmodule
isomorphism.
(L, A)) for we use the Hom(C, Z)
defined
as
Hence the composition
T : C gG A = (C @ A), z (C @ A)G : (Horn (C*, A))’
= Horn, (C*, A)
is an isomorphism (N* is an isomorphism because C 0 A is an induced Gmodule). From this it follows that Horn, (P,, A) z Pndl @o A, and hence that Hbq(HomG (L, A)) = H,,(G, A) for q 3 2. Finally, we have to consider the cases q = 0, 1. The mapping HOIll,
(P 1, A) +
HOm,
(PO, A)
(6.1)
.?*
is induced by the composition P, 2 Z + Pml. If we identify Horn, (P 1, A) with P, OG A by means of the isomorphism r, the mapping (6.1) becomes a mapping from P,, @o A to Horn, (PO, A), and from the definition of r it is not difficult to see that this mapping factorizes into P,~,A~A,~AG_tHomG(P,,A)
(6.2)
where the extreme arrows are the mappings induced by E. From this it follows that Hq(HomG (L, A)) = flq(G, A) for q = 0, 1. Remark. Since any Gmodule can be expressed either as a submodule or as a quotient module of an induced module, it follows from Prop. 6 and Theorem 3 that the Tate groups A4 can be “shifted” both up and down. If H is a subgroup of G, the restriction homomorphism Res : Hq(G, A) + Hq(H, A)
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M. F. ATIYAH AND C. T. C. WALL
has been defined for all q > 0. It is therefore defined for the Tate groups fig, q 2 1, and commutes with the connecting homomorphism 6. By dimensionshifting it then gets extended to all fiq (use the exact sequence (3.3) and the fact that A* is induced as an Hmodule). Similarly, the corestriction, which was defined in the first place for H, (i.e. keqel, q 2 1) gets extended by dimensionshifting to all fig (use (1.4) likewise). PROPOSITION 7. Let H be a subgroup of G, and let A be a Gmodule.
Then (i) Res : I?,,(G, A) + fjo(H, A) is induced by N&/n : A, + AR, where N;,,(a)
= c s; Ia
and (sJ is a system of coset representatives of G/H; (ii) Cor : fi’(H, A) + fi’(G, A) is induced by No,n : A“ + AG, where NGIH(a) = 7 sia* We shall prove (i), and leave (ii) to the reader. First of all, since 8 : &‘(G, A) + H’(G, A’) is induced by 6 : H’(G, A) + H’(G, A), and since Res : H’(G, A) + H”(H, A) is the embedding AG + AH and is compatible with 6, it follows that Res : fi’(G, A) + fi’(H, A) is induced by AG + A”. Now let v: fio(G, A) + l?,(H, A) be the map induced by NAIH. We have to check that the diagram fio(G, A) : 8’(G, fi,;fr,
A’)
A) : A’(&‘)
is commutative. Let a E A be a representative of Z E fj,(G, A), so that N,(a) = 0. Lift a to b E A,, then No(b) has zero image in A and is Ginvariant, hence belongs to (A’)G E (A’)H. The class of N,(b) mod. N,(A’) is Res 0 8(E). On the other hand, v(Z) is the class mod. I,A of N&,,(a), which lifts to N;,,,(b), and 8 0 v(Z) is represented by NH 0 N;,,(b) = N,(b). Note: For q = 2 and A = Z we have Z?‘(G, Z) = H,(G, Z) z G/G’; Res : G/G’ + H/H’ is classically called the transfer and can be defined as follows. G/G’ is dual to Horn (G, C*), hence the transfer will be dual to a homomorphism Horn (H, C*) + Horn (G, C*). This homomorphism
is given by
P I+ det (i*p)ldet td), where i,p is the representation of G induced by p, and det denotes the corresponding onedimensional representation obtained by taking determinants (Horn (G, C*) is here written multiplicatively).
COHOMOLOGY PROPOSITION
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8. If (G : H) = n, then
Cor 0 Res = n. Prooj For I?’ this follows from Prop. 7(ii): Res is induced by the embedding AC + AH, and Cor by NC/n :AH+AG; and N,,,(a) = na for all a E A’. The general case then follows by dimensionshifting. COROLLARY 1. If G has order n, all the groups @(G, by n.
A) are annihilated
Take H = (1) in Prop. 8, and use the fact that fiq(H, A) = 0
ProoJ
for all q. COROLLARY 2. If A is a jinitelygenerated Gmodule, all the groups fiq(G, A) are finite. ProoJ The calculation of the fiq(G, A) from the standard complete resolution L shows that they are finitely generated abelian groups; since by Cor. 1 they are killed by n = Card (G), they are therefore finite. COROLLARY 3. Let S be a SyIow psubgroup of G. Then Res : Bq(G, A) + fiq(S, A) is a monomorphism
on the pprimary
component of Aq(G, A).
Let Card (G) = pa. m where m is prime to p. component of Aq(G, A), and suppose that mx = Cor 0 Res(x) = 0 by Prop. 8, since m = (G: S). On the other hand, Cor. 1; since (p”, m) = 1, it follosvs that x = 0. COROLLARY 4. If an element x of fiq(G, A) restricts Proof.
pprimary
Let x belong to the Res (x) = 0. Then we have pax = 0 by to zero in Aq(S, A)
for all Sylow subgroups S of G, then x = 0.
7. Cupproducts THEOREM 4. Let G be a jinite group. Then there exists one and only one family of homomorphisms @(G, A) 6 Aq(G, B) + fip+q(G, A @I B) (denoted by (a @ b) + a. b), defined for all integers p, q and all Gmodules A, B, such that: (i) These homomorphisms are functorial in A and B; (ii) For p = q = 0 they are induced by the natural product AG@BG+(A@B)G; (iii)
If 0 + A + A’ ) A” + 0 is an exact sequence of Gmodules, and if O+ A@ B+ A’@ Bd A”@ B0 is exact, thenfora”EpP(G, A”) and b E fiq(G, B) we have @a”).b = &Y’.b) (E~?~+~+‘(G,A@B));
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(iv) If 0 ‘+ B + B’ + B” + 0 is an exact sequence of Gmodules, and if O+A@B+A@B’+A@B” + 0 is exact, then for a E fip(G, A) and b” E aq(G, B”) we have a. (66”) = ( 1jP6(a. b”) (E fiP+n+l(G,
A @ B)).
be a complete resolution for G, as in 5 6. Let (PnL existence depends on constructing Gmodule homomorphisms qpP*q
* P,+,
The proof of
+Pp@Pq
for all pairs of integers p, q, satisfying the following two conditions: qp.q o d = (d 8 1) o ‘pp+l,q+(ljp(l
@ 4 0 qp,q+i;
(6 63 4 0 90.0 = 8,
(7.1) (7.2)
where E: PO + Z is defined by s(g) = 1 for all g E G. Once the (ppSq have been defined, we proceed as follows. Let f E Horn, (P,, A), g E Horn, (P,, B) be cochains, and define the product cochain f 0 g E Horn, (P,,,, A @ B) by f * 9 = UC3 s> 0 ‘pp.q.
Then it follows immediately
from (7.1) that
d(f. s> = WI. g+(ljPf. (dgj. (7.3) Hence iff, g are cocycles, so is f. g, and the cohomology class off . g depends only on the classes off and g: in other words, we have a homomorphism fiP(G, A) @ fiq(G, B) i fiP+q(G, A 0 B).
Clearly condition (i) is satisfied, and (ii) is a consequence of (7.2). Consider (iii). We have an exact sequence 0 + Horn, (P,, A) + Horn, (P,, A’)+ Horn, (P,, A”) + 0. Let tl” E Horn, (P,, A”) be a representative cocycle of the class a”, and lift a” back to tl’ E Horn, (P,, A’); da’ has zero image in Horn, (P,, r, A”) and therefore lies in Horn, (P,,,, A). The class of da’ in fiP+‘(G, A) is &a”). Hence if p E Horn, (P,, B) is a cocycle in the class b, then a”./3 represents the class a”. b; d(a’. j?) represents 6(a”. b); and (da’). j? represents (da”). b. But (since d/II = 0) we have d(a’./.?) = (da’) ./? from (7.3); hence s(a”. b) = (da”). b. The proof of (iv) is similar. Thus it remains to define the (P~,~,which we shall do for the standard complete resolution (Pq = Z[Gq+‘] if q > 0; P, = dual of PqSl if q > 1). Ifq a l,P, = P,*_, has a basis (as Zmodule) consisting of all (gr, . . . , g,*), where (gy,. . ., g,*) maps (gl,. . ., gq) E PqSl to 1 E Z, and every other basis element of Pq _ 1 to 0. In terms of this basis of P,, d: P, + PWqml is given by
COHOMOLOGY
andd:P,+P
I by GoI
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OF GROUPS
=s;G’s*‘
We define qpsq : Pp+4 + Pp @ P4 as follows: (1) ifp>Oandq>O, (Pp,qk70,. * * 2gp+q) = (903 * * * >sp) Q (gp, * * * 7gp+q); (2) if p 2 1 and 4 > 1,
cp,.q(& * *d7f+ql=
‘pp. p&:2
. . ..gq*)=C(gl.Sl,...,Sp)Q(Sp*1...,ST,g:,...,gq*);
(Ppq.p(d” Pp+q,
q(go,.
(Pq.p+q
( go,*
 .,s:>
= CG*
* *d?:JT,.~
* .,$I,>
= C(909.
* 9 sp)
= c
* .,gp,s1,.
(ST, * * . , $1
d;)
QDsp,
* .Jq)QJSp*,.
.  .,s1,gq);  .,a;
Q (sq, * * *, Sl, go,  * *, gp).
(In the sums on the righthand side, the si run independently through G.) The verification that the (pp.4 satisfy (7.1) is tedious, but entirely straightforward. This completes the existence part of the proof of Theorem 4. The uniqueness is proved by starting with (ii) and shifting dimensions by (iii) and (iv): the point is that the exact sequence (3.3), namely O+A’+A*+A+O, splits over Z, as the Zhomomorphism A + A, = A 0 A defined by a I+ 1 @ a shows; hence the result of tensoring it with any Gmodule B is still exact, and A, @ B = A 6 A 0 B = (A @ B),. Similarly for the exact sequence (I .4). Note the following properties of the cupproduct, which are easily proved by dimensionshifting : PROPOSITION
9.
(i) (a.b).c = a.(b.c) (identvying
(A @ B) @ C with A @ (B @ C)). dimbb.a (identifying A @I B with B Q A). (iii) Res (a. b) = Res (a). Res (b). (iv) Cor (a. Res (b)) = Cor (a). b. (ii) a.b = (l)dim@.
As an example, let us prove (iv). Here His a subgroup of G, a E Ap(H, A), b E Aq(G, B), so that both sides of (iv) are elements of fjp+q(G, A @ B). If p = q = 0, a is represented by say CIE AH, and by Prop. 7(ii) Cor (a) is represented by N,,,(a) = c Sitl E AG; b is represented by /I E BG, hence I
Cor (a). b is represented by NG/H(Oo
@ p
=
sia>
@ b
=
c
sita
@ 8
=
NG/H(a
@ p)*
On the other hand, a. Res (b) is represented by tl @ p E (A 8 B)H, hence
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M. F. ATIYAH
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C. T. C. WALL
Cor (a. Res (b)) by N,,,(a @ j?). This establishes (iv) for p = q = 0. Now use dimensionshifting as in the proof of the uniqueness of the cupproducts and the fact that both Cor and Res commute with the connecting homomorphisms relative to exact sequences of the types (3.3) and (1.4). We shall later have to consider cupproducts of a slightly more general If type. Let A, B, C be Gmodules, cp: A @ B + C a Ghomomorphism. we compose the cupproduct with the cohomology homomorphism cp* induced by q, we have mappings 4
fiP(G, A) Q fiq(G, B) + l?P+q(G, C);
explicitly,
a Q b I+ cp*(u. b). cp*(a. b) is the cupproduct
of a, b relative to cp.
8. Cyclic Groups; Herbrand Quotient If G is a cyclic group of order n, and s is a generator of G, we can define a particularly simple complete resolution K for G. Each K, is isomorphic by T = s 1 if i is even (resp. by to A, and d: Ki+l + K, is multiplication N if i is odd). The kernel of T is AG = N. A = image of N, and the image of T is IG = kernel of N. Hence for any Gmodule A the complex HOIll,
(K, A)
is N . . .
T
N
T
+AtA+AtA+...
and therefore tt’q(G, A) = l?‘(G, A) = AGINA, A2q+l(G, A) = fi,(G, A) = .A/Io A,
where NA is the kernel of N: A + A. In particular, H’(G, Z) = ZG/NZ = ZlnZ is cyclic of order n. THEOREM
5.
Cupproduct by a generator of H’(G, Z) induces an iso
morphism fiq(G, A) + fiq+2(G, A) for all integers q and all GmodulesA.
Proof. The exact sequences OtIG+h+Z+O, N
(8.1)
T
o+z+I\+I,+o,
(8.2)
give rise to isomorphisms 8O(G, Z) : H’(G, lo) : H2(G, Z).
Since both (8.1), (8.2) split over Z, they remain exact when tensored with A, and we are therefore reduced to showing that cupproduct by a generator of fi’(G, Z) induces an automorphism of Aq(G, A). By dimensionshifting
COHOMOLOGYOFGROIJPS
109
again, we reduce to the case q = 0. Since &‘(G, Z) = Z/nZ, a generator h of fi”(G, Z) is represented by an integer fi prime to n, and cupproduct with b is multiplication by /j’. Now /I is prime to n, hence there is an integer y such that j?y E 1 (mod. n); fi’(G, A) is killed by n, hence multiplication by /? is an automorphism of fi’(G, A). Let h,(d denote the order of fiq(G, A) (q = 0, 1) whenever this is finite. If both are finite we define the Herbrand quotient
PROPOSITION 10. Let 0 + A + B t C + 0 be an exact sequence of Gmodules (G a cyclic group). Then if two of the three Herbrand quotients h(A), h(B), h(C) are defined, so is the third and we have h(B) = h(A). h(C).
Proof. In view of the periodicity of the fiq, the cohomology is an exact hexagon: Ho(A) + Ho(B) L H’(C) H’(C) J s H’(B) t H’(A)
exact sequence
where Ho(A) means &‘(G, A), and so on. Suppose for example that H’(A), H’(A), Ho(B), H’(B) are finite. Let M, be the image of Ho(A) in H’(B), Then the sequence and so on in clockwise order round the hexagon. 0 + M2 + Ho(C) + M3 + 0 is exact, and M2, M3 are finite groups (M, because it is a homomorphic image of H’(B), M3 because it is a subgroup of H’(A)). Hence Ho(C) is finite, and similarly H’(C) is finite. The orders of are respectively m6m1, mlm,, . . ., m5m6 the groups H’(A), . . . , H’(C) (mi = order of M,), hence h(B) = h(A). h(C). PROPOSITION Il. If A is a jnite Gmodule, then h(A) = 1. Proof. Consider the exact sequences T
O+AG+A+A+Ao+O, .
0 + H’(A) + Ao 5 AC + Ho(A) + 0.
The first one shows that AC and Ao have the same order, and then the second one shows that Ho(A) and H’(A) have the same order. COROLLARY. Let A, B be Gmodules, f: A + B a Ghomomorphism with finite kernel and cokernel. Then if either of h(A), h(B) is dejined, so is the other, and they are equal.
110 Proof.
M. F. ATIYAH AND C. T. C. WALL
Suppose for example
that h(A) is defined.
From
the exact
sequences O+Ker(J)+A+f(A)+tJ 0 +f(A) + B + Coker (f) ) 0 it follows from Prop. 10 and 11 that h(f(A)) is defined and equal to h(A), then that h(B) is defined and equal to h(f(A)). PROPOSITION 12. Let E be a finitedimensional real representation space of G, and let L, L’ be two lattices of E which span E and are invariant under C Then if either of h(L), h(L’) is defined, so is the other, and they are equal.
For the proof of Prop. 12 we need the following lemma: LEMMA. Let G be a finite group and let M, M’ be two finitedimensional Q[G]modules such that MR = M @QR and ML = M’ Q. R are isomorphic as R[G]modules. Then M, M’ are isomorphic as Q[G]modules.
Proof. Let K be any field, L any extension field of K, A a Kalgebra. If Yis any Kvector space let V, denote the Lvector space V BK L. Let M, M’ be Amodules which are finitedimensional as Kvector spaces. An Ahomomorphism cp: M + M’ induces an A,homomorphism cp @ 1 : ML + ML, and rp H 40 0 1 gives rise to an isomorphism (of vector spaces over L) (HomA CM, M’)), zz HomAr. (ML, ML). (8.3) In the case in point, take K = Q, L = R, A = Q[G], so that AL = R[G]. The hypotheses of the lemma imply that M and M’ have the same dimension over Q, hence by choosing bases of M and M’ we can speak of the determinant of an element of Homorcl (M, M’), or of Hom,rc, (MR, ML). (It will of
course depend on the bases chosen.) From (8.3) it follows that if Ti are a Qbasis of Homot,, (M, M’), they are also an Rbasis of Hom,tc, (MB, Mi). Since MR, Mi are R[G]isomorphic, there exist ai E R such that det (c airi) # 0. Hence the polynomial F(t) = det(z ti5J E Q[tl,. . *, ~1, where ti are independent indeterminates over Q, is not identically zero, since F(a) # 0. Since Q is infinite, there exist bi E Q such that F(b) # 0, and then 2 bi5; is a Q[G]isomorphism of M onto M’. For the proof of Prop. 12, let M = L 0 Q, M’ = L’ @ Q. Then MR and ML are both R[G]isomorphic to E. Hence by the lemma there is a Q[G]isomorphism cp: L @ Q + L’ 0 Q. L is mapped injectively by cp to a lattice contained in (l/N)L’ for some positive integer N. Hence f = N. rp maps L injectively into L’; since L, L’ are both free abelian groups of the same (finite) rank, Coker (f) is finite. The result now follows from the Corollary to Prop. 11.
.
COHOMOLOGY
OF GROUPS
111
9. Cohomological Triviality A Gmodule A is cohomologically triviaI if, for every subgroup H of G, fiq(H, A) = 0 for all integers q. For example, an induced module is cohomologically trivial. LEMMA 1. Let p be a prime number, G a pgroup and A a Gmodule such that pA = 0. Then the foIlowing three conditions are equivalent: (i) A = 0; (ii) H’(G, A) = 0; (iii) H,(G, A) = 0. Proof. Clearly (i) implies (ii) and (iii). (ii) => (i): Suppose A # 0, let x be a nonzero element of A. Then the submodule B generated by x is finite, of order a power of p. Consider the Gorbits of the elements of B; they are all of ppower order (since the order of G is a power of p), and there is at least one fixed point, namely 0. Hence there are at least p fixed points, so that H’(G, A) = AG # 0. (iii) + (i). Let A’ = Horn (A, FP) be the dual of A, considered as a vectorspace over the field FP of p elements. Then H’(G, A’) = (A’)G = HOIll, (A, F,) is the dual of H,(G, A). Hence H’(G, A’) = 0, so that A’ = 0 and therefore A = 0. LEMMA 2. With the same hypotheses as in Lemma 1, suppose that H,(G, A) = 0. Then A is a free module over F,[G] = AlpA. Proof. Since pA = 0, we have p. H,(G, A) = 0 and therefore H,(G, A) is a vector space over FP. Take a basis e, of this space and lift each e, to a, E A. Let A’ be the submodule of A generated by the a,, and let A” = A/A’. Then we have an exact sequence
in which by therefore A” Hence they F,[G]module.
Ho (G, A’) : Ho (G, A) , Ho (G, A”) ) 0 construction tl is an isomorphism. Hence H,(G, A”) = 0 and = 0 by Lemma 1, so that the a, generate A as a Gmodule. define a Gepimorphism cp: L + A, where L is a free By construction, q induces an isomorphism
B : ffo(G L> , Ho(G A). Let R = Ker (9). Then since H,(G, A) = 0, the sequence 0 + H,(G, R) + H,(G, L) : H,(G, A) + 0 is exact; since j? is an isomorphism, H,(G, R) = 0 and therefore R = 0 by Lemma 1. Hence cp is an isomorphism.
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THEOREM 6. Let G be a pgroup and let A be a Gmodule such that pA = 0. Then the following conditions are equivalent:
(i) (ii) (iii) (iv)
A is a free F,[G]module; A is an induced module; A is cohomologically trivial; kg(G, A) = 0 for some integer q.
Clearly (i) * (ii) * (iii) * (iv). (iv) => (i). By dimensionshifting we construct a module B such that pB = 0 and fiq+‘(G, A) = A’‘(G, B) for all r. Hence H,(G, B) = 0 and therefore (Lemma 2) B is free over FJG]; hence Proof.
I?‘(G,
A) = fig4(G,
B) = 0
and therefore (Lemma 2 again) A is free over FJG]. THEOREM 7. Let G be a pgroup and A a Gmodule without Then the following conditions are equivalent:
ptorsion.
(i) A is cohomologically trivial; (ii) fig(G, A) = Ag+l(G, A) = 0 for some integer q; (iii) A/pA is a free F,[G]module. Proof. (i) => (ii) is clear. (ii) + (iii): From the exact sequence O+A:A+A/pA,O
we have an exact sequence fig(G, A) + Ag(G, A/PA) , fi4+‘(G, A), hence fig(G, A/PA) = 0. Hence, by Theorem 6, A/pA is free over F,,[G]. (iii) =z=(i): From the same exact sequence it follows that fig@, A) i: fig(H, A)
is an isomorphism for all integers q and all subgroups H of G. But fig(H, A) is a pgroup (Prop. 8, Cor. I), hence fig(H, A) = 0. COROLLARY. Let A be a Gmodule which is Zfree andsatisfies the equivalent conditions of Theorem 7. Then, for any torsionfree Gmodule B, the Gmodule N = Horn (A, B) is cohomologically trivial.
Proof.
Since A is Zfree, the exact sequence O+B:B+B/pB+O
gives an exact sequence O+N:N+Hom(A,B/pB)tO,
so that N has no ptorsion and N/pN z Horn (A/PA, B/pB). Since A/pA is a free F,[G]module, it is induced, hence is the direct sum of the
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COHOMOLOGYOFGROUPS
s. A’ (s E G), where A’ is a subgroup of A/PA. Hence N/pN is sum of the subgroup s. Horn (A’, B/pB) and is therefore induced. N is cohomologically trivial by Theorems 6 and 7. A Gmodule A is projective if Horn, (A, ) is an exact functor, lently if A is a direct summand of a free Gmodule. A projective is cohomologically trivial.
the direct Therefore or equivaGmodule
THEOREM 8. Let G be a finite group, A a Gmodule which is Zfree, Gp a Sylow psubgroup of G. Then the following are equivalent: (i) For each prime p, the G,module A satisfies the equivalent conditions of Theorem 7; (ii) A is a projective Gmodule.
Proof. (ii) * (i) is clear. (i) = (ii): Choose an exact sequence 0 + Q ) F + A ) 0, where F is a free Gmodule. Since A is Zfree, this gives an exact sequence 0 + Horn (A, Q) + Horn (A, F) + Horn (A, A) + 0 By the Corollary to Theorem 7, Horn (A, Q) is cohomologically trivial as a G,module for each p, hence H’(G, Horn (A, Q)) = 0 by Prop. 8, Cor. 4. Bearing in mind that H’(G, Horn (A, Q)) = (Horn (A, Q))G = Horn, (A, Q), and so on, it follows that Horn, (A, F) + Horn, (A, A) is surjective, hence the identity map of A extends to a Ghomomorphism A + F. Consequently A is a direct summand of F and is therefore projective. THEOREM 9. Let A be any Gmodule. Then the following are equivalent: (i) For each prime p, fiq(Gp, A) = 0 for two consecutive values of q (which may depend on p); (ii) A is cohomologically trivial; (iii) There is an exact sequence 0 + B, f B, + A + 0 in which B. and B, are projective Gmodules. Proof. (ii) * (i) is clear; so is (iii) => (ii), since a projective Gmodule cohomologically trivial. (i) * (iii): Choose an exact sequence of Gmodules O+B1+Bo+A+O,
is
with B, a free Gmodule. Then fiq(Gp, B,) z kqr(Gp, A) for all q and all p, hence fiq(Gp, B,) = 0 for two consecutive values of q. Also B, is Zfree (because B, is); hence, by Theorem 8, B, is projective. 10. Tate’s Theorem THEOREM 10. Let G be afinite group, B and C two Gmodules and f: B + C a Ghomomorphism. For each prime p, let G, be a Sylow psubgroup of G,
114
M. F. ATNAH
AND
C. T. C. WALL
and suppose that there exists an integer n,, such that f; : flq(Gp, B) + fiq(G,, C) is surjective for q = np, bijective for q = n,+ 1 and injective for q = n,+ 2. Then for any subgroup H of G and any integer q, f$ : &(H, B) f f&H, C) is an isomorphism. Proof. Let B* = Horn (A, B) and let i: B + B* be the injection (defined by i(b)(g) = g. b). Then (f, i) : B + C @ B* is injective, so that we have an exact sequence O+B+C@B*+D+O. Since B* is cohomologically trivial, the cohomology of C @ B* is the same as that of C. Hence the cohomology exact sequence and the hypotheses of the theorem imply that fiq(Gp, 0) = 0 for q = np and q = n,+ 1. It follows from Theorem 9 that D is cohomologically trivial, whence the result. THEOREM 11. Let A, B, C be three Gmodules and cp: A @I B + C a Ghomomorphism. Let q be a fixed integer and a a given element of Aq(G, A). Assume that for each prime p there exists an integer np such that the map B”(G,, B) + fin+q(G,, C) induced by cupproduct with Res,,,, (a) (relative to cp) is surjective for n = np, bijective for n = n,+ 1 and injective for n = n,+2. Then, for all subgroups H of G and aN integers n, the cupproduct with Res,,, (a) induces an isomorphism A”(H, B) + B”+q(H, C). (Explicitly, this mapping is b I+ cpz+,(ResclR (a). b).) Proof The case q = 0 is essentially Theorem 10. We have a E fi’(G, A): choose a E AC representing u (then a also represents Res,,, (a) for every subgroup H of G). Define f: B + C by fcB> = q(a @I /3); f is a Ghomomorphism, since a is Ginvariant. We claim that, for every b E A”(H, B), (10.1) P*(R%,n (a>. b) = f*(b). Indeed, this is clear for n = 0 (from the definition off), and the general case then follows by dimensionshifting. To shift downwards, for example, assume (10.1) true for n + 1, and consider the commutative diagram O+B’+B*+B+O YJ l@fJ 5f (10.2) o+c’+c*+c+o where B, = A @ B, C, = A 8 C, and the rows are exact. B,, C, are induced modules and therefore cohomologically trivial, hence the connecting homomorphisms 8 are isomorphisms, and the diagram
COHOMOLOGY OF GROUPS
115
fjt"(H, B) : AR+l(H, B') Jr* lf* 8 A"(H, C) + Bn+yH, C')
is commutative. Moreover, the rows of (10.2) split over Z, hence (10.2) remains exact (and commutative) when tensored with A (over Z). Let cp”: A @ B’ + C’ be the homomorphism induced by rp : A &I B + C. Then using the inductive hypothesis and the compatibility of cupproducts with connecting homomorphisms, we have 8 o f*(b) = f '* O 8(b) = #‘*(Res,,, (a). 8(b)) = cp,,* o &Res,,, (a). b) = 8 0 ‘p*(ResclH (a). b). Since 8 is an isomorphism, (10.1) is proved. Nowfsatisfies the hypotheses of Theorem 10, hencef,* is an isomorphism. This establishes Theorem 11 for the case q = 0. The general case now follows by another piece of dimensionshifting. To shift downwards from q+ 1 to q, for example, consider the exact sequence O+A’+A*+A+O where A, = A @ A; this gives rise to isomorphisms 8 : fiq(H, A) + Aq+ ‘(H, A’). Let u = Res,,, (a) E Aq(H, A); then U’ = 8(u) = Res,,, (&a)). Also 40: A 8 B + C induces cp’ : A’ @JB + C’. Consider the diagram A”(H, B) :
A”+q(H, A 0 B) Z
@+q(H, c)
Cl); A.$ 9B)“‘.+ A n+‘l+l(H, A’@&)jn+9+&, it is commutative, because 8 0 (3*(u. b) = gf* 0 S(u . b) = rp’*(&u). b) = q’*(u’. b); by the inductive hypothesis, the bottom line is an isomorphism, and b is an isomorphism; hence the top line is an isomorphism. THEOREM 12 (fate). Let A be a Gmodule, a E H2(G, A). For each prime p let G, be a Sylow psubgroup of G, and assume that (i) H’(G,, A) = 0; (ii) H’(G,,, A) is generated by Res,,cP (a) and has order equal to that of G,. Then for all subgroups H of G and all integers n, cupproduct with Res,,, (a) induces an isomorphism &(H, Z) + A” + ‘(H, A). Proof: TakeB=Z,C=A,q=2,n,=linTheorem11. Fern=1 the surjectivity follows from (i). For n = 0, fi’(G,, Z) is cyclic of order equal to the order of GP, so the bijectivity follows from (iii). For n = 1, the injectivity follows from the fact that H’(G, Z) = Horn (G,,, Z) = 0. Thus all the hypotheses of Theorem 11 are satisfied.
CHAPTER V
Profinite
Groups
K. GRUENBERG 1. The Groups ............... 1.1. Introduction ............. 1.2. Inverse Systems ............ 1.3. Inverse Limits ............. . . 1.4. Topological Characterization of Protinite Groups 1.5. Construction of Prokite Groups from Abstract Groups 1.6. Profinite Groups in Field Theory ....... 2. The Cohomology Theory ........... 2.1. Introduction ............. 2.2. Direct Systems and Direct Limits ....... 2.3. Discrete Modules ............ 2.4. Cohomology of Profinite Groups ....... 2.5. An Example: Generators of propGroups. .... 2.6. Galois Cohomology I: Additive Theory ..... 2.7. Galois Cohomology II: “Hilhert 90” ...... 2.8. Galois Cohomology III: Brauer Groups ..... References ................
. . . . . . . . .
. . . . . . . . . .
. . .
. .
. .
. . .
116 116 116 117 117 119 119 121 121 121 122 122 123 123 124 125 127
1. The Groups 1.1. zntroduction A profinite group is an inverse limit of finite groups. We begin by explaining this definition in some detail. [ For all basic facts concerning inverse limits (and direct limitsthese will be needed later) we refer to Chapter VIII in EilenbergSteenrod : Foundations of Algebraic Topology (ES).] All our topological groups are assumed to have the Hausdorff separation axiom. We recall that a morphism of topological groups means a continuous homomorphism. 1.2. Inverse
Systems
Let Z be a directed set with respect to a relation I. This means that reflexive, transitive and to every il, iz in I, there exists i in Z such that i and i 2 i,. An inverse system of topological groups over Z is an object (I; G’; where, for each i in Z, Gi is a topological group and, for each i I j 116
5 is 2 il xi), in I,
PROFINITE
GROUPS
117
zf is the identity on GI; and if from right to left). We shall often simply write our inverse system as (Gi). Suppose (G!,) (over Z’) is a second inverse system. Let 4: I’ + Z be an order preserving mapping and assume that to each i’ in I’ we are given a morphism +i,: G4u,, + G;. such that, whenever i’ < j’ in I’, n{ is a morphism:
Gj + Gi. Moreover,
i _<j I k, then xi rrj” = nf (read mappings
commutes.
Then we call (4; 4Y, i’ E I’) = @ a morphism
of (Gi) to (Gi.).
1.3. Inverse Limits We shall view finite groups as topological groups with the discrete topology. Let (Gi) be an inverse system offinite groups and form IIG, the Cartesian product, made into a topological group in the usual way by letting the kernels of the projections IIGi f Gi form a subbasis of 1. Let L be the subset of all (xi) in IIGi with the property that whenever i I j, ni(xj) = Xi* Then L is a subgroup of IIGi and we give it the induced topology. We say L is the inverse (or projective) limit of the system (Gi) (or, more loosely, of the groups Gi, i E Z). If all the groups Gi are finite pgroups we call L a propgroup. The notation will be L = lim Gi. Clearly, L is closed in IIGi: for if x 4 L, there exists i I j so that niXj # xr and the set of all elements in IIGi with iterm xi and jterm xi is open and contains x but contains no element of L. If CD: (Gi) + (Gi,) is a morphism of inverse systems, we may define a mapping $ of IIGi into IIG;, as follows: given x in IIGi and any i’ in I’, $(x) is to be the element in IIG$ whose i’term is $Y (x+(~,,). Obviously, @ is a group homomorphism and is continuous because, for each i’, the kernel of IIGi + Gi, is open. By restriction, we obtain a continuous homomorphism of $IJ Gi into lim G!,. 1.4. Topological Characterization of Projinite Groups The main result of this section will not be needed later in these two lectures (but the corollaries will be : the reader may take them on trust). Nevertheless, we give the proof in some detail because it is not too easy to extract from the existing literature.
118
K. GRUENBERG
THEOREM 1. A topological group is profinite totaI[v disconnected.
if, and only if, it is compact and
We recall two facts for which we refer to MontgomeryZippin : Topological Groups (MZ). (i) To say that a compact group is totally disconnected is equivalent to saying that 1 is the meet of all compact open neighbourhoods of 1: MZ, p. 38. (ii) In a compact, totally disconnected group every neighbourhood of 1 contains an open normal subgroup (and hence 1 is the meet of all open normal subgroups): MZ, p. 56. Transformation
Proof. Let (GJ be an inverse system of finite groups and write L = &IJ Gi, C = IIGi. Then C is compact (Tychonoff’s theorem) and hence L is also compact because L is closed in C. We must show next that L is totally disconnected. Consider first C. If xf linC,thenxi# lforsomei. PutUi=li,Uj=G,forj#iand U = IIU,. Then U is compact and open and contains 1 but not x. Hence C is totally disconnected (by (i), above). Now in a compact group, a subset is compact if, and only if, it is closed. Thus in C, 1 is the meet of all open and closed subsets containing 1. It follows that the same is true in L (induced topology) and hence L is totally disconnected. Next assume, conversely, that G is a compact, totally disconnected group. Let (Hi) be the family of all open, normal subgroups of G. Then (G/H,) forms an inverse system (we put i I i whenever Hi 2 Hj and use the natural epimorphism G/Hj f G/H,). Let L = lim G/H,. Since each G/H, is finite, L is profinite. The mapping 8: g + (gH,) is clearly a continuous homomorphism of G into L. Since 1 = n H, (by (ii), above), 0 is oneone. On the other hand, if a = (ai Hi) E L and S = n aI Hi, then S is not empty (by compactness) and so, if g E S, B(g) = a: thus 0 is surjective. Hence 8l is also a mapping; and it is continuous. We conclude that 0 is an isomornhism and thus G is profinite.
The last part of our argument also yields COROLLARY 1. For any profnite
group G,
G g &
G/U,
where U runs through all open, normal subgroups COROLLARY 2.
of G.
If H is a closed subgroup of G, Hr&H/HnU.
ProoJ If V is an open normal subgroup of H, V = 0 n H for some neighbourhood 0 of 1 in G. Hence 0 contains some open normal subgroup Uof G ((ii) above) and so U n H E V. Thus the family (U n H), for varying U,
PROFINITE GROUPS
119
is cofinal in the family of all open normal subgroups of H. The result follows now by Corollary 1 and ES, p. 220, Corollary 3.16. COROLLARY
3. If H is a closednormal subgroup of G, G/H “= l@t~G{UH.
Proof. We have only to show that G/H is profinite. Clearly, G/H is compact. It remains to check that G/H is totally disconnected. Take any x 4 H. For each h in H we may choose an open and compact neighbourhood Oh of h not containing x (because G is totally disconnected). Then H s U 0, and so, by the compactness of H, H c O,,, u . . u O,,, = S, say. Then S is open and compact and contains H but not x. 1.5. Construction of Profinite Groups from Abstract Groups
Let G be an abstract group and suppose (Hi; i E 1) is a family of normal subgroups such that given any Hi,, Hi>, there exists H, E Hi, n Hi,. If we partially order I by defining i I j whenever H, 2 Hi, then I becomes a directed set and (G/H,) is an inverse system of groups ($ being the natural homomorphism G/HI + G/Hi). The mapping 8: g + (gHJ is a homomorphism of G into L = Jinx G/H,. If Go = (I H,, then G/G0 becomes a topological group if we use as basis for open sets at 1 the groups Hi/G, (MZ, p. 25). The homomorphism 8 induces a continuous homomorphism: G/G, + L and L is the completion of G/G,, with respect to (Hi/G,). There are two important cases.
(i) (HJ is the family of all normal subgroups of finite index in G. We denote the profinite group lim G/Hi by G. For example, 2 is the inverse limit of all finite cyclic groups. (ii) (HJ is the family of all normal subgroups of index a power of p, where p is a given prime number. Here lim G/H, = G,, is a propgroup. For example, 2, (usually written Z$ is the inverse limit of all finite cyclic pgroups. This is the group ofpadic integers. Exercise: 2 = HZ,. P
1.6. Profinite Groups in Field Theory Let E/F be a Galois extension of fields. This means that E is algebraic over F and the group G = G(E/F) of all Fautomorphisms of E has no fixed points outside F. [ For simple facts about field theory (including finite Galois theory) we refer to Bourbaki: AZgSbre, Chapter V (B).]
120
K. GRUENBERG
Let (K,, i E Z) be the family of all finite Galois extensions of F contained in E Then E = U Ki (B, Section 10.1). Now (i) if K, G Kj, then we have a natural homomorphism ni: G(Kj/F) + G(K,/F) (B, Section 10.2); and (ii) the Fcomposite of K,, and K,, IO OK, is some Kj. Hence the finite Galois I I groups (G(K,/F)) form an inverse system and we may construct its inverse G(Ki/F) G(KJKi) 7 7”’ limit call it L G(Kj/F) 0 OF 3 . 1 PROPOSITION 1. G(E/F) g L.
Proof For each i in I, we have a homomorphism G(E/F) + G(KJF). These together yield a homomorphism 8: G(E/F) t nG(K,/F). The image of 8 is obviously contained in L. We assert 0 is an isomorphism onto L. If g # 1 in G, there exists x in E such that g(x) # x; and then there exists K, containing x. Now the image of g in G(K,/F) maps x to g(x) and thus is not the identity. Hence i3 is oneone. Take (gJ in L. If x E E and we set g(x) = gi(x), where x E K,, then this is an unambiguous definition of a mapping g of E into E. It is easy to check that g is, in fact, an Fautomorphism of E. Since B(g) = (gi), L is the image 0f 8.
We use the isomorphism 8 to transfer the topology on L to G. Thus G = G(E/F) is now a profnite group and if Vi = G(E/K,), (Vi) is a defining system of neighbourhoods
of 1 (MZ,
pp. 2526).
Example. If E is an algebraic closure of the field F, of p elements, then G(E/F,) r 2 (cf. Section 1.5). THEOREM 2 (Fundamental
theorem of Galois theory). Let E/F be a Galois extension with group G, 9 the set of all closed subgroups of G and 9 the set of aN fields between E and F. Then K f G(E/K) is a oneone mapping of 9 onto 9. The inverse of this is the mapping S + ES, where ES denotes the $eld of fixed points for the closed subgroup S. Proof: First Step. We show that if K E F, then G(E/K) E 9’. Let (Lj; j E .Z) be the family of all finite extensions of F contained in K. Then K = U Lj and thus G(E/K) = n G(E/Lj). Each Lj is contained in some finite Galois extension Kt and hence G(E/Lj) 2 G(E/Ki). By Proposition 1, G(E/K,) is open and therefore G(E/Lj) is also open in G(E/F). Hence G(EIL,) is closed and thus n G(E/LJ = G(E/K) is closed. Second step. If K E % then K = EC@“) (B, Section 10.2).
PROFINITE
Third step.
10 SO ’ TO Go
I
121
GROUPS
Let S be a closed subgroup of G, K = ES and T = G(E/K). Clearly S c T. But by the second step, ES = ET. Hence for every open normal subgroup Vi of T, if OE Li = BY’ LT”‘ = K = Lfvftvl. By finite Galois theory, we conclkde T/V, = SV,/Vi, i.e., T = SV,. Hence S is I dense in T. But S is closed by the first step and so S =I T. YK 10
I OF
vi 0
OE
lo
OL,
OL,
OK
OK
SV‘ 0 To
2. The Cohomology
T/V,0
Theory
2.1. Introduction
In order to define the cohomology of profinite groups we make use of two facts : (i) a profinite group is put together from finite groups ; and (ii) we know how to define the cohomology of finite groups. 2.2. Direct Systems and Direct Limits
We shall only be concerned here with the category of all discrete abelian groups (written additively). We consider a family (AJ of abelian groups, indexed by a directed set I. Assume that for each i 5 j in Z we are given a homomorphism ri : A, + A, and that (i) zf is the identity on Ai, (ii) if i < j pi k, then r; 7; = 7:. We call (I; Ai ; 7:‘) a direct system of abelian groups over I. Frequently we just write (A,). If (A;,) (over Z’) is a second direct system, let $: I + I’ be an order preserving mapping and to each i in 1, let et be a homomorphism: A, + A;,,,. If i I j always implies that
Al
commutes, then we call ($; (4) , (A;+
Jlr
‘A;,‘)
pi) = Y a morphism
of direct systems:
122
K.
GRUENBERG
Given (A J, let S be the disjoint union of the groups A *, i E I. If x E A I, Y E A,, write x  y to mean that there exists k 2 i, k 2 j such that rfx = 7;;~. Then  is an equivalence relation on S. We write lim AI for the set of equivalence classes dnd X for the class containing x. Now A = & Ai is made into an abelian group (the direct limit of the groups A i, i E Z) as follows. If 2, J E A, where x E A i, y E Aj, then find k so that k 2 i and k 2 j and define x” + J to be the class containing T:X + zty; also  2 is to be % . Clearly A is now an abelian group. If Y: (A,) + (A;,) is a morphism of direct systems, we obtain a homorphism of&A i into lint Af. by defining the image of X, where x E A,, to be the class Of It/i(X). 2.3. Discrete Modules Let G be a profinite group and A a (left) Gmodule. If Uis an open subgroup of G, we shall denote, as usual, the set of all fixed elements in A under U by AU. We shall consider only Gmodules A satisfying the condition A = UA”, where the union is taken over all open normal subgroups of G. Such modules are called discrete Gmodules. It is not difficult to see that the following three conditions on the Gmodule A are equivalent: (i) A is a discrete Gmodule; (ii) the stabilizer in G of every module element is an open subgroup of G; (iii) the pairing G x A + A is continuous, where A is viewed as a discrete space and G has its usual topology as a profinite group. 2.4. Cohomology of Projinite Groups Let A be a discrete Gmodule and let (Ui ; i E Z) be the family of all open normal subgroups of G. Then G g lim G/U, (Corollary 1 to Theorem 1) and A E lim A”’ (because A = lJ A”* and U A”’ is naturally isomorphic to )im A”?. Let q be a fixed nonnegative integer. For every i _<j, we obtain a homomorphism (inflation) Ai: H4(G/Ui,
A”‘) f Hq(G/Uj,
A”J)
in the standard way (cf. Chapter IV, Section 4). Clearly, we now have a direct system of abelian groups (I; Hq(G/U,,
Aul); Ai).
DEFINITION. lim Hq(G/Ui , Au’) is called the qth cohomology group of G in ;4 and written Hq(G,A).
PROFINITE
123
GROUPS
There is another way of defining these cohomology groups. We consider the additive group C” = C”(G,A) of all continuous mappings of G” into A and define a coboundary d: C” , Cn+’ by the standard formula: WXg,,
. ..> &+1) = g, f(g2, ***3gn+l)+
This yields a complex C(G,A)
i$l ( 1Yfk19 **3 gigi+
and the homology
***9gn+l)
+ (lY”fkl9 . . ..gJ* groups of it are precisely
Hq(G,A).
Of course, this must be proved. The argument is not difficult but (when written out in full) is both long and tedious. The crucial point is this. A mapping 4 of a profinite group H into a discrete space S is continuous if, and only if, there exists an open normal subgroup K of H and a mapping 1+4of the finite group H/K into S such that 4 is the product of the natural projection H + H/K and $. It follows that if f E C”(G,A), then there is an open normal subgroup U, of G such thatf is G” + (G/U,)”
Sincef is finitely valued, the image offlies If U = U, n U,, then f is
+ A. in AU2, for some open normal
G”+(G/U)“+Av+A andf’: (G/U)” f AU is an element of C”(G/U,AU). plausible that C”(G,A) = lim C”(G/U,A’). 2.5. An Example:
Generators
U,.
This makes it more than
of propGroups
Let G be a propgroup and FP the field ofp elements, viewed as a discrete Gmodule with trivial action. The formula above for df shows immediately that H’(G,F,) = Horn (G,F,). If G* = GP[G,G], then the righthand side is Horn (G/G*, F,). Assume G is finitely generated as a topological group. Then G/G* is finite and its dimension d, as a vector space over Fp, is the dimension of H’(G,F,). Let xlG*, . ., x,,G* be a basis of G/G*. If U is any open normal subgroup of G contained in G*, then xi, . . , x,, generate G module U (by the Burnside basis theorem for finite pgroups) and hence x1, . . , x, generate G topologically. Thus dim p, H’(G,F,) is the minimum number of generators of G. 2.6. Galois CohomoIogy I: Additive Theory
[ Our reference for elementary
Galois cohomology
is Chapter X of Serre:
Corps Locaux (S).]
As in Section 1.6, let E/F be a Galois extension of fields with group G = G(E/F). Denote by (Ki; i E Z) the family of all finite Galois extensions of F contained in E and set Ui = G(E/KJ. Then G = bG/U,.
K. GRUENBERG
124
The action of G on E turns the additive group of E into a Gmodule. Now E”’ = K, and E = u Ki so that E is a discrete Gmodule. Moreover, K, is a W,IF)  module and G(Ki/F) z G/U,. Thus we have Hq(G,E) PROPOSITION 2. Hq(G,E)
g lint Hq(G(K,/F),
K,).
(1)
= Ofir all q r 1.
By equation (1) this is an immediate consequence of Galois extension and G = G(E/F). Then Hq(G,E) = 0 for all q 2 1. Proof. The normal basis theorem states that E, as FGmodule, is free on one generator (B, Section 10.8), i.e., is isomorphic to the induced module on F. Hence the result (cf. Chapter IV, Section 6). LEMMA 1. Let E/F be a finite
Note that this argument yields the following stronger fact: COROLLARY. If E/F is a finite Galois extension, then the Tare cohomology groups fiq(G(E/F), E) = 0 for all integers q. (Recall that A4 = Hq for q 2 1.) 2.7. Galois Cohomology II: “Hilbert 90” We continue with the notation of Section 2.6. We saw that E as a Gmodule turned out to be cohomologically uninteresting. The situation is very different when we look at E* (the multiplicative group of E) as a Gmodule. Again, since (E*)” = KF and E* = U K:, E* is a discrete Gmodule; and Hq(G,E*) z lim Hq(G(K,/F), K;). (2: PROPOSITION 3. H’(G,E*) = 0. Proof. By equation (2), we need only prove this when E/F is finite. Let f be a Icocycle of G in E*. By the theorem on the independence of automorphisms (B, Section 7.5), there exists c such that
b = 1 f(x) . x(c) # 0. xeo Apply y in G to this: y(b) = X;oIyf
WllvxWl = X;of  1 (y)f (yx) . yx(c) (since f Cvx) = f(y) . yf (x))
= f ‘(Y)&f
(4 * z(c)
= f‘(y)b. Thus f (y) = b.y (b)‘, i.e., f is a coboundary.
/
PROFINITE GROUPS
125
COROLLARY (Hilbert Theorem 90). If G = G(E/F) is finite cyclic with generator g and a E E* is such that N&a) = 1, then there exists b in E* such that a = b/g(b). Proof. Since G is cyclic, H’(G, A) = .A/(1  g)A (cf. Chapter IV, Section 8). Here A = E* and so, in multiplicative notation, (1  g)A is (b/g(b), b E E*}. Since Hl(G,E*) = 0, the result follows.
2.8. Galois Let EJF, Suppose and hence U = G(EJj
Cohomology III: Brauer Groups EJF be two Galois extensions and write Gi = G(E,/F). j is an Fhomomorphism: E1 + Ez. Then j(E,)/F is Galois the restriction g + glj(E,) yields a morphism j: G, + G,. Let (El) ). Now inf
Hq(G1, E:) S Hq(GJU,
E;‘)
, Hq(G,,E,*)
and we shall write j* for the product of these two homomorphisms: j*: Hq(G,,E:) We assert j* is independent
E1 O&
+ Hq(G,,E,*).
of j. 7”
j(E,)
P’ 0
U
E G oG, ’ 1 Let j’ be another Fhomomorphism: E1 ) E,, yielding j’ : G, ) G1. Since j’(E,) = j(Ei) (B, Section 6.3), j’ = jg, for some g in G1. Hence (j’)* = j*g*. Now g: E1 f E1 will yield, by the above procedure, a morphism 3: G1 + G1. Clearly g is the inner antomorphism x + g‘xg of G1. Therefore g* is the identity on Hq(G,,EI*). (Cf. Chapter IV, Proposition 3.) Thus (j’)* = j* and hence j* is indeed independent of j. Now suppose E,, Ez are two separable closures of F. Then there always exist Fisomorphisms: E1 + E, and all these yield the same isomorphism Hq(G,,E,*) + Hq(G,,E,*). From the cohomological point of view, it is therefore immaterial which separable closure one uses and we shall simply write Hq(F) for the group Hq(G(E/F),E*), where E is any given separable closure of F. The groups Hq(F) depend functorially on F. DEFINITION. The Brauer group of the field F is the group H’(F). THEOREM 3. If EJF is a Galois extension containing a Galois extension K/F, then we have the exact sequence F
0 + H’(G(K/F),K*)
I o
I OF
+ H’(G(E/F),
E*) + H’(G(E/K),E*).
K. GRUENRERG
126
COROLLARY I. If K/F is a Galois extension, the following sequence is exact:
0 + H’(G(K/F),K*) Proof. Theorem
+ H’(F) + H’(K).
Take E to be any separable closure of F containing 3.
An immediate
K and apply
consequence is the following corollary.
COROLLARY 2. If (KJ is the family of all$nite Galois extensions of F in a separable closure of F, then H’(F) = U H2(G(Kt/F),Kt*).
To prove the theorem, we first translate it into a result about abstract profinite groups. Let G = G(E/F), H = G(E/K) and A = E”. Then, by Galois theory, G/H z G(K/F) and AH = K*. The mapping H’(G,A) + H2(II,A) is restriction and the mapping H2(G/H,AH) + H’(G,A) is inflation. (Restriction and inflation for profinite groups are defined precisely as for abstract groups.) Theorem 3 is now seen as a consequence of the following result, together with Proposition 3. PROPOSITION 4. Let H be a closed normal subgroup of the profinite group G and A be a discrete Gmodule such that H’(H,A) = 0. Then the following sequence is exact: inf res 0 + H’(G/H,An) + H’(G,A) ) H’(H,A).
Outline of proof. The condition H’(H,A) H’(HU~lUi,
A”)
= 0 is equivalent to = 0
for all open normal subgroups Vi of G. Hence, for each Vi, 0 , H’(G/HUI
, AHUt) ) H2(G/Ui,
A”) * H2(HUJUi,
A”)
is exact (Chapter IV, Proposition 5 with q = 1). If i 2 j, we have an exact row for i, one for j and the inflation mappings connecting the two and producing a commutative diagram. All these diagrams together constitute an “exact sequence of direct systems”. The result we want now follows by taking the direct limit and using two facts: (1) lint is an exact functor on the category of direct systems over a fixed indexing set (ES, p. 225); (2) lint H/H n U, r Hand @ G/Hut z G/H (Corollaries 2 and 3 to Theorem 1, above). Alternatively, one may prove the result directly by a method identical to that used for establishing the abstract case.
almost
PROFINITE GROUPS
127
REFERENCES
The basicreferencefor profinite groups is Serre, JP., “Cohomologie Galoisienne”. Springer Vetlag,Berlin (1965). Books referred to in this paper: Bourbaki, N., “Algkbre,” Chapter V Hermann, Paris (B) (1950). Eilenberg, S. and Steenrod, N., “Foundations of Algebraic Topology” (ES). Princeton University Press,Princeton, New Jersey, U.S.A. (1952). Montgomery, D. and Zippin, L., “Topological Transformation Groups” (MZ). Interscience,New York and London (1955). Serre, JP., “Corps Locaux”. Hermann, Paris (S) (1962).
CHAPTER VI
Local Class Field Theory JP. SERRE ................ Introduction 1. The Brauer Group of a Local Field .......... ........... 1.l Statements of Theorems. 1.2 Computation of H2(K,,/K) ........... 1.3 Some Diagrams .............. 1.4 Construction of a Subgroup with Trivial Cohomology 1.5 AnUglyLemma. ............. .............. 1.6 End of Proofs 1.7 An Auxiliary Result ............. Appendix. Division Algebras Over a Local Field ...... 2. Abelian Extensions of Local Fields .......... 2.1 Cohomological Properties ........... 2.2 The Reciprocity Map ............ ...... 2.3 Characterization of (a, L/K) by Characters ........ 2.4 Variations with the Fields Involved 2.5 Unramified Extensions ............ ............. 2.6 Norm Subgroups 2.7 Statement of the Existence Theorem ........ ........ 2.8 Some Characterizations of (a, L/K) 2.9 The Archimedean Case ............ 3. Formal Multiplication in Local Fields ......... ............. 3.1 TheCaseK=Q, 3.2 Formal Groups .............. 3.3 LubinTate Formal Group Laws ......... 3.4 Statements 3.5 Construction of i, and ia], . : : : : : : : 3.6 First Properties of the Extension K, of K. ...... 3.7 The Reciprocity Map ............ 3.8 The Existence Theorem ............ ........ 4. Ramification Subgroups and Conductors ............ 4.1 Ramification Groups 4.2 Abelian Conductors ............. ............. 4.3 Artin’s Conductors 4.4 Global Conductors ............. ............ 4.5 Artin’s Representation 128
.
.
:
:
129 129 129 131 133 . 134 135 136 136 137 139 139 139 140 141 141 142 143 144 145 146 146 147 147 148 : 149 151 152 154 155 155 157 158 159 160
,
LOCAL
CLASS
FIELD
THEORY
129
Introduction We call a field K a local field if it is complete with respect to the topology defined by a discrete valuation v and if its residue field k is finite. We write q = pf = Card (k) and we always assume that the valuation o is normalized; that is, that the homomorphism a: K* + Z is surjective. The structure of such fields is known: 1. If K has characteristic 0, then K is a finite extension of the padic field Qp, the completion of Q with respect to the topology defined by the padic valuation. If [K: Q,] = n then n = ef where f is the residue degree (that is, f = [k : F,,] and e is the ramification index u(p). 2. If K has characteristic p (“the equal characteristic case”), then K is isomorphic to a field k((T)) of formal power series, where T is a uniformizing parameter. The first case is the one which arises in completions of a number field relative to a prime number p. We shall study the Galois groups of extensions of K and would of course like to know the structure of the Galois group G(K,/K) of the separable closure KS of K, since this contains the information about all such extensions. (In the case of characteristic 0, KS = R). We shall content ourselves with the following : 1. The cohomological properties of all galois extensions, whether abelian or not. 2. The determination of the abelian extensions of K, that is, the determination of G modulo its derived group G’. Throughout this Chapter, we shall adhere to the notation already introduced above, together with the following. We denote the ring of integers of K by OR, the multiplicative group of K by K* and the group of units by U,. A similar notation will be used for extensions L of K, and if L is a galois extension, then we denote the Galois group by G(L/K) or GLIR or even by G. If s E G and a EL, then we denote the action of s on a by ‘a or by s(a). In addition to the preceding Chapters, the reader is referred to “Corps Locaux” (Actualites scientifiques et industrielles, 1296; Hermann, Paris, 1962) for some elided details. In what follows theorems etc. in the four sections are numbered independently. 1. The Brauer
Group
of a Local Field
1.1. Statements of Theorems In this first section, we shall state the main results; the proofs of the theorems will extend over $5 1.21.6. We begin by recalling the definition of the Brauer group, Br (K), of K.
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JP. SERRE
(See Chapter V, 5 2.7.) Let L be a finite galois extension of K with Galois group G(L/K). We write H’(L/K) instead of H2(GyK, L*) and we consider the family (LJ,,, of all such finite galois extensions of K. The inductive (direct) limit lii H’(L,/K) is by definition the Brauer group, Br(K), of K. It follows from the definition that Br (K) = H’(K,/K). In order to compute Br (K) we look first at the intermediate field K,,, K c K,,, c KS, where K,, denotes the maximal unramified extension of K. The reader is referred to Chapter I, 5 7 for the properties of K,,,. We recall in particular, that the residue field of K,, is 2, the algebraic closure of k, and that G(K,,/K) = G(k/k). We denote by F the Frobenius element in G(KJK); the effect of F on the residue field k is given by 1 H Aq. The map u H F” is an isomorphism 2 + G(K,,/K) of topological groups. From Chapter V 3 2.5, we recall that 2 is the projective (inverse) limit, lim Z/nZ, of the cyclic groups Z/nZ. Since K,, is a subfield of KS, H2(K,,/K) is a subgroup of Br(K) = H’(K,/K). In fact: THEOREM 1. H’(K,,/K) = Br (K). We have already noted above that H’(K,,/K) = Hz@, K,‘). THEOREM 2. The valuation map v : K,‘r + Z defines an isomorphism H’(K,,/K) ) HZ@, Z). We have to compute Hz@, Z). More generally let G be a profinite group and consider the exact sequence O+Z+Q+Q/Z+O
of Gmodules with trivial action. The module Q has trivial cohomology, since it is uniquely divisible (that is, Zinjective) and so the coboundary 6 : H’(Q/Z) + H’(Z) yields an isomorphism H’(Q/Z) , H’(G, Z). Now H’(Q/Z) = Horn (G, Q/Z) and so Horn (G, Q/Z) E Hz (G, Z). We turn now to Horn (2, Q/Z). Let 4 E Horn (2, Q/Z) and define a n$p y : Horn (2, Q/Z) + Q/Z by 4 H 4(l) E Q/Z. It follows from Theorem 2 that we have isomorphisms H’(K,,/K) : Hz@, Z)‘: Horn (2, Q/Z) L Q/Z. The map inv, : H’(K,,/K) + Q/Z is now defined by invK=yo6loo. For future reference, we state our conclusions in: COROLLARY. The map inv, = y 0 6l Q v defines an isomorphism between the groups H’(K,,/K) and Q/Z. Since, by Theorem 1, H’(K,,/K) = Br (K), we see that Iye have defined an isomorphism inv, : Br (K) + Q/Z.
LOCAL CLASSFIELD
If L is a finite extension by inv,. THEOREM
131
THEORY
of K, the corresponding
map will be denoted
3. Let L/K be a jinite extension of degree n. Then
inv, 0 Res,,, = n . inv,. In other words, the following diagram is commutative
Br(K) ilwR ! QIZ . .
(For the definition of Res,,,, and to Chapter V 3 2.7.) COROLLARY
1.
ResK/L, Br(L) invr. ! n
f
QIZ
the reader is referred to Chapter IV
$4
An element CIE Br (K) gives 0 in Br (L) if and only if
ncc= 0. COROLLARY 2. Let LJK be an extension of degree n. Then H’(L/K) is cyclic of order n. More precisely, H’(L/K) is generated by the element uMK E Br (K), the invariant of which is I/n E Q/Z.
Proof. This follows from the fact that H’(L/K)
is the kernel of Res.
1.2 Computation of H’(K,,/K) In this section we prove Theorem 2. We have to prove that the homomorphism H’(K,,/K) + Hz@, Z) is an isomorphism. PROPOSITION 1. Let K,, be an unramified extension of K of degree n and let G = G(KJK).
Thenfor all q E Z we have:
(1). Hq(G, U,) = 0, where U, = U,“; (2). the map v : Hq(G, K,*) + Hq(G, Z) is an isomorphism.
(Theorem 2 is evidently a consequence of (2) of Proposition H2K,/K) = Hz@, KJ.) Proof. The fact that (1) implies (2) follows from the cohomology
1, since sequence
Hq(G, U,,) ) Hq(G, K,*) + Hq(G, Z) + Hq+‘(G, U,)
It remains to prove (1). Consider the decreasing sequence of open subgroups u, 1 u, =J u,’ 2 . . . defined as follows: x E Ui if and only if v(x 1) > i. Now let rc E K be a uniformizing element; so that Vi = 1 +x*0,,, where 0, = 0,. Then U, = lim U,,/Uj. The proof will now be built up from the three following lemma: LEMMA 1. Let k,, be the residuefield of K,,. Then there are galois isomorphisms lJ,JU, z k,* and, for i > 1, U#Uj’ ’ z k.’ .
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JP. SERRE
(By a galois isomorphism we mean an isomorphism which is compatible with the action of the Galois group on either side.) Proof. Take a E U,, and map a H E where B is the reduction of a into k.. By definition, Uj = 1 +x0,; so if a E U,’ then I = 1, and the first part of the lemma is proved. To prove the second part, take a E Ui and write a = 1 +&I where /3 E 0,. Now map a H 8. We have to show that in this map a product au’ corresponds to the sum p + j?. By definition, au’ = l+n’(/I+jI’)+. . ., whence au’ H p + j’. Finally, the isomorphisms are galois since “a = 1 +n’.“j?. LEMMA 2. For alI integers q andfor
all integers i L 0, Hq(G, Vi/ Ujt ‘) = 0.
Proof. For i = 0, LJt = U,, and the first part of Lemma Hq(G, UJ U.‘) = Hq(G, k:)
1 gives
= Hq(GknIk, k:).
Now for q = 1, H’(Gk,,k, k,*) = 0 (“Hilbert Theorem 90”, cf. Chapter V, $2.6). For q = 2, observe that G is cyclic. Since k.* is finite, the Herbrand quotient h(k,*) = 1 (cf. Chapter IV, $ 8, Prop. 11); hence the result for q = 2. For other values of q the result follows by periodicity. For i > 1, the lemma follows from Lemma 1 and the fact that k.’ has trivial cohomology. The proof of Theorem 2 will be complete if we can go from the groups u;/uy to the group U, itself and the following lemma enables us to do this. LEMMA 3. Let G be a finite group and let M be a Gmodule. Let M’, i > 0 and Ma = M, be a decreasing sequence of Gsubmodules and assume that M = lim M/M’; (more precisely, the map from M to the Iimit is a btjection). Then, gfor
some q E Z, Hq(G, M’JM”
‘) = 0 for all i, we have Hq(G, M) = 0.
Let f be a qcocycle with values in M. Since Hq(G, M/M’) = 0, there exists a (q I)cochain $i of G with values in M such that f = S$, +fi, where fi is a qcocycle in M’. Similarly, there exists lclz such that fl = S$, +f2, fi E M2, and so on. We construct in this way a sequence ($., f,) where tj. is a (q I)cochain with values in M”l and f, is a qcocycle with values in M”, andf, = 6.1,4.+ i +f, + i. Set IJ?= $I + e2 + . . . . In view of the hypotheses on M, this series converges and defines a (ql)cochain of G with values in M. On summing the equations fn = 6$,,+1+f.+ i, we obtain f = SJ/, and this proves the lemma. Proof.
We return now to the proof of Proposition 1. Take M in Lemma 3 to be U,. It follows from Lemma 3 and from Lemma 2 that the cohomology of U, is trivial and this completes the proof of Proposition 1 and so also of Theorem 2.
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THEORY
1.3 Some Diagrams PROPOSITION 2. Let L/K be ajinite extension of degreen and let L,,, (resp. K,,) be the maximal unramljied extension of L (resp. K); so that K,, c L,,. Then the folIowing diagram is commutative.
fwL/w
2%
fw”,/L)
invK invr. I 1 Q/Z 24 Q/Z
Proof. Let lYK = G(K,,/K) and let FK be the Frobemus element of lYK; let rL and FL be defined similarly. We have FL = (FK)f where f = [I: k] is the residue field degree of L/K. Let e be the ramification index of L/K, and consider the diagram:
H2G
K,*,) 22
H2(G,
Z) 6’
I RCS
(1) l
e.Res I
Horn Um
(2)
Q/Z)EL
.?.R.ZS I
(3)
Q/Z “1
H2(L, Z> !.I?+ Horn (r,, Q/Z)yr. Q/Z, fmL9 L,*,) 2L, where Res is induced by the inclusion I?, + JYx, and yK (resp. y,) is given by rp H cp(F,) (resp. cpH cp(F,)). The three squares (I), (2), (3) extracted from that diagram are commutative; for (l), this follows from the fact that vL is equal to e.vK on K,*,; for (3), it follows from FL = FL, and n = efi for (2), it is obvious. On the other hand, the definition of inv, : H’(f’,, K,*,) + Q/Z is equivalent to: inv~=y,~6‘~v,, and similarly : inv, = yL 0 6l 0 VL. Proposition 2 is now clear. COROLLARY 1. Let H’(L/K),, be the subgroup of H2(K,,/K) consisting of those a E H’(K,,/K) which are “killed by L.” (that is, which give 0 in Br (L)). Then H2(L/K),, is cyclic of order n and is generated by the element uLIK in H’(K,,/K) such that inv, (uLlg) = l/n. Proof. H’(L/K),,
Note that a less violent = H’(L/K)
definition
of H2(L/K),I
is provided
by
n H2(K,,/K).
Consider the exact sequence 0 4 H2(L/K),, 4 H2(K,,/K) Relr, H2(L”,/L). The kernel of the map H’(K,,/K) + H’(L,,/L) is H2(L/K),, and this goes to 0 under inv L : H’(L,,,/L) + Q/Z. On the other hand, it follows from Proposition 2 that inv, 0 Res = n. inv,. The kernel of the latter is (l/n)Z/Z and so H’(L/K),, is cyclic of order n, and is generated by u,,, E H’(K,,/K) with inv, (uL,x) = I/n.
JP. SERRE
134
COROLLARY 2. The order of H’(L/K) Proof.
H2(L/K)
is a multiple of n.
contains a cyclic subgroup of order n by Corollary
1.4 Construction
1.
of a Subgroup with Trivial Cohomology
Let L/K be a finite galois extension with Galois group G, where L and K are local fields. According to the discussion in Proposition 1, the Gmodule U, has trivial cohomology when L is unramified. PROPOSITION 3. There exists an open subgroup, cohomology. That is, Hq(G, V) = 0 for all q.
V, of U, with
trivial
Proof. We shall give two proofs; the first one works only in characteristic 0, the second works generally. Method 1. The idea is to compare the multiplicative and the additive groups of L. We know that L+ is a free module over the algebra K[G]. That is, there exists a EL such that [SalssG is a basis for L considered as a vector space over K. Now take the ring 0, of integers of K and define A = cssc OK. “a. This is free over G and so has trivial cohomology. Moreover, by multiplying a by a sufficiently high power of the local uniform&r zx, we may take such an A to be contained in any given neighbourhood of 0. It is a consequence of Lie theory that the additive group of L is locally isomorphic to the multiplicative group. More precisely, the power series ex = 1+x+. . . +x”/n!+. . ., converges for u(x) > V(P)& 1). Thus in the neighbourhood v(x) > v(p)l(p  1) of 0, L* is locally isomorphic to L+ under the map XI+ eX. (Note that, in the same neighbourhood, the inverse mapping is given by log (1 +x) = xx2/2+x3/3. . . .) Now define V = eA; it is clear that V has trivial cohomology. The foregoing argument breaks down in characteristic p; namely at the local isomorphism of L+ and L*. Method 2. We start from an A constructed as above: A = CSEo 0,. “a. We may assume that A c 0,. Since A is open in OL, $OL c A for a suitable N. Set M = z;A. Then M.M c n,M if i > N+l. For M. M = 7ti?q. A c niiOL and if i > N+l then ~~‘0 KL
c X,.&A
c n,M.
Now let V = 1+ M. Then Y is an open subgroup of U,. It remains to be proved that V has trivial cohomology. We define a filtration of V by means of subgroups V’ = 1 +nkM, i Z 0. (Note that V’ is a subgroup since (1 +&x)(1 +&y) = 1 +&(x+y+&xy), etc.) This yields a decreasing filtration V = V” 3 v’z v2 3 . . . . As in 5 1.2, Lemma 2, we are reduced to proving that Hq(G, Vi/Vi+l) = 0 for all q. Take x = 1+&j?, /IEM and associate with this its image fl E M/n,M. This is a group isomorphism
LOCAL CLASS FIELD THEORY
135
of Yi/v’+’ and M/n,M and we know that the latter has trivial cohomology, since it is free over G. This completes our proofs of Proposition 3. We recall the definition of the Herbrand quotient h(M). Namely, (See h(M) = Card (fiO(M))/Card (H’(M)), when both sides are finite. Chapter IV, 5 8.) COROLLARY 1. Let L/K be a cyclic extension of degree n. Then we have h(U,) = 1 and h(L*) = n. Proof. Let V be an open subgroup of U, with trivial cohomology (cf. Prop. 3). Since h is multiplicative, h(U,) = h(V) . h( UJV) = 1. Again, L*JU, z Z. So h(L*) = h(Z). h(UJ. Now h(U,) = 1 and h(Z) = n, since fi’(G, Z) = n and H’(G, Z) is trivial. Hence h(L*) = n. COROLLARY 2. Let L/K be a cyclic extension of degree n. Then H’(L/K) is oforder n = [L: K]. Proof: We have Card (iY’(G, L*)) h(L*) = Card (H’(G, L*))’ Now Corollary 1 gives h(L*) = n. Moreover, H’(G, L*) = 0 (Hilbert Theorem 90). Hence Card (H’(G, L*)) = n. But H’(G, L*) is H’(L/K), whence the result. 1.5 An Ugly Lemma LEMMA 4. Let G be a finite group and let M be a Gmodule and suppose that p, q are integers with p 2 0, q > 0. Assume that: (a) H’(H, M) = 0 for all 0 < i < q and all subgroups H of G; (b) if H c K c G, with H invariant in K and K/H cyclic of prime order, then the order of Hq(H, M) (resp. I?‘(H, M) if q = 0) divides (K: H)P. Then the same is true of G. That is, Hq(G, M) (resp. I?‘(G, M) is of order dividing (G : l)p. Proof. Since the restriction map Res : fiq(G, M) + fiq(Gp, M) is injective on the pprimary components of fiq(G, M), where Gp denotes a Sylow psubgroup of G, we may confine our attention to the case in which G is a pgroup. We now argue by induction on the order of G. Assume that G has order greater than 1. Choose a subgroup H of G which is invariant and of index p. We apply the induction hypothesis to G/H. We know from (b) that, for q > 0, the order of Hq(G/H, MH) divides (G : H)P = pp and by the induction hypothesis Hq(H, M) divides (H: 1)“. Now it follows from (a) that we have an exact sequence (Chapter IV, 5 5). 0 , Hq(G/H, MH) Inf , Hq(G, M) Resf Hq(H, M). Thus Hq(G, M) has order dividing pp. (H : l)p = (G : 1)“.
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JP. SERRE
For g = 0, we recall (see Chapter IV, 8 6) that fiO(G, M) = MG/NGM. Then we have the exact sequence MH/NHM
NG’n, MGINo M +
(Ma)G’HjNo,H MH
where NGIH denotes the norm map and the second map is induced by the identity. The remainder of the argument now runs as before. 1.6 End of Proofs PROPOSITION 4. Let L/K be a finite galois extension with Galois group G of order n = [L : K]. Then H’(L/K) is cyclic of order n and has a generator u,,~ E H’(K,,JK) such that inv, (u,& = l/n.
Proof. In Lemma 4, take M = L*, p = 1 and g = 2. Condition (a) is satisfied by “Theorem 90” and (b) is true by Prop. 3, Cor. 2. Hence H’(G, L*) has order dividing (G: 1) = n. But by Prop. 2, Cor. 1, H’(L/K) contains a cyclic subgroup of order n, generated by u,,, E H’(K,,,/K) and such that inv, (u& = l/n. Whence Proposition 4. It follows from this proposition that H2(L/K) is contained in H’(K,,/K). We turn now to the proof of Theorem 1. The theorem asserts that the inclusion Br (K) 2 H’(K,,/K) is actually equality. Now by definition, Br (K) = u H’(L/K), where L runs through the set of finite galois extensions of K. But as remarked above, H’(L/K) c H’(K,,/K). Hence Br (K) c H’(K,,/K), as was to be proved. Evidently, Theorem 3 follows from Theorem 1 and Proposition 2.
1.7 An Auxiliary Result We have now proved all the statements in 5 1.1 and we conclude the present chapter with a result which has applications to global fields. Let A be an abelian group and let n be an integer > 1. Consider the cyclic group Z/nZ with trivial action on A. We shall denote the corresponding Herbrand quotient by h,(A), whenever it is defined. We have hnG4 =
Order (A/nA) Order .A 
where .A is the set of tl E A such that ncr = 0. (Alternatively, begin with the map A 5 A and take h,(A) to be: order (Coker (n))/order (Ker (n)).)
we could
Now let K be a local field. Then for u E K there is a normalized absolute value, denoted by jalx (see Chapter II, § 11). If a E OK, then lc& = l/Card (O&O,).
LOCALCLASSFIELDTHEORY
137
PROPOSITION 5. Let K be a localfield and let n 2 1 be an integer prime to the characteristic of K. Then h,(K*) = n/l&
Proof. Suppose that K has characteristic 0. We have h,(K*) = h,(Z). h,( U,>. Now h,(Z) = n; so we must compute h,(U,). As in Proposition 3, we con
sider a subgroup V of U which is open and isomorphic to the additive group of 0,. We have h,(U,) = h,(V). h,(U,/V) and since U,/V is finite, UWV) = 1. We have h,(V) = MO,) and h,(O,) = Card (O&OK) = I/[&. Whence
h,(K*) = n. (~/l&J = dl& Suppose now that K has characteristic p. We take the same steps as before. First, h,(K*) = n. h,( U,). Now consider the exact sequence 0 * U; ) U, + k* + 0
where Vi is a propgroup (cf. Lemma 1). Since n is prime to p it follows that h,(Ui) = 1 and that h,(k*) = 1. So n, h,(U,) = n. Whence the result. We note that the statement of the proposition is also correct for R or C. In these cases we have lnlR = Inl, I& = lnl2 and one can check directly that, for R, h,(R*) = n/In1 = 1 and, for C, h,(C*) = n/l& = l/n.
APPENDIX
Division Algebras Over a Local Field It is known that elements of Brauer groups correspond to skew fields (cf., for instance, “Stminaire Cartan”, 1950/51, Exposes 6/7), and we are going to use this correspondence to give a description of skew fields and the corresponding invariants. Most results will be stated without proof. Let K be a local field and let D be a division algebra over K, with centre K and [D : K] = n2. The valuation v of K extends in a unique way from K to D (for example, by extending first to K(a), c1E D, and then fitting the resulting extensions together). The field D is complete with respect to this valuation and, in an obvious notation, 0, is of degree n2 over OK. Let d be the residue field of D; we have n2 = ef where e is the ramification index andf = [d: k]. Now e < n; for there exists a E D such that Q(U) = e ’ and a belongs to a commutative subfield of degree at most n over K, The residue field d is commutative, since k is a finite field, and d = k(z) for some a E D. Hence f < n. Together with n2 = ef, the inequalities e < n and f < n yield e = n andf = n.
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Since [d: k] = n, we can find B E d such that k(E) = d. Now choose a corresponding a E OD and let L = K(a). Evidently [L : K] < n, since L is a commutative subfield of D. On the other hand, cii is an element of I (the residue field of L) and I = d; hence [I: k] = n, It follows that [L : K] = n and L is unramified. We state this last conclusion as: D contains a maximal commutative subfield L which is unramijied over K. The element 6 E Br (K) corresponding to D splits in L, that is 6 E H’(L/K). So any element in Br (K) is split by an unramified extension and we have obtained a new proof of Theorem 1. Description of the Invariant The extension L of K constructed above is not unique, but the SkolemNoether theorem (Bourbaki, “Algebre”, Chap. 8, $ 10) shows that all such extensions are conjugate. The same theorem shows that any automorphism of L is induced by an inner automorphism of D. Hence there exists y E:D such that yLy I = L and the inner automorphism x H yxy’ on L is the Frobenius F. Moreover y is determined, up to multiplication by an element of L”. Let vL be the valuation uL : L* + Z of L; so that a, : D* t (I/n)Z extends uL on D. The image i(D) of u,(y) in (I/n)Z/Z c Q/Z is independent of the choice of y. One can prove that i(D) = inv, (6), where 6 E Br (K) is associated with D. We can express the definition of i(D) in a slightly different way. The map x H y”xy” is equal to F” on L and so is the identity. It follows that y” commutes with L and y” = c E L*. Now 1 UD(Y)= ; u,(f) = f UD(C)=  c(c)* n Hence we have u,(y) = (l/n&(c) = i/n where c = r&u. Application Suppose that K’JK is an extension of degree n. By Theorem 3, Cor. 2, an element 6 E Br (K) is killed by K’. Hence: any extension K’/K of degree n can be embedded in D as a maximal commutative sub$eld. This may be stated more spectacularly as: any irreducible equation of degree n over K can be solved in D. EXERCISE Consider the 2adic field Qz and let H be the quaternion skew field over Q2. Prove that the ring of integers in H consists of the elements a+ bi+cj+dk where a, b, c, d E Z, or a, b, c, d E 3 (mod Z,). Make a list of the seven (up to conjugacy) quadratic subfields of H.
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2.1 Cohomological Properties
Let L/K be a finite galois extension of local fields with Galois group G = G(L/K) of order n. We have seen (9 1.1, Theorem 3, Cor. 2.) that the group H2(L/K) = H’(G, L*) is cyclic of order n and contains a generator u,,, such that inv, (u~,,J = l/n E Q/Z. On the other hand, we know that H’(G, L*) = 0. Now let H be a subgroup of G of order m. Since H is the Galois group of L/K’ for some K’ 1 K, we also have H’(H, L*) = 0 and H’(H, L*) is cyclic of order m and generated by u,,,,. To go further, we need to know more about u,,,,. Now we have the restriction map Res : Br(K) , Br(K’) and this suggests that u,,,, = Res (u&. To see that this is the case, we simply check on invariants. We have invr(Resu,,,)
= [K’ : K] inv, (u&
= [K’ : K] . X = i = inv,, (uLIK, .
We can now apply Tate’s theorem (Chapter IV, 5 10) to obtain: THEOREM 1. For all q E Z, the map ccH u. uLIK given by the cupproduct is an isomorphism of fiq(G, Z) onto l?Q+2(G, L*). A similar statement holds if H is a subgroup of G corresponding to an extension L/K’. The mappings Res and Cor connect the two isomorphisms and we have a more explicit statement in terms of diagrams. The diagrams
STATEMENT.
flq(G, Z> “ux , @+ 2(G, L*) Res Rcs 1 I AyEI,
Z)
“L’n’
, m+2(H,
L*)
“L/X #+2(G, L*) fiq(G, Z> __, COI
C0r T
lP(H,
T Z)
“WC’
l
Aq+2(H,
L*)
are commutative. Proof. As above, uLIK. = Res (u&.
We must show that Res,,,. (uLIR.@) = u,,,, .Res,,, (a). The lefthand side is Res,,,. (u& . Res,,,. (a) (see CartanEilenberg, “Homological Algebra”, Chap. XII, p. 256) and so commutativity with Res is proved. For the second diagram we have to show that Cor (uLIK,. p) = uLIK . Car(P). = u,,, . Cor (p) (CartanEilenberg, Now Cor (uLIK,. /I) = Cor (Res (u,&.P) lot. cit.) and this proves the commutativity of the second diagram. 2.2 The Reciprocity Map
We shall be particularly concerned with the case q = 2 of the foregoing discussion. By definition fi‘(G, Z) is H,(G, Z) and we know that
140 H,(G,
JP. SERRE
On the other hand, fi’(L/K) = K*/NLIICL*, denotes the norm. In this case, Theorem 1 reads as follows.
Z) = G/G’ = Gab.
where N,,,
THEOREM 2. The cupproduct onto K*/NLIKL*.
by uL/K defines an isomorphism
of Gab(L/K)
We give a name to the isomorphism just constructed, or rather to its inverse. Define 8 = t&IK to be the isomorphism of K*/NLIKL* on to Gab, which is inverse to the cupproduct by ULIx. The map 0 is called the local reciprocity map or the norm residue symbol. If a E K* corresponds to 5?E K*/NL,KL*, then we write eL,K(a) = (a, L/K). The norm residue symbol is so named since it tells whether or not a E K+ is a norm from L*. Namely, (a, L/K) = 0 (remember that 0 means 1 !) if and only if a is a norm from L*. Observe that if L/K is abelian, then Gab = G and we have an isomorphism 8 : K*/NL,KL*
) G.
Let L/K be and we seek a % = (a, L/K). degree 1 of G map 6 : H’(G,
a galois extension with group G. We characterization of (a, L/K) E Gab. For Let 1 E Horn (G, Q/Z) = H’(G, Z) and let 6~ E H’(G, Z) be the image of Q/Z) + H2(G, Z) (cf. 9 1.1) Let
2.3 Characterization
E E K*/NLIK(L*)
of (a, L/K)
= &‘(G,
by Characters
start from an a E K* ease of writing we set be a character of x by the coboundary
L*)
be the image of a. The cupproduct Cr.6~ is an element of H2(G, L*) c Br (K). PROPOSITION 1. With the foregoing notation, we have the formula x(sA = invx (z .6x). Proof. By definition s;.u,,, = E E fi’(G, L*), s, being identified with an element of H 2(G, Z). Using the associativity of the cupproduct, this gives ti.6X = u,,,.S,.6X = u,,,.(&.6X) = u,,,.‘+,.X) with S,.X E fl‘(G, Q/Z). Now I?‘(G, Q/Z) !+ fi’(G, Z) = Z/nZ and we identify fi‘(G, Q/Z) with Z/nZ. Moreover, the identification between Hm2(G, Z) and Gab has been so made in order to ensure that s,. x = x(s,) (see “Corps Locaux”, Chap. XI, Annexe pp. 184186). Write s,.x = r/n, r E Z. Then &r/n) E fi’(G, Z) and 6(r/n) = r. Hence u,lK.(s.bx) = r.uLKK and the invariant of this cohomology class is just r/n = x(s,J. So Proposition 1 is proved. As an application we consider the following situation. Consider a tower of galois extensions K c L’ c L with G = G(L/K) and H = G(L/L’). Then, if x’ is a character of (G/H)“b and x is the corresponding character of Gab, and if ccE K* induces s, E G’b and si E (G/Hyb under the natural map s, w s:, we have x(s,) = x’(si). This follows from Prop. 2 and the fact that the inflation map transforms x’ (resp. Sx’) into x (resp. Sx).
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This compatibility allows us to define s, for any abelian extension; in particular, taking L = K”b, the maximal abelian extension of K, we get a homomorphism OK: K* f G(Kub/K) defined by c(H (E, Kab/K). 2.4 Variations with the Fields Involved
Having considered the effect on (a, L/K) of extensions of L we turn now to consider extensions of K. Let K’/K be a separable extension and let Kab, Kfab be the maximal abelian extensions of K, K’ respectively. We look at the first of the diagrams in the Statement of $2.1 and the case q = 2. Taking the projective limit of the groups involved, we obtain a commutative diagram : K* eK f G”K” VI
incl
Here V denotes the transfer (Chapter IV, 5 6), G$ denotes G(K’Ob/K) and Gib denotes G(Kab/K) = Gab(Klab/K). Similarly, using the second of the diagrams in the Statement, we obtain a commutative diagram : K’* “IC + G”,s NIL/K I I 5 K*
eK
l
G”K”
where i is induced by the inclusion of G,, into G,. [Note that if K’/K is an inseparable extension, then in the first of these diagrams the transfer, V, should be replaced by qV where q is the inseparable factor of the degree of the extension K’/K. The second diagram holds even in the inseparable case.] 2.5 UnramiJied Extensions
In this case it is possible to compute the norm residue symbol explicitly in terms of the Frobenius element : 2. Let L/K be an unramified extension of degree n and let be the Frobenius element. Let a E K* and let v(u) E Z be its norF E GLIR malized valuation. Then (a, L/K) = PVC’). Proof. Let x be an element of Horn (GLJK, Q/Z). By Prop. 1, we have: PROPOSITION
x((cc,L/K)) = inv, (Z. 8~).
The map inv,:
Z!Z2(GLIK, L*) + Q/Z has been defined as a composition:
H2(G~,~, L*) 2 H’(GL,K, Z)“: H1(G,m Q/Z>2 Q/Z. We have ~(5.6~) = v(a). 6x, hence : inv, (E. 6x) = y&‘ov(E. 6x) = v(a) .7(x) = v(a)x(F) = x(F”(“)).
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This shows that X((% L/K)) = X(F”‘“‘) for any character x of GLIR; hence (a, L/K) = F”(‘). COROLLARY. Let E/K be a finite abelian extension.
The norm residue symbol K* + GEIR maps U, onto the inertia subgroup T of GEIK. Proof. Let L be the subextension of E corresponding to T. By Prop. 2,
the image of UK in GLIK is trivial; this means that the image of UK in GEIR is contained in T. Conversely, let t E T, and let f = [L : K]; there exists a E K* such that t = (a, E/K). Since t E T, Prop. 2 shows that f divides u,(a); hence, there exists b E E* such that UK(a) = v,(Nb). If we put u = a. Nb‘, we have u E UK and (u, E/K) = (a, E/K) = t. 2.6 Norm Subgroups DEFINITION. A subgroup M of K* is called a norm subgroup ly there exists a finite abelian extension L/K with M = NLIKL*. Example: Let m > 1 be an integer, and let M,,, be the set of elements a E K* with OK(a) E 0 mod m ; it follows from Prop. 2 (or from a direct computation of norms) that M,,, is the norm group of the unramified extension of K of degree m.
Norm
subgroups are closely related to the reciprocity map 0, : K* + Gzb = G(lPb/K) defined in 9 2.3. By construction, 0, is obtained by projective limit from the isomorphisms K*/NL* + GLIK, where L runs through all finite abelian extensions of K. If we put: R = lim.K*/NL*, we see that 13, can be factored into K’XG$ where i is the natural map, and 0 is an isomorphism. Note that R is just the completion of K* with respect to the topology defined by the norm subgroups. This shows that norm subgroups of K* and open subgroups of GKb correspond to each other in a oneone way: if U is an open subgroup of Ggb, with fixed field L, we attach to U the norm subgroup 0; ‘( U) = NLIRL* ; if M is a norm subgroup of K*, we attach to it the adherence of O,(M); the corresponding field L, is then the set of elements in Kab which are invariant by the O,(a), for a E M. We thus get a “Galois correspondence” between norm subgroups and finite abelian extensions; we state it as a proposition: PROPOSITION 3. (a) The map L H NL* is a bijection of the set of finite abelian extensions of K onto the set of norm subgroups of K*.
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(b) This bijection reverses the inclusion. (c) N(L.L’) = NL n NL’ and N(L n L’) = NL.NL’. (d) Any subgroup of K* which contains a norm subgroup is a norm subgroup. (For a direct proof, see “Corps Locaux”, Chap. XI, 5 4.) Nonabelian extensions give the same norm subgroups as the abelian ones : PROPOSITION 4. Let E/K be a Jinite extension, and let L/K be the largest abelian extension contained in E. Then we have: Proof This follows easily from the properties of the norm residue symbol proved in 5 2.4; for more details, see ArtinTate, “Class Field Theory”, pp. 228229, or “Corps Locaux”, p. 180. (These two books give only the case where E/K is separable; the general case reduces to this one by observing that NL = K when L is a purely inseparable extension of K.) COROLLARY (“Limitation theorem”). The index (K* : NE*) divides [E : K]. It is equal to [E : K] if and only if E/K is abelian. Proof This follows from the fact that the index of NL* in K* is equal to [L: K]. 2.7 Statement of the Existence Theorem It gives a characterization of the norm subgroups of K*: THEOREM 3. A subgroup M of K* is a norm subgroup if and only if it satisfies the following two conditions: (1) Its index (K* : M) is$nite. (2) M is open in K*. (Note that, if (1) is satisfied, (2) is equivalent to “ M is closed”.) Proofof necessity. If M = NL*, where L is a finite abelian extension of K, we know that K*IM is isomorphic to G,,,; hence (K* : M) is finite. Moreover, one checks immediately that N: L* + K* is continuous and proper (the inverse image of a compact set is compact); hence M = NL* is closed, cf. Bourbaki, “Top. G&r.“, Chap. I, $10. As remarked above, this shows that M is open. [This last property of the norm subgroups may also be expressed by saying that the reciprocity map tJx:K*+G;b is continuous.] Proof of suficiency. See § 3.8, where we shall deduce it from Lubin Tate’s theory. The usual proof, reproduced for instance in “Corps Locaux”, uses Kummer and ArtinSchreier equations.
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We give now some equivalent formulations. Consider the reciprocity map 0x : K* + Gib. By Prop. 2, the composition K* 2 Ggb+ G(K,,/K) is just the valuation diagram :
map u : K* + Z.
= 2
Hence we have a commutative
04JK+K*+Z+0 04
JO JO lid. I, +G~b&+O,
where I, = G(Kab/K,,) is the inertia subgroup of Ggb, and G(K,,/K) is identified with 2. The map 0 : UK + IK is continuous, and its image is dense (cf. Cor. to Prop. 2); since UK is compact, it follows that it is surjectiue. We can now state two equivalent formulations of the existence theorem. THEOREM 3a. The map 8 : U, + IX is an isomorphism. THEOREM 3b. The topology induced on U, by the norm subgroups is the natural topology of U,.
The group IK is just lim. U,/(M n U,>, where A4 runs through all norm subgroups of K*; the ezvalence of Theorem 3a and Theorem 3b follows from this and a compacity argument. The fact that Theorem 3 * Theorem 3b is clear; the converse is easy, using Prop. 2. COROLLARY. The exact sequence0 + U, + K* + Z + 0 gives by completion the exact sequence:
04J,&+2+0.
Loosely speaking, this means that fz is obtained from K* by “replacing Z by 2”. 2.8 Some Characterizations of (a, L/K)
Let L be an abelian extension of K containing K,,, the maximal unramified extension. We want to give characterizations of the reciprocity map 9 : K* + GLIK. Since K, c L, we have an exact sequence 0 + H + GLIR + 2 + 0 where H = G(L/K,,,) and 2 is identified with G(K,,,/K). Choose a local uniformizer rc in K and write cm = t?(n) = (n, L/K) E GLIK. We know that b, maps onto the Frobenius element FE GK,,,k. Moreover, we can write GLIK as a direct product of subgroups GLIK = H. I, where I, is generated by un. Corresponding to this we have L = K,, @ K,, where K, is the fixed field
of CT,= 0(x). In terms of a diagram, the interrelationship is expressed by
between the fields
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L<
/L\
THEORY
145
‘K /”
\K/
where K,, and K, are linearly disjoint. PROPOSITION
5. Let f: K* + G be a homomorphism and assumethat:
(1) the composition K* : G ) G(K,,/K),
where G + G(K,,,/K) is the natural map, is the valuation map v : K* + Z; (2) for any untformizing element n E K, f(n) is the identity on the corresponding extension K,. Thenf is equal to the reciprocity map 8. Proof. Note that condition (1) can be restated as: for CIE K*, f(m) induces on K,, the power of the Frobenius element, F”@‘. We know that f(n) is F on K,,, and that 0(x) is F on K,,,. On the other hand, f(n) is 1 on K, and e(n) is 1 on K,. Hence f(n) = e(n) on L. Now K* is generated by its uniformizing elements rru (write rc”u as (7m).7r1 ). Hence f = 8. PROPOSITION 6. Let f: K* + G be a homomorphism and assumethat (I) of Proposition 3 holds, whilst (2) is replaced by:
(2’) if ME K*, if K’IK is a finite subextension of L and if u is a norm from K’*, then f(a) is trivial on K’. Then f is equal to the reciprocity map 8. Proof. It suffices to prove that (2’) implies (2). That is, we have to prove that if rc is a uniformizing element, then f(n) is trivial on K,. Let K’/K be a finite subextension of K,. We want to prove that n E NK’*. But e(n) is trivial on K, and so on K’. This implies n E NK’*. 2.9 The Archimedean Case
For global classfield theory it is necessary to extend these results to the (trivial) cases in which K is either R or C. Let G = G(C/R). In the case K = C, the Brauer group is trivial, Br (C) = 0. On the other hand, Br (R) = H’(G, C*) = R*/RP and so Br (R) is of order 2. The invariant inv, : Br (R) + Q/Z has image (0, l/2} in Q/Z and inv, : Br (C) + Q/Z has image (0). The group H’(G, C*) = H’(C/R) is cyclic of order 2 and is generated by u E Br (R) such that inv, (u) = l/2. Under the reciprocity map (or rather its inverse) we have an isomorphism G = H‘(G,
Z) + H’(G, C*) = R*/R*,.
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3. Formal Multiplication in Local Fields The results given in this chapter are due to LubinTate, Annals of Muthematics, 81 (1965), 380387. For our purposes, the main consequences will be: (1) the construction of a cofinal system of abelian extensions of a given local field K; (2) a formula giving (a, L/K) explicitly in such extensions; (3) the Existence Theorem of fj 2.7.
In order to illustrate the ideas involved, we begin with the case K = Q,. The results to be proved were already known in this case (but were not easily obtained) and they will be shown to be trivial consequences of LubinTate theory. 3.1 The Case K = Q, THEOREM 1. Let QT’ be the field generated over Q, by all roots of unity. Then Qiyc’ is the maximal abelian extension of Q,. In order to determine (CI,L/K) it is convenient to split QFc’ into parts. Define Q., to be the field generated over Q, by roots of unity of order prime to p (so Q,, is the maximal unramified extension of Q,) and define QPODto be the field generated over Q, by p”th roots of unity, u = 1, 2,. . . (so QPm is totally ramified). Then Q., and QPm are linearly disjoint and Q;yc'= Q,. Qpm= Q,, 63 Qp. We have a diagram:
\a,’ Now G(Q,,/Q,) = 2 an d ‘f1 rr E G(Qpm/Qp) then c is known by its action on the roots of unity. Let E be the group ofp”th roots of unity, u = 1,2,. . . . As an abelian group, E is isomorphic to le Z/p”Z = Q,/Z,. We shall view E as a Z,module. There is a canonical map Z, + End (E), defined in an obvious way and this map is an isomorphism. The action of the Galois group on E defines a homomorphism G(Qpm/Qp) + Aut (E) = U,, and it is known that this is an isomorphism. (See Chapter III, and “Corps Locaux”, Chap. IX, § 4, and Chap. XIV, § 7.) If u E Up, we shalldenote by [u] the corresponding automorphism of Qpm/Q,. THEOREM 2. If a = p” . u where u E Up, then (a, QF”/Q,) = a, is described by: (1) on Qnr,a, induces the nth power of the Frobenius automorphism; (2) on Qp, a, induces the automorphism [ul].
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Of these (1) is trivial and has already been proved in § 2.5, Prop. 2. The assertion (2) can be proved by (a) global methods, or (b) hard local methods (Dwork), or (c) LubinTate theory (see § 3.4, Theorem 3). Remark. Assertion (2) of Theorem 2 is equivalent to the following: if w is a primitive p”th root of unity and if u E Up then b”(W) = WYl = 1 +
XD,
where w = 1+x. 3.2 Formal Groups
The main game will be plicative group law F(X, binomial expansion. The variables and we begin by
played with something which replaces the multiY) = X+ Y+XY and something instead of the group law will be a formal power series in two studying such group laws. Let A be a commutative ring with 1 and let FE A[[X, Y]].
DEFINITION. We say that F is a commutative formal group law $(4 FGK F(K 2))
= JfSK
Y), 2);
(b) F(0, Y) = Y and F(X, 0) = X; (c) there is a unique G(X) such that F(X, G(X)) (4 W, Y) = I;(K W; (e) F(X, Y) 3 X+ Y (mod deg 2).
= 0;
(In fact one can show that (c) and (e) are consequences of (a), (b) and (d). Here, two formal power series are said to be congruent (mod deg n) if and only if they coincide in terms of degree strictly less than n. Take A = Ox. Let F(X, Y) be a commutative formal group law defined over Ox and let mK be the maximal ideal of OK. If x, y E mx then F(x, y) converges and its sum x*y belongs to Ox. Under this composition law, m, is a group which we denote by F(Q). The same argument applies to an extension L/K and the maximal ideal mL in 0,. We then obtain a group F(ntJ defined for any algebraic extension of K by passage to the inductive limit from the finite case. If F(X, Y) = X+ Y+ XY then we recover the multiplicative group law of l+m,. The elements of finite order of F(m,,) form a torsion group and G(KJK) operates on this group. The structure of this Galois module presents an interesting problem which up to now has been solved only in special cases. 3.3 LubinTate FormaI Group Laws Let K be a local field, q = Card (k) and choose a uniformizing
ITCOp Let & be the set of formal (1) f(X) s aX (mod. deg. 2); (2) f(X) E XQ (mod. n).
power series f with:
element
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(Two power series are said to be congruent (mod. 7~) if and only if each coefficient of their difference is divisible by 71. So the second condition means that if we go to the residue field and denote by j(X) the corresponding element of k[[X]] thenf (X) = X4.) Examples. (a) f(x)
= 72x+ xq;
(b) K =
Qp,
The following of Prop. 5.
‘II = P,
; 0
(X) = px +
four propositions
XZ+...+pXP‘+XP.
will be proved in 5 5 as consequences
PROPOSITION 1. Letfcz 5,. Then there exists a unique formal group law F; with coeficients in A for which f is an endomorphism. (This means f(F/(X, Y)) = FJf(X), f( Y)), that is f OFf = F, 0 (f xf).) PROPOSITION 2. Let f E 5, and Ff the corresponding group law of Prop. 1. Then for any a E A = 0, there exists a unique [a]/ E A[[X]] such that:
(1) [aIf commutes with f, (2) [a]/ = aX (mod. deg. 2). Moreover, [a], is then an endomorphismof the group law Ff.
From Prop. 2 we obtain a mapping For example, consider the case K =
Qp,
then F is the multiplicative
f=pX+
A + End (Ff) defined by a H [aIf. ; 0
law X+ Y+XY,
[alf=(l+X)“l=f
xz+...+xp;
and ‘: X’. i=l 0 z
PROPOSITION 3. The map a H [a]/ is an injective homomorphism of the ring A into the ring End (F/>. PROPOSITION 4. Let f and g be members of 3,. group laws are isomorphic.
Then the corresponding
3.4 Statements
Let K be a local field and let z be a uniformizing element. Let f E 5, and let I;; be the corresponding group law (of Prop. 1). We denote by Mf = Ff(mKS) the group of points in the separable closure equipped with the group law deduced from F. Let a E A, x E M/ and put ax = [al/x. By Prop. 3, this defines a structure of an Amodule on M/. Let E, be the torsion submodule of MI; that is the set of elements of M/ killed by a power of rc.
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THEOREM 3. The following statements hold.
(a) The torsion submodule EJ is isomorphic (as an Amodule) with K/A. (b) Let K, = K(E/) be the jieId generated by E/ over K. Then K, is an abelian extension of K. (c) Let u be a unit in K*. Then the element o, = (u, K,/K) of G(K,/K) acts on EY via [ul]/. (d) The operation described in (c) defines an isomorphism U, + G(KJK). (e) The norm residue symbol (71,KJK) is 1. (f) The fields K,,, and K, are linearly disjoint and Kab = K,,, .K,.
We may express the results of Theorem 3 as follows. Pab Id .,\
We have a diagram:
\ ,Kz
\K/ Here G(K,,,/K) = $ and G(K,/K) = U,. Moreover every u E K* can be written in the form c( = rc”.u and on gives o (the Frobenius) on K,,/K whilst o, gives [u ‘1 on KJK. Example. Take K = QP, rc = p and
f =
pX +
:
X2+...+XP.
The
0 group law; E, is the set of p”th roots
formal group law is the multiplicative of unity; K, is the field denoted by Qpm in 9 3.1and and 2.
we recover Theorems 1
3.5 Construction of Ff, [a],
In this section we shall construct the formal group law Ff and the map
a i+[al/. PROPOSITION 5. Let f, g E &, let n be an integer and let 41(Xl,. . ., X,) be a linear form in Xl,. . ., X, with coeficients in A. Then there exists a unique 4 E A[[X,, . . ., X,]] such that:
(a) 4 = r$r (mod deg 2); (b)f,$=4.(gx...xg). Remarks. (1) The property (b) may be written ’ * 3dX”N. f(ddXl, * * * 9 X”>> = 4MXl),. (2) The completeness of A will not be used in the proof. Moreover, the proof shows that 4 is the only power series with coefficients in an extension of A, which is torsion free as an Amodule, satisfying (a) and (b). Proof. We shall construct I$ by successive approximations. More precisely, we construct a sequence ($‘p’) such that 4(P) E A[[X,, . . . , X,]], #p) satisfies (a) and (b) (mod deg p+ l), and c#@) is unique (mod deg pf 1).
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JP.
SERRE
We shall then define 4 = lim 4(P) and this will be the 4 whose existence is asserted. We take #‘I = 4i. Suppose that the approximation 4r + . . . +4, = #p) has been constructed. That is, f 0 4(P) z 4(P) 0 (g x . . . x g) (mod deg p + 1). For convenience of writing, we shall replace g x . . . x g by the single variable g. Now write $(P+‘) = #p)+ 4,+ 1. Then we may write fo c##‘) E c$(p) o g+E p+l (mod deg p+2), (“the error”) satisfies Ep+ 1 E 0 (mod degp+
where E,,, $Jp+l); we have f. ~#&p+l) =fo (t#@)+~$,+~) =fo
~$(p)+n~~+~
1).
Consider
(mod degp+2)
(the derivative off at the origin is z) and 4(P) 0 g++p+l
0 g = 4(P) 0 g+“p+14p+l
(mod deg p +2).
Thus f. @‘+l)~(p+l)
(mod deg p+2).
o g s Ep+l+(n~~+l)~p+l
These equations show that we must take 4 P+l = E,+,/~(l~3. The unicity is now clear and it remains to show that 4,+i has coefficients in A. That is, E,,, = 0 (mod n). Now for 4 E F,[[X]J, we have r#&#?) = (4(X))q and together with&Q E X4 (mod rr) this gives f 0 ~#(~)4(~)
.fe
(cjcp)(X))4#P)(X4)
= 0
(mod it). and the proof is completed
So, given 4(p) we can construct a unique #p+l) by induction and passage to the limit. Proof of Proposition 1. For each f E $J,, let F/(X, Y) be the unique solution of F/(X, Y) 3 X+ Y (mod deg 2) and f 0 F/ = F, 0 (f x f) whose existence is assured by Prop. 5. That F, is a formal group law now requires the verification of the rules (a) to (e) above. But this is an exercise in the application of Prop. 5: in each case we check that the leftand the righthand sides are solutions to a problem of the type discussed there and we use the unicity statement of Prop. 5. For example, to prove associativity note that both F/(F,(X, Y), 2) and F/(X, F,( Y, Z)) are solutions of H(X,Y,Z)=X+Y+Z
(moddeg2)
and fwcn Proof of Proposition
f(Y), f(Z)) = f(H(X, r, Z)). 2. For each u E A and f, g E 5, let [u],,Q)
unique solution of
C&X0
= UT
(mod deg 2)
be the
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THEORY
and
SC+, ,ux = C4/, &mh (that is fo [a]/,e Now we have
= bl,, 0id. WAe k$ = blf,f.
~f<M/, em Cd/, goN= ld/,,wx9
Y)).
For each side is congruent to aX+a Y (mod deg 2) and if we replace X by g(X) and Y by g(Y) in either side, then the result is the same as if we substitute the sides in question in $ Thus [a],,@ is a formal homomorphism of Fg into Fp If we take g = f, this shows that the [+‘s are endomorphisms of F,. Proof of Proposition 3. In the same way as outlined above, one proves that and
I3 + blf, IJ= Ff om/.,x
CKIf.3
Wlf,h = b1f.e oC%* It follows from this that the composition of two homomorphisms of the type just established is reflected in the product of corresponding elements of A. Taking f = g, we see that the map a H [a]/ is a ring homomorphism of A into End (E/>. It is injective because the term of degree 1 of [a]/ is aX. Proof of Proposition 4. If a is a unit in A, then [a]/,, is invertible (cf. the proof of Prop. 2) and so FB r Fi by means of the isomorphism [a],,. Note that [z]/ = f and [l]/ is the identity (proved as before). This completes the proofs of the propositions 1, 2, 3, 4. 3.6 First Properties of the Extension K, of K From now on, we confine our attention to subfields of a fixed separable closure K’ of K. Given f E &, let F/ be the corresponding formal group law and let EJ be the torsion submodule of the Amodule F’(m,). Let E; be the kernel of [n”],; so that E, = v Ej’. Let K,” = K(E;) and K, = u K:. If Gn,n denotes the Galois group of K(E/3 over K, then G(K,/K) = lim G,,“. PRO~OSI~O% 6. (a) The Amodule E/ is isomorphic to K/A; (b) the natural homomorphism G(KJK) + Aut (EI) is an isomorphism. Proof. We are free to choose f as we please, since, by Prop. 4, different choices give isomorphic group laws. We take f = .zX+ X4. Then a E EJ if and only if f cm)(a)= 0, where f (“) denotes the composition f. . . . Of n times; that is f (“) = [z”]~ If a E tttK, then the equation rcX+ X4 = a is separable and so solvable in KS, its solution belonging indeed to Q,. This shows that iUJ is divisible.
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JP. SERRE
’
Hence EJ is divisible also. This already implies that E, is a direct sum of modules isomorphic to K/A. Let us consider the submodule E; of Ef consisting (see above) of those a E M, such that [~]~a = 0. The submodule Ej is isomorphic with A/mKS, since it is an Amodule with q elements. This is enough to show that Ef is isomorphic to K/A. An automorphism u E G(K,/K) induces an automorphism of the Amodule E,. But since Ef 2 K/A and End, (K/A) = A this gives a map G(K,JK) + Aut (E,) = U,. This map is injective by the definition of K, and it remains to be proved that it is surjective. Take n 2 1 and define E; and Ki as above. We have an injection G(Kz/K) + UK/U{, where Ug = 1 +d’A. Let ME ET be a primitive element; that is an element of E; such that [n”lfa = 0, but [x”‘]~cx # 0. Finally, we define C#Ias follows: 4 =p/p1) =f(f’““)/f’“1’. Nowf
= X4+71X; sof/X
= Xq‘+n.
Hence
fP ‘9 which is of degree qn~qf~“” = (f’“lYoql+n~ n1 and which is irreducible, since it is an Eisenstein polynomial. All primitive elements CI are roots of 4. Thus the order of G(K,“/K) is at least (q  l)q“ ‘. On the other hand, this is actually the order of the group U,lUL. Hence G(K”/K) = U,llJ;. It follows that G(K,/K)
= lim G(K:/K) 
= lim U&J: 
= UK,
and this completes the proof of Prop. 6. The same proof also yields: COROLLARY. The element TTis a norm from K(a) = K:. Proof. The polynomial C/Jconstructed above is manic and ends with 71. Hence iV(cc) = 71. 3.7 The Reciprocity Map
We shall study the compositum L = K,,,K%of K,,, and K,, and the symbol We need to compare two uniformizing elements rr and
(a, L/K), CIE K”. co = nu,uE U,.
Let R,, be the completion of K,, (remember: K,,, is an increasing of complete fields but is not itself complete) and denote by A^,, the integers of I?,,. By definition i?,, is complete; it has an algebraically residue field and rc is a uniformizing parameter in I?,,. We take and g E 3,.
union ring of closed f E 5.
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THEORY
153
LEMMA 1. Let a E G(KJK) be the Frobenius automorphism and extend it to I?,,, by continuity. Then there exists a power series 4 E A,,[[X]] with d(X) s EX (mod deg 2) and E a unit, such that
(4 “4 = 4 0 blf; (b) 4oF/ = Fg4W; (c) f#~o [a], = [a], O4 for all a E A. Proof. Since cr 1 is surjective on & and on firm, (cf. “Corps Locaux”, p. 209), there exists a 4 E &[[X]] such that d(X) E EX (mod deg 2) where E is a unit and “4 = I$ Q[u]~. This is proved by successive approximation and we refer the reader to LubinTate for the details. This particular 4 does not necessarily give (b) and (c) but can be adjusted to do so; the computations are given in LubinTate (where they appear as (17) and (18) in Lemma 2 on p. 385). Note that together the above conditions express the fact that $J is an Amodule isomorphism of I;; into F#. Computation of the norm reciprocity map in L/K. Let L, = K,, . K,. Since K,,, and K, are linearly disjoint over K, the Galois group G(LJK) is the product of the Galois groups G(KJK) and G&,/K). For each uniformizing element rc E A we define a homomorphism r, : K* f G(L,/K) such that: (a) r,(x) is 1 on K, and is the Frobenius automorphism o on K,,; (b) for u E UK, m(u) is equal to [uI]~ on K, and is 1 on K,,. We want to prove that the field L, and the homomorphism r, are independent of rr. Let w = nu be a second uniformizing element. First, L, = L,. For by Lemma 1, Ff and FB are isomorphic over R,,. Hence, the fields generated by their division points are the same. So R,,. K, = R,,. K,. On taking completions we find that Kz = Kz. In order to deduce that K,, . K, = K,, . K, from this, we require the following: LEMMA 2. Let E be any algebraic extension (finite or infinite) of a local field and let CIE l?. Th en, if ccis separable algebraic over E, a belongs to E.
Proof. Let ES be the separable closure of E and let E’ be the adherence of E in ES. We can view a as an element of E’. Hence it is enough to show that E’ = E. Let s E G(E,/E). Since s is continuous and is the identity on E, it is also the identity on E’. Hence G(EJE) = G(E,/E’) and by Galois theory we have E’ = E. It follows from Lemma 2 that L, = L, and so L, = L is independent of n. We turn now to the homomorphism r,: K* + G(L/K). We shall show
154
JP.
SEW
I
r&o) = rJo). This will imply that m(o) is independent of n and so the r,‘s coincide on the local uniformizers. Since these generate K*, the result will follow. We look first at rD(w). On K,,,, ~~(0) is the Frobenius automorphism Q. On K, it is 1. On the other hand, r%(w) is c on K,,; so we must look at T,(O) on K,. Now K, = K(E,), where g E &,. Let 4 E J[[X]] be as in Lemma 1; 4 determines an isomorphism of Ef onto E,,. So if 3, E E,, then we can write Iz = 4(p) with p E Ep We look at r,(0)3, and we want to show that this is 1. As already remarked, rJcl))(l) = r,(o)+@). Write s = m(w). We want to show that “A = rl; that is “40 = &). Now, r,(o) = m(x). r%(u) and the effects of m(n) and r&) are described in (a) and (b) above. Since 4 has coefficients in R,,, “4 = “4 = 4 0 [u]/ by (a) of Lemma 1. But thar
“(4Q> = w/4 = “dau‘I&N. Hence
wPl= 4 0Cd, 0b‘l/O
= 4w
So r, is the identity on K, and it follows that r, is independent of 71. Thus r : K* + G(L/K) is the reciprocity map 8 (5 2.8, Prop. 5). All assertions of Theorem 3 have now been proved except the equality L = Kab, which we are now going to prove. 3.8 The Existence Theorem Let Kab be the maximal abelian extension of K; it contains K,,. The Existence Theorem is equivalent to the following assertion (9 2.3, Theorem 3a). If Ix = G(K”b/K,,) is the inertia subgroup of G(Kub/K), then the reciprocity map 0 : U, + ZK is an isomorphism. Let L be the compositum K,. K, and let ZL = G(LIK,,) be the inertia subgroup of G(L/K). Consider the maps 6 e tJ,+Z,+I; where 0 is the reciprocity map and e is the canonical map ZK + ZL. Both 0 and e are surjections. On the other hand, the composition e 0 0 : UK ) Ix has just been computed. If we identify Z& with Ux it is u H ul. Hence the composed map e 0 8 is an isomorphism. It follows that both 8 and e are isomorphisms. As we have already noted, the first isomorphism is equivalent to the Existence Theorem. The second means that L = Kab, since both L and K”” contain K,,. [Alternative Proof. Let us prove directly that every open subgroup M of K*, which is of finite index, is a norm subgroup corresponding to a
j
LOCALCLASSFIELDTHEORY
155
finite subextension of L. This will prove both the existence theorem (3 2.7, Theorem 3) and the fact that L = K“‘. Since A4 is open, there exists n > 1 such that Uz c 44; since it4 is of finite index, there exists m > 1 such that rP E M; hence M contains the subgroup V,,, generated by U,” and n;“‘. Now let K,,, be the unramified extension of K of degree m, and consider the subfield L,,,, = Ki. K,,, of L. If u E UK, and a E Z, we know that (ux”, L&K) is equal to [ul] on Ki and to the ath power of the Frobenius element on K,,,; hence (an”, L,,,/K) is trivial if and only if u E U< and a 3 0 mod m, i.e. if and only if urc’ E V,,.,. This shows that V.,, = NL,,,, and, since M contains V,,,, M is the norm group of a subextension of L,,,, Q.E.D.] 4. Ramification
Subgroups and Conductors 4.1 Ram$cation Groups Let L/K be a galois extension of local fields with Galois group G(L/K). We recall briefly the definition of the upper numbering of the ramification groups. (For details, the reader should consult Chapter I, 9 9, or “Corps Locaux”, Chap. IV.) Let the function ic: G(L/K) + {Z u co} be defined as follows. For s E G(L/K), let x be a generator of OL as an OR algebra and put i&s) = v~(s(x)x). Now define GU for all positive real numbers u by: s E G, if and only if i,Js) > u+ 1. The groups G, are called the ramification groups of G(L/K) (or of L/K). In order to deal with the quotient groups, it is necessary to introduce a second enumeration of the ramification groups called the “upper numbering”. This new numbering is given by G” = G,, where u = 4(u) and where the function I$ is characterized by the properties: (a> 4(O) = 0; (b) 4 is continuous; (c) 4 is piecewise linear; (d) 4’(u) = l/(G, : GJ when u is not an integer. The G”‘s so defined are compatible with passage to the quotient: (G/H) is the image of G” in G/H (“Herbrand’s theorem”). This allows one to define the G”‘s even for infinite extensions. On the other hand, we have a filtration on UK defined by Uz = 1 +m$. We extend this filtration to real exponents by Ul = Ufl if n 1 < u < n. (It should be noted that v in this context is a real number and is not to be confused with the valuation map!) THEOREM 1. Let L/K be an abelian extension with Galois group G. Then the local reciprocity map 9 : K* + G maps U,V onto G” for all v 2 0. Proof. (1) VeriJication for the extensions K,” of 5 3.6. Let u E Vi with i < n and u 4 Uk”. Let s = 0(u) E G(c/K). We have
156
JP.
SERRE
i(s) = +&L2),
where I is a uniformizing element. We choose a primitive root a for 1; that is, an a satisfying [?]/a = 0 but [n”‘lfa # 0. Observe that s,(a) = [u ‘If a and u r = l+n’v (see 8 3.3, Theorem 3), where v is a unit. These imply that s,ct = [l +7civ]fa = F&t, [n’v]p). Jf we write p = [nivlfcr, then /J is a primitive (n  i)th root (that is, [n*‘lz/I = 0, [Yli]f/I # 0), and we have
for some yij E OK. Accordingly, S,(O()c( = P + C and Now a is a uniformizing
Yija’Pj
element in K,” whilst /? is a uniformizing
Figure
element
1.
in K”,’ and K;/KEsi is totally ramified. Its degree is ql. So we have determined the i function of 19(u); namely, if UE U’ but u 4 Uifl, then i@(u)) = qi. This says that if q’’  1 < u < qi 1 then the ramification group G, is e( U&). We turn now to the upper numbering of the G,‘s. That is, we define a function 4 = &F/R, corresponding to the extension K;, which satisfies the conditions (a) to (d) above. Namely,
LOCAL CLASS FIELD THEORY
157
Then G” = GU with v = 4(u). The graph of d(u) is shown in Figure 1. Ifq’‘1 < u < q’l,then@(u) = l/(q’q’l)and(U,: Ui)=q’q’‘. So if i 1 < v < i, then G” = 0(&) for o < n. The general case (2) Verzjication in the general case. Having proved Theorem 1 for K; it follows for K, = u K,” by taking projective limits. Hence also for K,. K,,, since both extensions have the same intertia subgroup. Since K, . K,, is the maximal abelian extension, the result is true in general. This concludes the proof of Theorem 1. COROLLARY. The jumps in the jiltration (G”} of G occur only for integral values of v. Proof. This follows from Theorem 1, since it is trivial for filtrations of U, and Theorem 1 transforms one into the other. [This result is in fact true for any field which is complete with respect to a discrete valuation and which has perfect residue field (theorem of HasseArf), cf. “Corps Locaux”, Chap. IV, V.] 4.2 Abelian Conductors Let L/K be a finite extension and let 8 : K* , G(L/K) be the corresponding reciprocity map. There is a smallest number n such that &Vi) = 0. This number n is called the conductor of the extension LJK and is denoted by fWK)PROPOSITION 1. Let c be the largest integer such that the ramification group G, is not trivial. Then f(L/K) = $LIR(~)+ 1. Proof. This is a trivial consequence of Theorem 1 and the fact that the upper numbering is obtained by applying 4. Now let L/K be an arbitrary galois extension. Let x : G + C * be a onedimensional character and let L, be the subfield of L corresponding to Ker (x). The field Lx is a cyclic extension of K and f(LJK) is called the conductor of x and is denoted by f(x). PROPOSITION 2. Let {G,} be the ramzjication subgroups of G = G(L/K) and write gr = Card (Gi). Then
(XI> =izo EClX(Gi)) where x(GJ = g; 1 c x(s) is the “mean value” of x on G,. SGG,
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JP. SERRE
Proof. We have x(GJ = 1 if x is trivial on Gi (that is, equal to 1 everywhere) and x(G,) = 0 if x is nontrivial on Gi. Hence (the reader is referred to “Corps Locaux”, Chaps. IV and VI for the details)
where cX is the largest number such that the restriction x]Gfz # 1. Now f(x) = f(L,/K) is equal to &,,x(c) + 1, where c is defined as m Prop. 1 for the extension LJK. Since &,x is transitive, it suffices to show that c = ~L,Lz(cX) and this is a consequence of Herbrand’s theorem (§ 4.1). 4.3 Artin’s Conductors Let L/K be a finite galois extension with Galois group G = G(L/K). Let x be a character of G (that is, an integral combination of irreducible characters). Artin defined the conductor of x as the number for> = igo E (X(l)  X(GJ)* If x is irreducible of degree 1, this j’(x) coincides with the previous f(x). We define Artin’s character a, as follows. For s E G, set ifs#l u&s) = f. iG(S) adl)
= f,J
GAS).
Here f is the residue degree [I: k] (not to be confused with the conductor!), and io is the function defined above. PROPOSITION 3. Let g = Card (G). Then f(x) = (% xl = ~s~ox(s)aGo. Proof. GrGi+l
The proof depends on summation on successive differences and is left as an exercise. (See “Corps Locaux”, Chap. VI, 9 2.) PROPOSITION 4. (a) Let K c L’ c L be a tower of galois extensions, let x’ be a character of G(L’/K) and let x be the corresponding character of G(L/K).
nenfol)
= f&9.
(b) Let K c K’ c L and let JI be a character of G(L/K’) the corresponding induced character of G(L/K). Then
and let II/* be
m*> = w . %@K~,K) +fx*,R: x4% where fKpIKis the residue degree of K’IK and bKpIK is the discriminant of K’/K. Proof. The proof depends on properties of the io function and on the relation between the different and the discriminant, and can be found in “Corps Locaux”, Chap. VI.
159
LOCAL CLASSFELDTHEORY THEOREM 2 (Artin). Let x be the character f(x> is a positive integer.
of [I representation
Proof. Let x be the character of the rational It follows from representation theory that
representation
of G. Then
M of G.
x(1) = dimM and x(GJ = dim MO’. Thus in
each term is positive (2 0) and so f 01) 2 0. It remains to be proved that f(& is an integer. According to a theorem of Brauer, x can be written x = c m i e: where ml E 2 and $: is induced by a character $r of degree 1 of a subgroup H, of G. Hence, since f($:> = ICII(l)vK(bRpIK) +fKpIK .f($J, f($:> is an integer provided that f($J is. But since $r has degree 1, f($J may be interpreted This proves as an abelian conductor and so is obviously an integer. Theorem 2. 4.4 Global Conductors Let L/K be a finite galois extension of number fields and let G = G(L/K) be the Galois group. If x is a character of G, then we define an ideal f(x) of K, the conductor of x, as follows. Let p be a prime ideal in K and choose a prime ideal !J3 in L which divides p. Let Gp = G(L.s,/KJ be the corresponding decomposition subgroup. Let f(x, p) be the Artin conductor of the restriction of x to G+, as defined above. We have f(x, p) = 0 when p is unramified. The ideal f(x) = g pf’(X*P) is called the (global) conductor of x. In this notation, Prop. 4 gives: PROPOSITION5. Let K’/K be a subextension of L/K. Let 9 be a character of H = G(L/K’) and let @” be the induced character of G(L/K). Then
fob*) = e:Ii. hr,dfW where bRnIIL is the discriminant of K’IK. We apply Prop. 5 to the case $ = 1 and we denote the induced character #* by sc,, (it corresponds to the permutation representation of G/H). Since f($) = (1) we obtain: COROLLARY. We have f(sc,a, L/K) = bK.,*
160
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In the case H = 1 we have s,,, = ro, the character of the regular representation of G, and the corollary reads bL,K = n fw”‘,
n
where x runs through the set of irreducible characters of G. This is the “Fiihrerdiskriminantenproduktformel” of Artin and Hasse, which was first proved by analytical methods (L functions). In the abelian case it reads : b L/K =x JI,xx~* In the quadratic case it reduces to the fact that the discriminant to the conductor. 4.5 Artin’s
is equal
Representation
We return to the local case. THEOREM 3. Let L/K be aJinite galois extension of localjelds with Galois group G. Let a, be the Artin character of G dejined above (cJ 3 4.3). Then ao is the character of a complex linear representation of G called “the” Artin representation.
Proof. The character a, takes the same values on conjugate elements and so is a class function. It follows that ao is a combination c, m,x, with complex coefficients mX, of the irreducible characters x. Since
mx = caG9 x) =fo1), we know (Prop. 3 and Theorem 2) that m, is a positive integer. Hence the result. Now let V, be an irreducible representation corresponding to x. We can define Artin’s representation Ao by: AC = ~fol) v,, where the summation is over all irreducible characters x. Remark. This construction of Ao is rather artificial. Weil has posed the problem of finding a “natural” Ao. THEOREM 4. Let I be a prime number not equal to the residue characteristic. Then the Artin representation can be realized over Q,.
Proof. See JP. Serre, Annals of Mathematics, 12 (1960), 406420, or “Introduction a la theorie de Brauer” Uminaire I.H.E.S., 1965166. Examples exist where the Artin representation cannot be realized over Q, R or Qp, where p is the residue characteristic. This suggests that there is no trivial definition of the Artin representation. Assume now that L/K is totally ramified. Let u, = r, 1; we have
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161
Q(S) = 1 ifs # 1 and U&Y) = Card (G)1 ifs = 1. Now a, = u,+b, where b, is a character of some representation. Note: a, = uo if and only if L/K is tamely ramified. So b, is a measure of how wild the ramification is. THEOREM 5. Let I be a prime number not equal to the residue characteristic. Then ‘there exists a finitely generated, projective Z,[G] module Bo,, with character b, and this module is unique up to isomorphism. Proof. This follows from a theorem of Swan, “Topology”, 2 (1963), Theorem 5, combined with Theorem 4 above and the remark that b,(s) = 0 when the order of s is divisible by 1. [See also the I.H.E.S. seminar quoted above.] For applications of Theorem 5 to the construction of invariants of finite Gmodules, see M. Raynaud, “S&m. Bourbaki”, 1964/65, expose 286. These invariants play an important role in the functional equation of the zeta functions of curves.
CHAPTER VII
Global
Class Field Theory J. T. TATE
1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. List
Action of the Galois Group on Primes and Completions . Frobenius Automorphisms . . . . . . . . . . Artin’s Reciprocity Law Chevalley’s Interpretation b; Ideles 1 : 1 : : : : Statement of the Main Theorems on Abelian Extensions . Relation between Global and Local Artin Maps . . . . Cohomology of Ideles . . . . . . . . . . . Cohomology of Idele Classes (I), The First Inequality . . Cohomology of Idele Classes (II), The Second Inequality . Proof of the Reciprocity Law . Cohomology of Idele Classes (III), ‘The Fundamental Class’ Proof of the Existence Theorem .’ . . . . . . . of symbols . . . . . . . . . . . . . . .
. 163 . . 164 . . 165 . . 169 . . 172 . . 174 . . 176 . . 177 . . 180 . . 187 . . 193 . . 201 . . 203
.
Throughout these lectures, K will be a global field, as has been defined in Chapter II, § 12. We treat the number field case completely, but in the function field case there is one big gap in our proofs, in that the second inequality and the accompanying key lemma for the existence theorem are proved only for extensions of degree prime to the characteristic. (The reader interested in filling the gap can consult the ArtinTate notes,? pp. 2938.) As in Local Class Field theory,,there are several aspe : (1) The cohomology theory of Galois extensi s of K. (2) The determinat’ To n of the abelian analysis. extensions of K. (3) Gserie oqd We will discuss the first two, leaving the third to Heilbronn (Chapter VIII), except for a few remarks. Sections l6 constitute a statement and discussion of the reciprocity law and the main theorems on abelian extensions, with no mention of cohomology. We hope that this preliminary discussion will serve both as orientation and bait for the reader. In sections 712 we give the main proofs, based on the determination of the Galois cohomology of idele classes and the Brauer group of K. This chapter is strictly limited to the central theorems. In the exercises at the end of the book the reader will find a few concrete examples and further t Harvard, Dept. of Mathematics. 1961. 162
GLOBAL CLASSFIELD
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THEORY
results. There are some references to recent literature scattered in the text but we have made no attempt to give a systematic bibliography. A list of symbols used in this chapter is given at the end. 1. Action of the Galois Group on Primes and Completions Let L be a finite Galois extension field of Kwith Galois group G = G(L/K). 1.1. First of all we have a few lines on our notation and language. If a E L and a E G, then the action of a on a will be denoted by aa or a*, according to the situation. If r E G we use the convention a(zu) = (ar)a and so (a’)” = &Jr). A prime is an equivalence class of valuations, or a normalized valuation, of K; we usually denote a prime by the letter u or w. A prime may be either archimedean or discrete; if v is discrete we write 0, for its valuation ring and !& for the maximal ideal of 0,. We reserve the symbol ‘$3 for prime ideals.. Let w be a prime of L, then with the definition ]&,, = ]a‘& it follows that aw is another prime of L and a(rw) = (ar)w. If Dw is the valuation ring of w, then an,,, = Daw. A Cauchy sequence for w, acted on by a, gives a Cauchy sequence for aw and conversely a Cauchy sequence for aw, acted on by ai, gives a Cauchy sequence for w; so 0 induces by continuity an isomorphism a,,, : L, ‘$I L,, of the completions of L with respect to the primes w and aw respectively. If w is over the prime o of K so is aw and this map is a K,isomorphism. Clearly, a, o r,,, = (ar),. The decomposition group G, of w is the subgroup G,={aoGIaw=w}
,
of G. Note that G,, = {a E Glarw = rw> = rG,r’ ,y” _(1) , “\ thus the decomposition group of w is determined p to conjugacy by the’\ prime v. By what we have said a is a K,aut 9ff orphism of L,,, if a E G, and so we have an injection i of G, into G(L,/K,). 1.2. PROPOSITION. (i) LJK, is Galois and the injection i : G, + G(L,/K,) is an isomorphism. (ii) If w and w’ are two primes of L over the prime v of K, there exists a a E G such that aw = w’. Proof. Letting [X] denote the cardinality of a set X, we have
CC,.1G W,/~,) G CL,: &I, and these inequalities are equalities if and only if (i) is true. Let and let (a,), 1 < i < r, be a system of representatives for the of G, in G. Put w1 = aiw for 1 < i < r.. These are distinct lying over v; let w1 for r+ 1 < i < s be the remaining such, if
r = (G : G,) cosets alGw primes of L any. Then
164 [G] = r[Gw] = i~l[G,,J
J. T. TATE
<&CL,,.,
: KJ G i&[L‘
: KU] = [L : K] = cc]1
so we have equality throughout. Hence r = s, which implies (ii) and [G,] = [L, : K,], which implies (i). [The fact that the sum of the local degrees is equal to the global degree follows from the bijectivity of the map L 63 K, + fi L,, on taking dimensions
over K,; see Chapter II 5 10.
The surjzctivity,‘w&ch is all we have used, is an easy consequence of the weak approximation theorem.] Write !U& for the set of primes of K. Then, since %XL + ‘9JIK is surjective (every prime D of K can be extended to a prime w of L), Proposition 1.2 amounts to saying that 5?& N 9&/G, i.e. the primes of K are in 1l correspondence with the orbits under G of the primes of L, and for each prime w of L, its stabilizer G, is isomorphic to the Galois group of the corresponding local field extension L,/K,. 2. Frobenius Automorphisms 2.1. Suppose that w is a discrete, unramified prime of L over the prime v of K. (This is true of “almost all” primes V, w, i.e. for all but finitely many.) Then G = G, N WXG) = W(wYWN, (1) where k(v) (resp. k(w)) denotes the residue class field of K (resp. L) with respect to 2)(resp. w). Since these residue class fields are finite the Galois is cyclic with a canonical generator, group G(k(~)/&)) F: XHXNU,  ~~where NV = [k(v)] is the “absolute norm”. Henee we see from (1) that t@ere is a unique element trw E G, which is characterized by the.property ew E G, and saw = !aN” (mod ‘p,) ow is called the Frobenius automorphism for all a E D,. This automorphism associated with the prime w. An immediate consequence of this definition is 2.2. PROPOSITION a,, = z1o,T.
Thus the Frobenius automorphism is determined by o up to conjugacy and we define F&U) = (conjugacy class of ew, w over v) = (the set of a,‘s for w over u). If S is a finite set of primes of K containing the archimedean primes and the primes ramified in the extension L/K, then FLIK is a map of 1111, S into the conjugacy classes of G(L/K).
GLOBAL CLASS FIELD THEORY
165
2.3. PROPOSITION. Let (T E F&U) have order f, so that it generates the subgroup
Reciprocity
Law
3.1. First of all we give some notation. S will usually denote a finite set of primes of K including all the archimedean primes. If we are considering a particular finite extension L/K, then S will also include the primes of K
,/ I ’
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ramified in L. We will denote by Is the free abelian group on the elements of %RxS (a subgroup of the group of ideals, see Chapter II § 17). Assume now that L/K is a finite abelian .extension. Then the comugacy classes of G = G(L/K) are single elements and so l$x is a map from !32x  S into G. By linearity we can extend this to a homomorphism (to be denoted by FLIR also) of 1’ into G, putting where the n,, are integers and n, = 0 except for a finite number of the V. The first proposition of this section concerns the change in the map FLlr when the fields are changed. Suppose that L'/K' and L/K are abelian field extensions with Galois group G’ and G respectively, such that L’ 3 L and K' 3 K, and let 8 be the natural map G’ + G (every automorphism of L'IKI induces one of L/K). Let S denote a finite set of primes of K including the archimedean ones and those primes ramified in L' and let S’ be the set of primes of K' above those in S. Then
3.2. PROPOSITION. The diagram IS’ 
h*/r*
G’
commutes, where N denotes “norm”. Proof.
By linearity,
it is clear that it is enough to check that
~&~,x:‘V~ = FL,K(&,R ~‘1 for an arbitrary prime u’ of K' such that u’ 4 S’. Let NK~,xuu) = fu, where u is the prime of K below u’; thus f = [k(u’) : k(u)]. Let c’ = FLp,Rr(u')and o = F&u). We must show &a’) = 0’. Now Q and c1 are determined by their effect on the residue fields. Let w’ be a prime of L’ above u’ and let w be the prime of L below w’. For x E k(w) c k(w’) we have f’ = XNu’ = JNv)f = a/ , as required. If a E K* (i.e. a is a nonzero element of K), then we write W = J&
us
where n, = u(a) for all u 4 S; thus (a)” is an element of Is. We can now state the reciprocity law in its crudest form 3.3. RECIPROCITY LAW (Crude form). If LJK is a finite abelian extension, and S is the set of primes of K consisting of the archimedean ones and those
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THEORY
ramified in L, then there exists E > 0 such that if a E K* and [a 11, < e for aN I) E S, then F((a)‘) = 1. In words, if a E K* is sufficiently near to 1 at all primes in a large enough set S, then F((a)s) = gsF(u)“@) = 1. In the number field case the subgroup (K:)” is open in K* for all n > 0, and we claim that the condition ]a I], < E can be replaced by a E (K:) for v E S, where n = [L : K]. Indeed if the latter is satisfied then, by the weak approximation theorem, there exists b E K* such that lab“ 11, < e for all v E S, and then F((a)S) = F((b”ab“)s)
= F((b)S)“F((ab“)S)
= 1.
Thus, in the case of number fields, although the set S depends on L, the neighbourhoods of 1 at the primes of S depend only on the degree n of L over K. In particular, for archimedean primes there is no condition needed unless v is real and n is even, in which case the condition a > 0 in K, is sufficient. Using the approximation theorem in L instead of K, one can replace the condition “a is a local [L : K]th power in S” by “a is a local norm from L to Kin S”, but for that we shall use the technique of idtles (see 4.4 and 6.4 below). The shift in emphasis from nth powers to norms was decisive, and is due to Hilbert. 3.4. Example. The reciprocity law for cyclotomic extensions. This reciprocity law may be verified directly in the cyclotdmic case k = Q, L = Q(n, where [ is a primitive mth root of unity. This particular result will be used later in one of our proofs of the general result (see 5 10, below) so we give some details. In the rational case it is conventional to denote primes by p and the associated valuations by up. The set Swill consist of the archimedean prime and those primes p which divide m (see Chapter III). For p 4 S and w above p, the powers of C have distinct images in the residue class field k(w). so we have PROPOSI~ON.
F(v,)C = cpfor all p $ S. From this we deduce COROLLARY.
If a E Z, a > 0 and (a, m) = 1, then F((a)sil: = c.
Consequently, all p E S, then a a = b/c with (b, so F((b)‘)I; = [* that‘ F((a)‘) = 1.
if a is a positive rational number with ]a 11, < ]m], for is a padic integer for all p dividing m, and we can write m) = (c, m) = 1, and b z c (mod m); hence lb,= c’ and = Cc = F((c)‘)C. By linearity this gives F((a)‘)C = f so
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For another example of an explicit description of FLIR, and for the connection between Artin’s general reciprocity law and the classical quadratic reciprocity law, see exercise 1. 3.5. Remark. The cyclotomic case was easy because one can use the roots of unity to “keep book” on the effect of the F(u)‘s for variable u. A similar direct proof works for abelian extensions of complex quadratic fields using division points and modular invariants of elliptic curves with complex multiplication, instead of roots of unity. In the general case no such proof is known (cf. the 12th problem of Hilbert), although Shimura and Taniyama and Weil have made a great contribution, using abelian varieties instead of elliptic curves. (See ShimuraTaniyama, “Complex Multiplication of Abelian Varieties and its Applications to Number Theory”, Publ. Math. Sot. Japan, No. 6, 1961, and more recently Shimura, “On the Field of Definition for a Field of Automorphic Functions: II”, Annals of Math, 81 (1965), 124165.) The proof of the reciprocity law in the general case is very indirect, and can fairly be described as showing that the law holds “because it could not be otherwise”. 3.6. Remark. In the function field case, as Lang has shown ((‘Sur les series L dune variete algebrique”, RUN. Sot. Math. Fr. 84 (1956), 385407), the reciprocity law relates to a geometric theorem about the field K = k(C) of a
curve C. Serre has carried out in detail the program initiated book, “Groupes algebriques et corps de classes”, Hermann, analogue of the reciprocity law is described as follows. Let rational map of a nonsingular curve C into a commutative G; let S be the finite set of points of C where f is not regular. a homomorphism of the group of divisors Is into G and
by Lang. In his Paris (1959) an f : C ) G be a algebraic group Then f induces
THEOREM. If #J E K takes the value 1 to a high order at each point of~S,
then
f((+)) = 1.
This theorem is due to Rosenlicht, and independently, It was Serre and Lang who applied it to class field theory.
but later, Serre.
Let K be a global field, S be a finite set of primes of K including all the archimedean ones and G a commutative topological group. A homomorphism 4 : Is + G is said to be admissible if for each neighbourhood N of the identity element 1 of G there exists E > 0 such that &(a)“) E N whenever a E K* and la 1 Iv < E for all v E S. If G is a discrete group, we simply take N to be (1). Thus 3.7. Definition.
3.8. Reformulation
of the Reciprocity
law: FLIK is admissible.
In this context the finite group G = G(L/K) is discrete. If G is the circle group, then 4 is admissible if and only if it is a Grijssencharakter (if it maps into a finite subgroup, it is a Dirichlet character). Dirichlet and Hecke
GLOBAL CLASS FIELD THEORY
formed produce Lseries Lseries, x of the
169
their Lseries with such characters; Artin was originally forced to his reciprocity law in order to show that in the abelian case his defined in terms of characters of the Galois group were really Weber in other words that #(u)) was admissible for each linear character abelian Galois group.
4. Chevalley’s
Interpretation
by Id&es
The set of elements of the idele group JR (see Chapter II 5 16) which have the value 1 at all the vth components, v E S, is denoted by J,“. If x E JR it has a nonunit component at only a finite number of vcomponents; if the (additive) valuation of the vth component X, of x is n, E Z we write (x)S =“~sn”u
E IS.
4.1. PROPOSITION. Let K and S be as before, G be a complete commutative topoIogica1group and 4 an admissiblehomomorphism of Is into G. Then there exists a unique homomorphism$ of JR + G such that
(i) $ is continuous; (ii) +(K*) = 1 ; (iii) I&) = 4((x)“) for all x EJ& Conversely, if $ is a continuous homomorphism of JR + G such that $(K*) = 1, then J/ comesfrom some admissiblepair S, 4 as defined above, provided there exists a neighbourhood of 1 in G in which (1) is the only subgroup. Remark. It is clear that if such a J/ exists then it induces a continuous homomorphism of the idele class group C, N J,/K* into G. This induced
homomorphism will also be denoted by I,+. Furthermore, if such a 1,4exists for a given 4 and S, then by the unicity statement, it is unchanged if S is enlarged to a bigger set S’ and 4 replaced by its restriction 4’ to Is’ c Is. Similarly, two 4’s on Is which coincide on I” for some finite S’ 3 S are actually equal on Is (cf. Exercise 7). For applications G can be thought of as a discrete group or the circle group. Proof. Suppose we have an admissible map 4 : Is + G. If such a $ were to exist, for any a E K* and x E JK we would have ti(x> = tW4
= 4V@x>dJ/((~&),
where (ax)l is the idele with the same vcomponent as ux for all v E S and value 1 elsewhere and (a~)~ is the idele with the same ucomponent as ax for all v 4 S and whose vcomponent is 1 at all v E S (thus (ux)~ E JE). By the (weak) approximation theorem (see Chapter II, 96) we can find a sequence
170
J. T. TATB
(4) of elements a, E K*, such that a, + xl, tw
= :;
as n + co, at all o E S. Then
4w(a, x)1). Ma, 47 = ii+: md>.
Hence given I$ we define a function t+Qby (1)
Hx) = Lt
44% 4”).
As n, m + co we have a,Ja,,, + 1 at all primes v E S, and consequently S
4((%4S) f$((a,x)s)
l
=
4
:
>>
+
l
in G, because C$is admissible. Thus the limit exists, since G is complete, and the limit is independent of the sequence {a,}, because it exists for all such sequences. Moreover $ is continuous. If the components of x are units at primes Y # S, then we have $(x) = lim &(u.)~); and if in addition the components of x are sufficiently close to 1 at primes u E S, then so will be those of a, for large n, and by admissibility, 4((u,)“) will be close to 1 in G. The last two conditions (ii) and (iii) are trivially verified by taking a, = xl and 1 for all n respectively. Now suppose we are given a continuous homomorphism 1,9: JR + G such that J/(K*) = 1. We will find a set S so that (a) the restriction of tj to JR comes from a function on 1’ and (b) if we call this function 4, then C$is admissible. For any finite set S of primes of K let Us be the set of idbles in JK for which the ucomponent is 1 at all v E S and a unit of K, for u 4: S. By taking S arbitrarily large we can make Us an arbitrarily small neighbourhood of the identity of JR. If N is a neighbourhood of (1) we can choose S sufficiently large so that I,+(Us) C N, since 1,4is continuous. Then taking iV small enough, we see that $(U’) = (1) for some set S by the “nosmallsubgroup” hypothesis. We choose such a set S. Now Ji/U’ is canonically isomorphic to 1’ and so II/, when restricted to J,“, induces a continuous homomorphism 4 of Is into G. It remains to verify that 4 is admissible; in words, given a neighbourhood iV, then ~((a)‘) E N whenever a E K* is near enough to 1 at all o E S. But in this case (a>” is near to a in JR and so by continuity tj((a)‘) is near to +(a), which is 1 since a E K*. 4.2. COROLLARY. The reciprocity law holds for a finite ubelian extension L of K fund only if there exists a continuous homomorphism 1,4of JR + G(L/K) such that (i) 1+4is continuous. (ii) J/(K*) = 1. (iii> 444 = &~K(W) f or aN x E J,“, where S consists of the archimedean primes of K and those ramified in L.
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Such a map $ = rtQLIKwhose existence we have just postulated is called the Artin map associated with the extension L/K. It has been defined as a map JR + G(L/K); but since it acts trivially on K* it may be viewed as a map of the idble class group C, = J,/K* into G(L/K). The reciprocity law for finite abelian extensions will be proved later (see 5 10). In the meantime certain propositions will be proved and remarks made which depend on its validity. Suppose that L’IK’ and L/K are abelian field extensions with Galois groups G’ and G respectively and that L’ 3 L, K’ I> K. Let 8 be the natural map G’ + G. Then in terms of idtles and Artin maps Proposition 3@becomes 4.3.
PROPOSITION.
If the rec$rocity
JK
law holds for L/K and L’IK’, !h/K’  G’
then
is a commutative diagram. Proof. Let S be a large finite set of primes of K, and S’ the set of primes of K’ above S. We have then a diagram
The nonrectangular parallelograms are commutative by the compatibility of ideal and idele norms, and by Proposition 3.2. The triangles are commutative by (4.2)(iii). Thus the rectangle is commutative, i.e. the restrictions of JI L/K 0 NIYJK and 8 0 JIL.,x. to JR‘1 coincide. But those two homomorphisms take the value 1 on principal ideles by 4.2 (ii), so they coincide on (K’)*J$, which is a dense subset of JK’ by the weak approximation theorem (Chapter II, 5 6). Since the two homomorphisms are continuous, they coincide on all of J,* which is what we wished to prove. For proving Proposition 6.2 below, which is in turn needed for the first of our two proofs of the reciprocity law in $10.4, we need the following VARIANT. Suppose L/K satisfies the reciprocity law, and K c iU c L. Then rl’,,rW~,r
Jd = W/M)
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J. T. TATE
Consider diagram (2) with L’ = L, K’ = M, but with the upper horizontal arrow $L,,KK’ = @L,M removed. It shows that
~,KWM,RJ%) = G’ = W/W. Consequently the same is true with Js replaced by M*J& set is dense in JM we are done. 4.4. COROLLARY. If the reciprocity
law holdsfor
~,RWL,RJL)
and since that
L/K, then
= 1.
It follows that I,~~,~(K*~V~,~J~) = 1; the next theorem states (among other things) that K*NLIKJL is the kernel of $L,R. 5. Statement of the Main Theorems on Abelian Extensions 5.1. MAIN THEOREM ON ABELIAN EXTENSIONS (TakagiArtin). (A) Every abelian extension L/K satisfies the reciprocity law (i.e. there is an Artin map $&. (B) The Artin map I+?~,~ is surjective with kernel K*NLIK(JL) induces an isomorphism of CK/NLIK(CL) on to G(L/K). (C) IfM 2 L r> K are abelian extensions, then the diagram
W%,K j,
CM 
SMIK
and hence
G(M/K) ,e
commutes (where 0 is the usual map and j is the natural surjective map which exists because NMIKCM c NLIKCL). (D) (Existence Theorem.) For every open subgroup N of finite index in C, there exists a unique abelian extension L/K (in a fixed algebraic closure of K) such that NLIKCL = N. The subgroups N of(D) are called Norm groups, and the abelian extension L such that NLIRCL = N is called the class field belonging to N. In the
number field case every open subgroup of C, is of finite index in C,. 5.2. A certain amount of this theorem may be deduced readily from the rest. First, given (A) and (B), then (C) is a special case of 4.3 (put K’ = K and L’ = A4). 5.3. Secondly, the uniqueness, though not the existence, of the correspondence given in (D) follows from the rest. Given the existence, let L and L’ be two finite abelian extensions of K in a fixed algebraic closure of K and let M be the compositum of L and L’ (which is again a finite abelian extension of K). Now consider the commutative diagram above, under (C). Since the horizontal arrows are isomorphisms (by (B)) we see that Ker 0 = G(M/L) is the iso
GLOBAL
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THEORY
morphic image, under I++~,~, of the group NLIKCL/NM,K CM. Thus L, as the fixed field of the group Ker 0, is uniquely determined as a subfield of M, by NL,,C,. Applying the same reasoning with L replaced by L’, we see that if NLfIRe CL, = NLIKCL, then L = L’. For some special examples of class fields (Hilbert class fields) see exercise 3. For the functorial properties of the Artin map when the ground field K is changed, see 11.5 below. 5.4. The commutative diagram of (C) allows us to pass to the inverse limit (see Chapter III, Ql), as L runs over all finite abelian extensions of K. We obtain a homomorphism t,bK: CK + lim G&/K) II G(Kab/K), .L where Kab is the maximal abelian extension of K; and then, by (D), G(Kab/K) N lim (&IN), ‘N where the limit is taken over all open subgroups N of finite index in C,. Thus we know the Galois groups of all abelian extensions of K from a knowledge of the idele class group of K. The nature of the homomorphism $R : C, + G(Kab/K) is somewhat different in the function field and number field cases. The facts, which are not hard to derive from the main theorem, but whose proofs we omit, are as follows: 5.5. Function Field Case. Here the map eg is injective and its image is the dense subgroup of G(Kab/K) consisting of those automorphisms whose restriction to the algebraic closure & of the field of constants k is simply an integer power of the Frobenius automorphism F; (see ArtinTate notes, p. 76). 5.6. Number Theory Case. Here $K is surjective and its kernel is the connected component D, of C,. So we have obtained a canonical isomorphism CK/DK N G(Ksb/K).
However, as Weil has stressed (“Sur la theorie du corps de classes”, J. Math. Sot. Japan, 3, 1951), we really want a Galoistheoretic interpretation of the whole of C,. The connected component DK can be very complicated
(see ArtinTate,
p. 8.2).
5.7. Example.
extension of Q. is isomorphic
Cyclotomic Fields. Consider Q/Q, the maximal cyclotomic Let $ = lim Z/nZ; by the Chinese remainder theorem this
to fl Z,, whire Z, is the ring of padic integers.
2 acts on
any abelian torsioi group (for ZlnZ operates on any abelian group whose exponent divides n) and the invertible elements of 2 are those in n UP, P
where UP is the set ofpadic
units in Z,.
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J. T. TATE
Now consider the torsion group ,u consisting of all roots of unity. If [ E ,Y we can define r” for all u E n UP; u induces an automorphism on ,u. The idble group JQ is isomorphic
ti the direct product Q* x R*, x fl UP. (In
fact, if x = (xoo, x2, x3,. . . } E JQ we have x = a. {t, u2, us,. . . }, khere a = (sign x,) n pup(+) E Q*, and where t > 0, and up E UP forp = G, 3,. . . ; moreover, this decomposition is unique, because 1 is the only positive rational number which is a padir unit for all primes p.) Hence Co is canonically isomorphic to RT x n UP, so there is a map of Co onto n UP, which is the Galois group of the ma$mal P
cyclotomic extension. What in fact happens is the following. If x E Co and x H u by this map, then c@(X)= p’ (this re suIt is an easy exercise, starting from 3.4, and is independent of parts (B) and (D) of the main theorem). Thus the kernel of e is RT, which is the connected component D, of Co. We have now used up the whole of C,/D,; so if we grant part (B) of the main theorem, we see that every abelian extension of Q must already have appeared as a subfield of Q”“, and that part (D) holds for abelian extensions of Q. The connected component R*, of Co is uninteresting; similarly, C, has an uninteresting connected component when K is complex quadratic, essentially because there is only one archimedean prime. It may well be that it is the connected component that prevents a simple proof of the reciprocity law in the general case. 6. Relation Between Global and Local Artin Maps We continue to deduce results on the assumption that the reciprocity law (but not necessarily the whole main theorem of 3 5) is true for an abelian extension L/K. 6.1. For each prime u of K, we let K, denote the completion of K at 1). If L/K is a finite Galois extension, then the various completions L, with w over v are isomorphic. It is convenient to write L” for “any one of the completions L,,, for w over u”, and we write G” = G(L”/K,) for the local Galois group, which we can identify with a decomposition subgroup of G (see 1.2). In the abelian case this subgroup is unique, i.e. independent of the choice of w. Assume that L/K is abelian and that there is an Artin map J/L,K : Jg + G(L/K) = G. For each prime v of K we have k=a
Kc&J, JU
G,
GLOBALCLASSFIELDTHEORY
175
where i,, is the mapping of an x E Kc onto the element of Jg whose v component is x and whose other components are 1, andj, is the projection onto the vth component. Call $, = $=,k 0 i,,; so 9, : Kt + G. In fact 6.2. PROPOSITION. If K, c &! c L”, then J/u(N4,&*) c G(L”/d). In particular, JI,(Kz) c e, and JIu(NLv/K,(L”)*) = 1. Proof. Let A4 = L n JX be the tied field of G(L”/&) in L, so that G(L/iU) is identified with G(L’/d) under our identification of the decomposition group with the local Galois group. Then 4 = M,,,, where w is a prime above v, and the diagram L “:&IT
is commutative.
7
JfM,K
JK K” By the ‘variant” of 4.3 we conclude that
VL(N,,K~~*) = $L,K%,K = W/M) = G(LDl4, 6.3. We shall call I/I”: KC: + G” the local Artin homomorphism, or by its classical name: norm residue homomorphism. If x = (x,) E Jg, then we have
and consequently, by continuity,
we have
(this product is actually finite since if x, is a vunit and v is not ramified, then it is a norm of L”/K,). Thus knowledge of all the local Artin maps $, is equivalent to knowledge of the global Artin map eLIR. Classically, the local maps Ic/, were studied via the global theory and, in particular, were shown to depend only on the local extension L”/K,, and not on the global extension L/K from which they were derived. Nowadays one reverses the procedure, giving fist a purely local construction (cf. Chapter VI) of We will take these maps 8, from Serre maps 8, : K: + G, = G(L”/K,). and show that n 8, satisfies the characterizing properties for II/, in particular, that n 6,(u) = “1 for all u E K* (see § 10). The” local theory tells us that the Main Theorem of 5.1 is true locally if we replace C, by K:, r/j by $, and G(L/K) by G(L”/K,). In particular K:/NL”* 2: G(E’/K,) andin this isomorphism the ramification groups correspond to the standard filtration of K:/NL”*. Going back to the global theory we get a complete
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J. T. TATE
description of prime decomposition in terms of idele classes, even in the ramified case. For the question of abelian and cyclic extensions with given local behaviour and the GrunwaldWang theorem, see ArtinTate notes Chapter 10, and Wang, “On Grunwald’s Theorem”, Annals, 51 (1950), pp. 471484. 6.4. We can now give an apparently stronger statement of the reciprocity law formulated in 3.3. RECIPROCITY LAW (Strong form). Let L/K be abelian, and let S consist of the archimedean primes of K and those ramified in L. If an element a E K+ is a norm from L” for all v E S, then FLIK((a)‘) = 1. For ifj”(a) is a norm for v E S we can write j”(a) = iVLYIR,(b”) for some b” EL”. Then by Corollary 4.2 1 = $((a)“>. v~shM4)
= FL,K((~)~). v~s$VWLY~dbv~~ = FLlK((4S>9
by 6.2. For the concrete description of the local Artin maps $, by means of the norm residue symbols (a, b)” in case of Kummer extensions, and the application to the general nth power reciprocity law, see exercise 2. 7. Cohomology
of Id&s
7.1. L/K is a finite Galois extension (not necessarily abelian) with Galois group G. Write AL for the addle ring of L; then JL is the group of invertible elements in AL = L OKAK, and G acts on L BKAK by ~JHQ 0 1; so G acts on JL. However, we want to look at the action of G on the Cartesian product structure of JL. Suppose x E JL, then x = (x,), where w runs through !&; 0 E G induces 6, : L, + L,, (see 1.1) and (ax),, = c,,,x,, that is, the diagrams anL;L, L*,, L* iv+1 ~ iW lwl5” ~ Ltw ye JL 
JL
JL 
JL
commute. (Note that the image of L$ in JL is not a Ginvariant the smallest such subgroup containing Lz is wg LE.)
subgroup;
7.2. PROPOSITION. Let v E !?& and w” E 91L with w” over v. Then there are mutually inverse isomorphisms
177
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and H’(G
2 Uw) +
W%w
Uwo>,
where U, denotes the group of units in L,. The assertions remain valid when W is replaced by A’. The proof is immediate from Shapiro’s lemma (see Chapter IV, Q4) in view of Proposition 1.2 of $ 1. Thus the cohomology groups H’(G,, Lz) are canonically isomorphic for all w over v, so it is permissible to use the notation H’(G”, (L”)*) for any one of these. 7.3. PROPOSITION. (a) JR N Jf, the group of id2les of L left fixed by all elements of G.
@I B’(G JL) =” &~W’,
WI*),
where u denotes the direct sum. Proof. (a) is clear from Chapter II, 0 19. To prove (b) we observe that
and S is a finite set of primes of K containing all the ramified primes in L/K and the archimedean primes. The limit is taken over an increasing sequence of S with lim S = !l.& The cohomology of finite groups commutes with direct limits, and any cohomology theory commutes with products, so it is enough to look at the cohomology of the various parts. By 7.2 and Chapter VI, 5 1.4, n n U has trivial cohomology if S contains all the v#SLb 9 ramified primes. Hence
B’(G, JL,s>= ~Jj,Xf”G”,(Lu)*), by 7.2. Let S + %Xx; we find fi’(G, Jd E u &(G”, (L”)*). 7.4. COROLLARY. (a) H’(G, Jd = 0. (b) H’(G, JJ II v (iZ/Z),
where n, = [Lo : K,].
Here, the determination of H’ is just Hilbert’s “Theorem 90” for the local fields (see Chapter V, 0 2.6 and Chapter VI, 0 1.4). The second part follows from the determination of the Brauer group of K, in Chapter VI, 0 1.6. 8. Cohomology of Idkle Classes (I), The First Inequality We recollect the exact sequence 0 + L* + JL + C, + 0. The action of G on CL is that induced by its action on J,.
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8.1. PROPOSITION. C, N C”,. Proof. The above exact sequence gives rise to the homology sequence OB H’(G, L*) , H’(G, JL) + H’(G, CL) + H’(G, L*), that is O+K*+J,+C,G+O. 8.2. Remark. Our object in the abelian case is to define = G. + L/K: CKINL,KCL+WIK) By the Proposition above CK/NLIKCL = fi’(G, CL), and on the other hand G = Ae2(G, Z). Comparison with Chapter VI, 9 2.1, suggests that the global theorem we want to prove about the cohomology of CL is essentially the same as the local theorem Serre proves about the cohomology of L*. This is in fact the case. Abstracting the common features, one gets the general notion of a “class formation”. [cf. the ArtinTate notes.] We recollect that if G is cyclic and A a Gmodule the Herbrand quotient is defined by h(G, A) = [H2(G, A)]/[H’(G, A)] if both these cardinalities are finite (see Chapter IV, f 8). [H’(G, 41 and W1(G 41 8.3. THEOREM. Let LJK be a cyclic extension of degree n. Then h(G, CL) = n. Proof. We take a finite set S of primes of K so large that we can write JL = L* .JLss, where
More precisely, S is to include the archimedean primes of K, the primes of K ramified in L and all primes of K which “lie below” some primes whose classes generate the ideal class group of L. Denote by T the set of primes of L which are above primes in S. Hence CL N JL/L* N JL,s/(L* n JL s) = JL s/L, where LT = L* n JL,s is the set of Tunits of L, i.e. those elements of L which are units of L, for w 4 T. It follows that
MC,) = NJ,, ~)/WT), if the righthand side is defined (we note that it is impossible to use the above equation with the S missing, since then the righthand side is not defined). First of all we determine h(J,,,). Since S contains all ramified primes, the group fl fl U, has trivial cohomology, as remarked in 7.3. Hence UC.9( vlw ) ~(JL,,) = h fl fl L* (vESL,” d> =L!:Kl”“); so by 7.2 we have h(JL,s) = n n,, where the n, are the local degrees (see USS
Chapter VI, 0 1.4). This was the “local part” of the proof. The “global part” consists in determining h(L,); in order to prove that
GLOBAL CLASSFIELD
h(C,) = n we have to show that nh(L,)
THEORY
179
= fl n,. We do this by constructing VSS
a real vector space, on which G operates, with two lattices such that one has Herbrand quotient nh(L,) and the other has quotient n n,. vE.9 Let V be the real vector space of maps f: T * R, so V N R’, where t = [T], the cardinality of T. We make G operate on V by defining (OS>(W) = f(a‘w) (so that (af>(crw) = f(w)), for all f E V, e E G and w E T. Put N = {f E VJf(w) E Z for all w E T}. Clearly, N spans V and is Ginvariant. We have N N where Z, N Z for all w, and the i!s($! zw)9 action of G on N is to permute the Z, for all w over a given u E S. Hence.
by Shapiro’s lemma again. Therefore
h(N) =v~Cfio(G”, Z>l/CfJ’(G”,Z>l = VOS n n,. Now define another lattice. Let 1 be a map: L, + V given by A(a) = f,, where f,(w) = log ]& for all w E T. The unit theorem (or at any rate its proof!) tells us that the kernel of A is finite and its image is a lattice M” of V spanning the subspace V” = {f E VIE f(w) = 01. Since the kernel of A is finite, h(L,) = h(M”) (see Chapter IV, 5 8). Now V = V” +Rg, where g is defined by g(w) = 1 for all w E S,. We define the second lattice M as M’+Zg. Then M spans V and both M” and Zg are invariant under G. Hence h(M) = h(M’).h(Z) = nh(M”) = nh(L,). Now A4, N are lattices spanning the same vector space, so h(N) = h(M) by Chapter IV, 5 8. Hence n n, = h(N) = h(M) = nh(L,), as required. ” 8.4. CONSEQUENCE.
If LJK is cyclic of degree n, then
CJKIK*N~IXJLIb n. This inequality, called in the old days the second inequality, was always proved by nonanalytic methods having their origins in Gauss’ theory of the genera of quadratic forms, of which our present ones are an outgrowth. For us it is theJirst inequality, since the other inequality is deduced with the aid of this one. 8.5. CONSEQUENCE. of JK such that
If L/K is a finite abelian extension and D is a subgroup
(a> D = h&JL9
(b) K * D is densein JK, then L = K. Proof. We may suppose that L/K is cyclic, since if L 2 L’ 2 K and L’/K is cyclic, then D c N=,,J, c NL,,KJLS. Serre has proved that locally the
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norms &,I~~ Lz are open subsets of K: which contain
so NLIRJL (which is simply fl NLwlxK.Lz,) and K*NLIKJL
U, for almost all a; are open, hence
closed in JR, and the latter is wdensesince its subset K*D is dense. So it is the whole of JR, that is, [JK/K*NLIRJL] = 1; so n = 1 by the previous consequence. 8.6. Remark. We emphasize that in the Galois case an element x = (x.) E JX is in N ,.,KJL if and only if it is a local norm everywhere, i.e. x,, E N,,,,V(L”)* for all u E !l&. 8.7. CONSEQUENCE. If S is a finite subset of lozK and L/K is a finite abelian extension, then G(L/K) is generated by the elements FLIR(v) for v 4 S (i.e. the map FLIR : Is + G(L/K) is surjective; I$ 3.3). Proof. Take G’ as the subgroup of G(L/K) generated by the F,,,(v) with u # S; let M be the fixed field of G’. For v 4 S, the FLIR(v) viewed in GWIK) N G/G’ are all trivial, so for all v # S, M,,, = K, if w E DIM is over v. Trivially, every element of K,* is a norm of this extension. Take D = Ji (ideles with X, = 1 for v E S) ; every element of D is a local norm, i.e. D c NIMIKJM. By the weak approximation theorem (see Chapter IT, 5 6) K*J,” is dense in JK. So by 8.5 we have M = K and G’ is the whole of G. 8.8. COROLLARY. If L is a nontrivial abelian extension of K, there are infinitely many primes v of K that do not split completely (i.e. for which ~L,rm z 1). For we have just seen that such primes exist outside of any finite set S. 9. Cohomology of Idkle Classes (II), The Second Inequality Here we deduce what in the nonanalytic treatment is the second inequality. This inequality can be proved very quickly and easily by analysis (see Chapter VIII, Theorem 5), and classically was called the first inequality. We give Chevalley’s proof (Annals, 1940). 9.1. THEOREM. Let L/K be a Galois extension of degree n, with Galois group G.
Then (1) [Z?‘(G, CJ] and [I?‘(G, C,)] divide n, (2) fi’(G, CL) = (0). Proof. The proof will be in several steps. Step 1. Suppose that the theorem has been proved when G is cyclic and n is prime. By the Ugly Lemma (see Chapter VI, $ 1.5) it follows that [Z?‘(G, CL)] d’1v1 ‘d es n and A’(G, C,) = (0). Using this triviality of fi’, it follows, again by the ugly lemma, that [fi’(G, C,)] divides n.
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Step 2. Now we assume that G is cyclic of prime order n; in this case we know that A0 N A2 and by the first inequality 8.3 that [A’] = n[fi’]; so it will be enough to show that [fi’(G, CL)] = [C, : NLIKCL] divides n. We will make the one assumption that in the function field case n is not equal to the characteristic of K. (The other case is treated in the ArtinTate notes, Chapter 6.) Step 3. We now show that we may further assume that K contains the nth roots of unity. In fact, if we adjoin a primitive nth root of unity c to K, we get an extension K’ = K(c) whose degree m divides (nl), and so is prime to the prime n. So
n
L
K
The degree of LK’ over K’ is n, and L and K’ are linearly disjoint over K. So there is a commutative diagram with exact rows (we drop the subscripts since they are obvious): CL 4
CK +
COII
COII
1 CL
I P CL
CK’
,
N __

+O
COll
1 a
&/NC,
I G
1 CKc/NCL. 0 I “1.
+
C,INC,
.+O
Here Con is the Conorm map; and the composite map N.Con is simply raising to the mth power (see Chapter II, 4 19, for this and the definition of the Conorm map). The group &/NC, is a torsion group in which each element has order n, for if a E C,, then a” is a norm, i.e. a” E NC,. Thus the map NKPIR Con K.IR : &/NC, P C,/NC, is surjective since (m, n) = 1. Hence the map NK,IR : C,.JNC,. + C,/NC, is surjective; so if [C,, : NC,,] divides n so does [C, : NC,]. Step 4. We are thus reduced to the case where n is a prime and K contains the nth roots of unity. In fact we shall prove directly in this case the more general result : Let K contain the nth roots of unity and L/K be an abelian extension of prime exponent n, with say G(L/K) = G N (Z/nZ>‘. Then (1)
[C, : N,,,C,]
divides [L : K] = nr.
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For although, as we have just seen, the case of arbitrary r does follow from the case r = 1, yet the method to be used does not simplify at all if one puts r = 1, and some of the constructions in the proof are useful for large r (see 9.2 and 9.5). By Kummer Theory (see Chapter III), we know that L = K(qa,, . . . , y/a,) for some ai, a,,. . ., a, E K. Take S to be a finite set of (bad) primes, such that (2) (i) S contains all archimedean primes, (ii) S contains all divisors of n, (iii) Jx = K*JK,s (by making S contain representatives of a system of generators for the ideal class group), (iv) S contains all factors of the numerator and denominator of any a,. Condition (iv) just means that all the ai are Sunits, that is, they belong to KS = Kn JK,s: they are units for all t, 4 S. Write M = K(qK,) for the field obtained from K by adjoining nth roots of all Sunits. By the unit theorem the group KS has a finite basis, so this extension is finite, and M is unramified outside S by Kummer Theory and condition (ii). Now M 3 L 3 K and By Kummer we have
K,=M*“nK,~L*“nK,~K*“nK,=K~. theory with [M: L] = n’, [L : K] = nr (given) and [M: K] = ns
(3) [KS : L*” n KS] = n’, [L*” n K, : Ki] = nr and [KS respectively. We claim that s = [S], the cardinality of S. By the there are [S] 1 fundamental units, and the roots of unity nth roots; so KS N Zcsl’ x (cyclic group of order divisible [KS : Kg] = nLsl = n’, where s = t + r. (4)
: Ki] = n’ unit theorem, include the by n) and
We recall we want to show that [C, : NLIIcCL] divides d, i.e. divides [L*” n KS : Ki]. So we need to show that NLIKCL is fairly largewe have to provide a lot of norms. If w is a prime of L above a v # S, then, since M/K is unramified outside S, the Frobenius map FIcIIL(w) is welldefined. By consequence 8.7, the F&W) generate G(M/L). Choose wl,. . ., w, so that P&w,) (i = 1,. . ., t) are a basis for G(M/L), and let vi,. . ., v, be primes of K below them. We assert that F,,,(w,) = F&vi) (i = 1,. . ., t). (In fact, each of the V’S is unramified, so F,,,(vi) is defined). The M/K decomposition group G,(M/K) is a cyclic subgroup of (Z/nZ>s, so is either of prime order n or trivial. The w’s were chosen so that the F.&w) were nontrivial, so the M/L decomposition group G,(M/L) is nontrivial; so the L/K decomposition group G,GIK)
= WWKYG,(MIL)
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is trivial, i.e. v splits completely in L (see Proposition 2). Therefore, G,, = G,, and it is generated by the FM,L(ui) = am,,.) Notice also that we have L,,=K,,foralli= l,..., t. Write T = {or,. . ., D,}. We claim that (L*)” n KS = (a E I(& E Ki for all D E T). (5) In fact, since L, = K, for all D E T and w above U, it follows trivially that L*” n KS is contained in the righthand side. Conversely, if a E KS, then ;/u E M. If further a E K”, for all v E T, then ;/a E K, for all v E T and so is left fixed by all FM,,(u) = FMIL(w); these generate G(M/L) so ;/a E L. This proves (5). Let where U, is the set of vunits in K,; so E c JK s u =. Also E c NLIKJL (see Remark 8.6)for every element of K:” is a norm, since K:/NLE N G, (see Chapter VI, 0 2.1), which is killed by n; we have K,* = Lz for all u E T, and so all the elements of these K: are norms, and the elements of U, are all norms for v unramified (see Chapter VI, $ 1.2, Prop. 1). Now
LWC/K Gl = [J&*&,K divides [JR : K*E] so that
because E c NLIxJL.
therefore [CK/N,I,CJ indices of groups is
JLI
The set S was chosen ((2)(iii))
JR = K*Jx,s = K*JR,sv T divides [K*JK,S”T: K*E]. A general formula
for
[CA:CB][CnA:CnB]=[A:B], so to prove (1) it will be enough to show that (7) [JK,SVT:EI/[KS,T:KnE]=n’ (where KS” T = K * nJK,@T). First, we calculate [JK,SVT : E]. so [J~,~ v T : E] = n [Kz : (K,*>n], by (5). From Chapter VI, 0 1.7 (cf. also “ES
the ArtinTate notes, p. xii), we see that the “trivial action” Herbrand quotient Ii = n/l& where ] 1, denotes the normed absolute value. But also h(K,*) = [K: : K,*“]/n because the nth roots of unity are in K:. This means that [K*  K*”” ] = n2/~& and (8) [JR.s"".T : E] = n*lg InI;’ = nZS, by the product formula since In]” = 1 if 0 $ S.
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We will also need in a moment the formula [U” : u:J = n/In]“. (9 which follows from the fact that h(U,) = l/l& (see Chapter VI, $ 1.7). By (8) we see that to prove (7), it will be enough to show that K* n E] = nzs = nS+‘, (10) [KS,.: As in (4), replacing S by S u T, we have [I&, T : K,“, =] = nS+‘, so it will be enough to show that K* n E = K,“, T’ Trivially, K* n E 3 K,“, r, so it remains to prove K*nEcKi,., (11) and this will result from the following lemma. 9.2. LEMMA. Let K contain the nth roots of unity. Let S be a subset of %R, satisfying parts (i), (ii), (iii) of (2) in the above proof, and let T be a set of primes disjoint from S, and independentfor KS in the sense that the map Ks + Vg U,/ Vi is surjective. Supposethat b E K* is an nth power in S, arbitrary in T, and a unit outside S v T. Then b E K*“. Proof of Lemma. Consider the extension K’ = K(vb); to deduce that K’ = K. Put
it will be enough
by arguments similar to ones used before (see after (6)), D c NRsIKJR,. Therefore, by the first inequality in the form of consequence 8.5, in order to prove that K’ = K it is sufficient to prove that K*D = JR. But by hypothesis, the map Ks + fl (U,/Un,> N J&D is surjective. Hence Y6T
J K,S = KsD, and JR = K*JR,s = K*D as required.
To deduce (11) from the lemma, we have to check that T is independent Let H denote the kernel of the map for S in the sense of the lemma. o rove that map is surjective it suffices to show that K, +&WJ3. T P (Ks : H) = n (U, : Uz). This latter product is just n’ by (9), because VET
1 for v E T. On the other hand, by (5) we have H = Ks n (L*>“, and consequently (Ks : H) = n’ by (3). The proof of the theorem is now complete. I&
=
9.3. Remark. Even the case of the Lemma 9.2 with T empty is interesting. “If S satisfies conditions (i), (ii), (iii) of (2), then an Sunit which is a local nth power at all primes in S is an nth power”. 9.4. CONSEQUENCE. If L/K is abelian with Galois group G, and there is an Artin map I/I : I?‘(G, C,) = C,/NC, + G, then $ must be an isomorphism.
GLOBAL CLASSFIELD
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185
In fact, consequence 8.7 of the first inequality already tells us that JI has to be surjective; if now [fi’(G, C,)] < [G], then $ can only be an isomorphism! 9.5. CONSEQUENCE. (Extracted from the proof of Theorem 9.1.) Let n be a prime and let K be afield, not of characteristic n, containing the nth roots of unity. Let S be a$nite set of primes of K satisfying the conditions (i), (ii), (iii)hzS), and let M = K(vKs). Th en, if the reciprocity law holdsfor M/K, K*NMIKJM = K*E, where E = n (K,*)” x fl U,. UES u+s Consider the case L = A4 of the proof of 9.1 (so that T is empty, 1 = 0 and s = r). Then the E of that proof is as given in (12), and E c N,,,J,. By (7) with L = M, we have [JK: K*E] = ns = [M: K]. On the other (12)
hand, if the reciprocity law holds, we know that [C, : N~,&J = [JK : K*NM,KJ&J = n”; hence (12) must hold. This result we put into the refrigerator; we will pull it out for the proof of the “existence theorem” in the final section $ 12. 9.6. CONSEQUENCE. Let L/K be a finite (not necessarily abelian) Galois extension. Since H’(G, C,> = 0, the exact sequence 0 + L* + JL f C, + 0 gives rise to a very short exact sequence 0 + H’(G, L*) P H’(G, JL). Now H2(G, J,.) = u H’(G”,(L”)*), by Proposition 7.3, so there is an injection VOrnK
(13)
0 + H'(G, L*) +" EgxH2(G", (I!')*).
We shall see later (from the fact that the arrow p1 in diagram (9) of 9 11 is an isomorphism, for example) that the image of this injection consists of those elements in the direct sum, the sum of whose local invariants is 0. We thus obtain a complete description of the structure of the group H2(G, L*). In terms of central simple algebras, (13) gives the BrauerHasseNoether Theorem, that a central simple algebra over K splits over K if and only if it splits locally everywhere. In particular, if G is cyclic, A2 N Go, and we have the Hasse Norm Theorem: If a E K*, and LJK is cyclic, then a E NLIKL* if and only if a E NLyIgyLv* or all v E .!UI=. Specializing further, take G of order 2, so L = K(Jb). NL,&+yJb)
= x2by2,
so (if the characteristic is not 2) we deduce that a has the form x2 by’ if and only if it has this form locally everywhere. It follows that a quadratic form Q(x, y, z) in three variables over K has a nontrivial zero in K if and only ifit has a nontrivial zero in every completion of K. Extending to n variables,
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we may obtain the MinkowskiHasse Theorem, that a quadratic form has a zero if and only if it has a zero locally everywhere, see exercise 4. One may consider the general problem, “if a E K* and a E NLv* for all v, is a E NL* ?” Unfortunately, the answer is not always yes! (See 11.4.) 9.7. We return to the sequence (13). We write H’(L/K) H’(L”/K,) for H2(G”, L”*). Thus (13) becomes
for H2(G, L*) and
0 + H2(L/K)
+ 11 H’(L”/K,). ” Serre (Chapter VI, 9 1.1, Theorem 3, Corollary 2) has determined H’(L”/K,); it is cyclic of order n, = [L”: K,], with a canonical generator. Thus (13’)
H'(G, JL) = u H2(E'/K,) "
N u (; Z/Z). v "
and 0 + H’(L/K) If a E u H2(L”/K,),
or u E H’(L/K),
+ u (; Z/Z). v ” we can find its local invariants inv, (a)
”
(more precisely inv, (j,(a)), where j, is the projection on the vcomponent of or), which will determine it precisely. We are interested in the functorial properties of the map inv,. Let L’ 2 L 1 K be finite Galois extensions with groups G’ = G(L’/K), and G = G(L/K) where H = G(L’/L).
N G’/H,
If CIE H2(G, Ja, then infl (a) E H’(G’,
JLS) and
inv, (infl a) = inv, (cx). (14) Indeed, choosing a prime w’ of L’ above a prime w of L above v, one reduces this to the corresponding local statement for the tower L$ I> L, 3 K,; cf. Chapter VI, 0 1.1. Thus nothing changes under inflation so we can pass in an invariant manner to the Brauer group of K, and get the local invariants for c1E Br (K) = H’(K/K), where R is the algebraic closure of K (see Chapter VI, 5 l), and more generally for a E H2(GIK,
JR) = $
H2(GLIK, J3,
where JR = lim JL, by definition, the limits being taken over all finite Galois T? . extensions L of K; cf. Chapter V.
GLOBAL CLASSFIELD
If now a E H’(G’,
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JLP), then res ga E H2(H, JLP) and inv, (res ga) = n,/” inv” (a),
(15) where w E ‘D, lies above u E ‘%RKand nwl” = [L, : K”] (again one reduces immediately to the local case, for which see Chapter VI, 9 1.1, or Serre’s “Corps Locaux”, Hermann, (1962), p. 175). Moreover, L/K need not be Galois here. Finally we mention the result for corestriction, though we will not use it. Again, L/K need not be Galois. If a’ E H’(H, JLS),then car gal E H’(G’, JLP) and inv” (car ga’) = C inv, (a’), (16) WI” where the sum is over all primes w E ‘%lZLover v E 1)32,(see “Corps Locaux”, p. 175). 9.8. COROLLARY. Let CIE Br (K) or H2(G(R/K), JR), where K is the separable algebraic closure of K. Let L be an extension of K in x. Then res f(a) = 0 if and only if [L, : K,] inv” (a) = 0 for every w over v (this is only a finite condition, since almost all the inv” (E) are zero). In the case when L/K is Galois, there is an exact sequence 0 
H2(L/K) infl
Br (K) ZL
and a E H’(L/K) if and only if the denominator for all w over 0.
Br (L),
of inv” (cc)divides [L, : K,]
10. Proof of the Reciprocity Law 10.1. Now let L/K be a finite abelian extension with Galois group G. We recall our discussion in $6 on local symbols in which we noted that if a global Artin map existed we were able to reduce it to the study of local symbols and remarked that, conversely, if the local Artin maps are defined we could obtain a global Artin map. We propose to carry out this latter program here using the local Artin (“norm residue”) maps defined in Chapter VI, Q2.2. Let the local Artin maps be denoted by 0” : K,* + G”; we define a map e:J,+G by
e(x) =” ~~KUd,
x EJK.
This is a proper definition, since (by Chapter VI, 0 2.3) 0,(x”) = FLy,K,(v)“(Xy) (v(x”) being the normalized valuation of x”) when v is unramified, and v(x”) = 0 if x” E U”; so 0,(x”) = 1 for all but finitely many v. (Indeed, even if L/K were an infinite extension, the product for 0(x) would be convergent.) It is clear that 0 is a continuous map.
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Take S,, c %JIKas the set of archimedean primes plus the primes ramified in L/K; then x E Jscsoimplies e(x) = ?‘((x)‘~). Thus, 8 satisfies two of the conditions for an Artin map ((i) and (iii) of Corollary 4.2); the other condition ((ii) of 4.2) is that e(a) = vEQ,O,(a) = 1 for all a E K*. So if we can prove this, we will have proved the reciprocity law. 10.2. To prove the reciprocity law, it will be convenient to state two related theorems, and to prove them both at once, gradually extending the cases for which they are true. THEOREM A. Every finite abelian extension L/K satisfies the reciprocity law, and the Artin map 8 : JR + G(L/K) is given by 8 = n 8.. cl THEOREM B. If u EBr (K), then c inv, (a) = 0. 06VRK Remarks. After what has been said above, Theorem A has been whittled down to the assertion that n e,(a) = 1 for all a E K*. (1) U~BRX The sum of Theorem B is finite since inv, (a) = inv, (j,. CQ= 0 for all but finitely many a. If a E Br (K), then ~~EH~(L/K) for some finite extension L/K, i.e. tl is split by a finite extension of K. Logically, the proof is in four main steps. Step 1. Prove A for an arbitrary finite cyclotomic extension LJK. Step 2. Deduce B for tl split by a cyclic cyclotomic extension. Step 3. Deduce B for arbitrary u E Br (K). Step 4. Deduce A for all abelian extensions. In practice, we first clarify the relation between Theorems A and B and deduce (step 2) that A implies B for cyclic extensions and (step 4) that B implies A for arbitrary abelian extensions. Then we prove Step 1 directly, and finally push through Step 3, by showing that every element of Br (K) has a cyclic cyclotomic splitting field. 10.3. Steps 2 and 4. The relation between A and B. A is about I? and B is about H2, so we need a lemma connecting them. Let L/K be a finite abelian extension with Galois group G. Let x be a character of G, thus x E Horn (G, Q/Z) = H’(G, Q/Z), where Q/Z is a trivial Gmodule. If v E ‘&Xx, denote by xv the restriction of x to the decomposition group G”. Let 6 be the connecting homomorphism 6 : H’(G, Q/Z) ) H’(G, Z). If x = (x,) E JR, let X be its image in JK/NLIKJL N f?‘(G, JL). Then the cup product (see Chapter IV, § 7) z.6~ E H2(G, J&
GLOBAL LEMMA.
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189
For each u we have inv, (z. 6x1 = x~(W,))~
and so T inv, (a. Sx) = x(0(x)). Proof. We refer to Chapter VI, 5 2.3. The projection j,: J, + (LO)* induces a map j, . res: y : H’(G, JJ , H’(G,, Jd + HZ&L CL”)*), and as restriction commutes with the cup product, so inv, (X. 6~) = inv, (j,. resG,”(X. 6~)) = inv, (( 1,. Z) . Sx,) = inv, (2,. 6~“) = x”(e”b”)), the final step coming from Chapter VI, 5 2.3. Tt follows immediately that x(eW> = x(IJ WJ)
= T xMxJ>
= T inv,WW.
To check Step 4, apply the lemma with x = a E K* c J= Denote by a” the image of a in fi’(G, L*). Then a”. 6% E A2(G, L*) c Br (K), as we need. The image of a”.6~ in H2(G, JL) is a.&, where d is the image of a in fl”(G,J,), and by the above lemma, c inv, (a.&) = x(6(u)); so if Theorem B is true for all a E Br (K), it follois that x(&z)) = 0, and since this is true for all x, that e(u) = 0. This is Theorem A. To check Step 2, take L/K cyclic. Choose x as a generating character, i.e. as an injection of G into Q/Z. Then cupping with 6~ gives an isomorphism fro 3 H2, so every element of H’(L/K, L*) is of the form c.6~. If Theorem A is true, then by the above lemma T inv, (Z .6x) = x(e(a)) = 0 for all a E K*, which is Theorem B. 10.4. Step 1. (Number Field Case.) We want to prove that if L/K is a cyclotomic extension, then n e,(a) = 1 for all a E K*. Let L/K be a finite “cyclotomic extension. Then we have L c K(c) for some root of unity 5, and it will suffice to treat the case L = K(c) because of the compatibility of the local symbols 8. relative to the extensions (K([))u/K, and L”/K,; cf. Chapter VI, 5 2.4. Next we reduce to the case K = Q. Suppose that M = K(n, with Galois
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J. T. TATE
group G’ ; define L = Q(l), with Galois group G. Then h4 = LK, and there is a natural injection i : G’ + G and norm map N: JK + JQ. The diagram B’  G’ JK
(where 0’ = n &,, and 0 = n 6,) is commutative, Y *)P = 3
('K,Q
since *,
NK./Q~
(see Chapter II, $ 11, last display formula) and since the diagrams KJ N 1
1
QP
,
G:
i
i
 GP
are commutative whenever v is above p (see Chaper VI, 3 2.1, cf. Proposition 3.1 above). Thus i OO’(X) = B(Nx,o(x)) for all x E JR and so, in particular, i Oe’(a) = e(N,o(U)) for all a E K. If Theorem A is true for L/Q, then O(b) = 1 for all b E Q, SO O’(o) = 1 for all a E K, because i is injective. First Proof for L/Q cyclotomic. We know that the reciprocity law, in the sense of 3.8, holds for L/Q by computation (see 3.4), i.e. we have an admissible map F: Is + G(L/Q) = G, for some S. Using the Chevalley interpretation (Proposition 4.1) we get an Artin map $ : JQ + G, and we obtain induced local maps JI, : Q: + G, (see 3 6). Using Proposition 4.3 we can pass to the limit and take L as the maximal cyclotomic extension Q”” of Q. This gives us local maps 9, : Qy + G(Qy/Q$, for all primes p. We want to show that these $p’s are the same as the Op’s of Chapter VI, 3 2.2; we do this by using the characterization given by Chapter VI, 5 2.8, Proposition 3. We have to check three things. Firstly, that Qy contains the maximal unramified extension Qy of Q,; this follows from Chapter I, 9 7, Application. Secondly, if a E Qp, then tip(a) 1Q, = F’p(‘), where r+,(a) is the normalized valuation of a, and F the Frobenius element of G(Qy/Q,); this is clear. Thirdly, if A/Q, is a finite subextension in Qy, and ccE Nlllo,,~Z*, then I/J,@) leaves .& pointwisefixed; this follows from Proposition 6.2. Hence +P = 0, for all finite primes p. We must not forget to check that $, is the same as 8, (see Chapter VI, $2.9). By Proposition 6.2, $m is a con
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tinuous homomorphism of R* into G, = G(C/R) E { f I}, and $,(ZVo,&*) = 1. Hence @m and 8, induce maps of R*/R$ into G(C/R) and 8, is onto, so we just have to check that $, is ontoin other words, we have to check that $, is not the null map. C = R(i), so consider the effect of $ on Q(i) 2 Q; the only ramification is at 2 and co. Therefore 1 = $(7)
= 1c/2(7)1c1,(7)~,(7)
= $,(7)*ti,(),
since 7 is a 2adic norm and so q?2( 7) = 1. Now 1,4,(7) is the map i+i7=  i, so $,(  1) is also the map i + i, i.e. is nontrivial. Second Proof for L/Q cyclotomic. We may proceed entirely locally, without using our results of the early sections, but using the explicit local computation of the norm residue symbol in cyclotomic extensions, due originally to Dwork. Let c be a root of unity; by Chapter VI, $2.9 cem(x)= [Sign (Xl, for x E R*, (1) and by Chapter VI, 5 3.1, if x E Qt, x = p’u, with u a unit in Q, and v an integer py when f has order prime to p. e,hw = (2) c g, when c has ppower order. { We need to check that flt9,(a) sufficient
to show that
= 1 for all a E Q* and to do this it is
fi 0,(q) = 1 for all primes
n 0,(  1) = 1. Furthermire,
that the effect
c1, p = co P z 1, 00
c, =
One checks explicitly
c1, p = 1 t9
pw
that
it is enough to consider the effect on c, an
Zrth power root of unity (Z a prime). is trivial, using the tables c“W 1) =
q > 0, and
p=4= p # 1, p # q (including tIql, P = 1, P # 4
[,
cq,
the case p = 03)
P f I, P = 4
(Since the Galois group is abelian, it does not matter in what order one applies the automorphisms 9,( l), resp. e,(q).) 10.5. Step 3. (Number FieId Case.) It is enough to show that every element of Br (K) has a cyclic, cyclotomic splitting field. In other words, for every a E Br (K), there is a cyclic, cyclotomic extension L/K such that for every
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o E ‘9Xx, the local degree [L” : K,] is a multiple of the denominator of inv, (tl) (see Corollary 9.8). Now inv, (a) = 0 for all but a finite number of primes and so we need only prove the LEMMA. Given a number field K of finite degree over Q, a finite set of primes S of K, and a positive integer m, there exists a cyclic, cyclotomic extension L/K whose local degrees are divisibIe by m at the nonarchimedean primes v of S and divisible by 2 at real archimedean primes v of S (in other words, L is complex). Proof. It is sufficient to construct L in the case K = Q (multiply m by the degree [K: Q]). Take r very large and q an odd prime. The extension L(q) = Qt’;/l> h as a Galois group isomorphic to the direct sum of a cyclic group of order (q 1) and cyclic group of order qrml, so has a subfield L’(q) which is a cyclic cyclotomic extension of Q of degree q’‘. Now
CL(q): wdl = 4 19 and so on localizing
at a fixed prime p # COof Q we have
[L(q)@’ : L’(q)(P)] < (q  1); since [L(q)(p) : Q,] + co as r + co (this follows for example from the fact that each finite extension of Q, contains only a finite number of roots of unity), it follows that [L’(q)(p) : Q,] + co as r + 0~). Therefore, since [L’(q)‘p’ : Q,] is always a power of q, it is divisible by a sufficiently large power of q if we take r large enough. Now let q = 2, and put L(2) = Q(2q1) for r large. L(2) has a Galois group isomorphic to the direct sum of a cyclic group of order 2 and a cyclic group of order 2’2. Let c be a primitive 2’th root of unity and set < = [cl and L’(2) = Q(c). The automorphisms of Q(c) over Q are of the form a,, : 5 H cP for p odd, and a,(r) = lP c“. Since Yzrl =  1, one seesthat cr p+2P1(r) = o,,(t); since either p or ~+2*l is E 1 (mod4), this implies that the automorphisms of Q(t)/Q are induced by those c,, where p EE 1 (mod 4) and that they form a cyclic group of order 2’2. Also, since 6 1{ =  <, Q(c) is not real, and so its local degree at an infiinte real prime is 2. Now [L(2) : L’(2)] = 2, and the same argument as above shows that for p # co we can make [L’(2)‘p) : Q,] d ivisible by as large a power of 2 as we like by taking r large enough. If now the prime factors of m are ql,. . . , qn and possibly 2, then for large enough r the compositum of L’(q,), . . . , L’(q,) and possibly L’(2) is a complex cyclic cyclotomic extension of Q whose local degree over Q, is divisible by m for all p in a finite set S. Cyclic cyclotomic extensions seem to be at the heart of all proofs of the general reciprocity law. We have been able to get away with a very trivial
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existence lemma for them, because we have at our disposal both cohomology and the local theory. In his original proof Artin used a more subtle lemma; see for example Lang, “Algebraic Numbers”, Addison Wesley, 1964, p. 60. (But notice that the necessary hypothesis that p be unramified is omitted from the statement there.) We may prove the reciprocity law for function fields on the same lines, but the special role of “cyclic cyclotomic extensions” in the proof is taken over by “constant field extensions”. Step 3 goes through, if we replace “cyclic cyclotomic extension” by “constant field extension”; we have only to take for the L in the lemma the constant field extension whose degree is m times the least common multiple of the degrees of the primes in S. For step 1, we check the reciprocity law directly for constant field extensions; in fact, if we denote by Q the Frobenius automorphism of k/k, where k is the constant field of K, then for each prime D of K the effect of F(v) on Z is just odes“, where deg t, = [k(u) : k] is the degree of u. Hence the effect on li of e(u) is fl c”(‘) de*” = &“(“) deg” = cdeg* = 1, since deg a = 0 for all a E K* (the number of zeros of an algebraic function the number of poles).
a is equal to
11. Cohomology of Id&le Classes (III), The Fundamental Class 11.1 Let E/L/K be finite Galois extensions of K; then we have an exact commutative diagram 0
0
I 0 pd
(1) 0 + 0 
lP(L/K,
i
H 2(E/K,
I
0
I L*) ,
fP(L/K,
E*)
H’(E/K,

H2(E/L, E*) a
I JL) 
H2(L/K,
J& 
H2(E/K,
H’(E/L, JJ +
H’(E/L,
I
I
i
C,) C,)
I C,),
where we have written H’(L/K, L*) for H’(G(L/K), L*), etc. In this diagram, the vertical lines are inflationrestriction sequences; these are exact since H’(E/L, E*) = (0) (Hilbert Theorem 90, Chapter V, 5 2.6), H’(E/L, JE) = (0) (Corollary 7.4) and H’(E/L, C,) = (0) (Theorem 9.1) [see Chapter IV, 5 5, Proposition 51. The horizontal sequences are exact, and come from the sequence 0 + L* + JL + C, + 0, since again H’(L/K, C,) = (0), etc. We pass to the limit and let E + K, where R is the algebraic closure of K, to obtain the new commutative diagram
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0
0 I 0h
H2(L/K,
1 H2(L/K,
Id*) yI+ I
(2) 0 +
0 JJ
El
1
H2(K,R*)
A*+
H2(K,JpJ
L+
I H2(L/K,
CL)
i
H2(K, C,)
I i I oH2(L,K*) 23% H2(L,JK) 2+ H2(L, C,), where we have written H2(K, R*) for H’(G@/K),K*), etc. Certain of the maps with which we shall be concerned below have been labelled in the diagram. 11.2. We are going to enlarge the above commutative diagram. For the Galois extension L/K we have the map inv, = c inv, : H’(L/K, JJ + Q/Z, ” and Theorem B of 10.2 tells us that the sequence H’(L/K, L*) y1, H”(L/K, JJ inv&. Q/Z (3) ois a comp1ex.t Since inv, (infla) = inv, (a) for all a E H’(L/K,J,) (see 9.7, (14)), we have a map inv, : H’(K, .7x) + Q/Z such that the diagram H’(L/K, JJ + invt Q/Z (4)
infl 1 H’W,
JK)
,
inn
l
Q/Z
is commutative, where i is the identity map. Furthermore, the sequence (5) 0 ...a H2(K, R*) A.+ H’(K, .J=)‘“X Q/Z is a complex. In a similar manner we have a complex 0 a H’(L, R*) =+ H2(L, JK) 2% Q/Z. (6) But now, inv, (res a) = nwlv inv, (a), where a E H’(K, JK) and w is a prime of L over v of K, and nwlv = [L, : K,] (see 5 9.7, (15)). Thus we have the commutative diagram inv2  Q/Z H2K JR) (7)
rcs I i f in*3 HZ@, JK) QIZ as the sum of the local degrees c nwlv = n = [L : K]. wlu t i.e. the image of each map is in the kernel of the next.
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Now let Image of s2 = Im .s2 in H’(K, C,) be denoted by H’(K, C& It follows that we have a map p2 (resp. &) and Im sg by H2(L, C&. induced by inv, (resp. inv,) of H2(K, C& into Q/Z (resp. H’(L, C,& into Q/Z). Thus for a E H2(K, Cg&, we have j12(u) = k,(b), where s2(b) = a (this is independent of the choice of b). We have now explained the two lower layers in diagram (9) below. We define (8)
H’(L/K,
CJrep = {a E H’(L/K,
CJjinfl
a E H’(K,
Cr&}.
Then nP2 infl a = 0, and so B2 induces a homomorphism /II : H’(L/K,
CJreo +  Z/Z n
such that B&J) = D2 W If a = sL b with b E H’(L/K,
4.
JL) then
Pi(a) = j12 (infl b) = inv, (infl b) = inv, (b). (Note the difference in construction of pt and B2; the point is that H’(L/K, C& =I Im s1 but they will not in general be equal.) We put all the information from (3)(8) into (2) to obtain a new commutative (threedimensional) diagram
O
O
(9)
in which i is the inclusion map, n is multiplication by n and the “bent” sequences are complexes, and the horizontal and vertical sequences are exact.
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J. T. TATE
11.2. (his) We propose to show that H2K Wren = H2W, Cd 1: Q/Z. (10) Now Im (invl)
in i Z/Z
is the subgroup
i
Z/Z,
where no is the lowest
common multiple of all the local degrees of L/K, by Corollary 7.4, and so since Im /I1 3 (invi) we have the inequalities n z [H’(L/K, Cd] 2 [H’(L/K, C&] 2 [ImA] 2 [Im WI)] = no, by the second inequality, Theorem 9.1. It follows that if n = no for this particular finite extension LJK, then we have equality throughout, so that /I1 is bijective and the sequence (11) 0 .+ H’(L/K,L*) 2. H2(L/K,&.) 2% Q/Z is exact (for if 0 = inv, (b) = /I1 s1 b, then s1 b = 0, and b E Im ri). Now if L/K is a finite cyclic extension, then n = no because the Frobenius elements I;L,x(u), whose orders are equal to the local degrees n,, generate the cyclic group G(L/K) by Consequence 8.7. So if, in particular, the extension L/K is cyclic cyclotomic, then (11) is an exact sequence. But the Lemma of $10.5 says that the groups H’(K, R*) and H’(K, JK) are the unions (of the isomorphic images under inflation) of the groups H’(L/K, L*) and H’(L/K, JL) respectively, where L runs over all cyclic cyclotomic extensions of K. Consequently, in our commutative diagram (9) the complexes 0 ,
H’(K,
g*) =.
H2(K, JR) ZL
Q/Z
and 0h H2(L, R*) y3+ H’(L, Jx) .=.‘. Q/Z are exact. Therefore ker (inv2) = ker (s2), so p2 (and similarly /Is) must be injective maps into Q/Z. They are surjective, since there exist finite extensions with arbitrarily high local degrees and consequently even inv2 and inv, are surjective. Hence both p2 and p3 are bijective maps. Now, letting L be an arbitrary finite Galois extension, we conclude that pi is a bijection:
;z/z; H2(LIK Cdreg but H2(L/K, CL)rel is a subgroup of H’(L/K, C,) which has order dividing So it is the whole of H2(L/K, C,>. Letting L ) K we see that H2(LIK Wreg = H2(L, Cd. Thus we can remove the subscripts “reg” from our diagram Also, we have proved the following RESULT. H’(L/K, uLlr with invariant
n.
(9).
C,> is cyclic of order n, and it has a canonical generator t, i.e. inv, (u,,,) = X.
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This element uL/x is called the fundamental class of the extension L/K. It was first exhibited by Weil (see the discussion in 11.6 below). The complete determination of the structure of H’(L/K, Cd is due to Nakayama. He and Hochschild were the first to give a systematic cohomological treatment of class field theory; see G. Hochschild and T. Nakayama “Cohomology in Class Field Theory”, Annals, 1952, and the references contained therein. The two lower layers of diagram (9) and the vertical arrows between them make sense for an arbitrary finite separable extension L/K of finite degree n, and in this more general case, that much of the diagram is still commutative, because the argument showing the commutativity of (7) did not require L/K to be Galois. Using this, and replacing L by K’, we see that if L 3 K’ 3 K with L/K Galois, then restricting uLix from L/K to L/K’ gives the fundamental class uL,x,. 11.3. Applications. The results we have obtained show that the idble classes constitute a class formation. In particular (cf. Chapter IV, 0 10) the cup product with the fundamental class #L/x gives isomorphisms B’(G(LIK), Z) =s B’+‘(G(LIK), CJ, for  cc < r < co, such that for L 1 K’ 3 K with L/K Galois the diagrams B’(G, Z) r fP+‘(G, C,) flr(G, Z) 7: l?+2(G, CJ resl car (12) resJ and COIN &(G’, Z) r &+‘(G’, CL) @(G’, Z) 3 i?,,:,‘, C,) are commutative, where G = G(L/K) and G’ = G(L/K’). Case r = 2. There is a canonical isomorphism (see Chapter IV, 6 3)
WIObt
CK/NL,KCLS
which is inverse to the Artin map. Using this as a definition in the local case, Serre deduced the formula inv (a. 6~) = x(@(a)) in Chapter VI, $2.3 ; we have proved the formula in the global case, so one can reverse the argument. (The isomorphism Gab II Hm2(G, Z) is to be chosen in such a manner that for x E Horn (G, Q/Z) N H1(G, Q/Z) and u E G, we have x. d = x(a) upon identifying
i Z/Z with H‘(G,
Q/Z) as usual.)
Reversing the horizontal arrows in (12) with r =  2, and letting L +K, we obtain the commutative diagrams rL ti CK p G(Kab/K) G(Kab/K) T and N?I (13) C0”l
i
iy
JI G((K’)“blK’), C Kp C K’ e G((K’>“b/K’) where the Il/‘s are the Artin maps and V is the “Verlagerung~“. The righthand t Called the “ transfer ” in Chapter IV, f 6, Note after Prop. 7.
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J. T. TATE
diagram expresses the socalled translation theorem, and in fact results also directly from 4.3, which in turn came from an almost obvious property 3.2 of the Frobenius automorphisms I;(v). The commutativity of the lefthand diagram (13) can also be proved by a straightforward but somewhat more complicated computation with the Frobenius automorphisms which was first made by Artin in connection with the “principal ideal theorem” (see exercise 3, and Serre, “Corps Locaux”, p. 130). Case r = 3. This leads to an isomorphism used by Roquette in Chapter IX, $2. 11.4. Application to the Cohomology of L *. The general idea is to determine the cohomology of L* from a knowledge of the cohomology of the ideles and the idble classes. Let L/K be a finite extension, with Galois group G. Then the exact sequence 0 + L* + JL + C, + 0 gives an exact sequence + 8’‘(G, JJ : tl’‘(G, CJ + &(G, L*) : B’(G, JJ + . . . in which the kernel off is isomorphic to the cokernel of g. We know f;t’l(~, ~3 =v~xPl(~o,c*) =.&=P~(G”, z). (see Proposition
7.3), and 8’‘(G,
CJ = flr3(G,
Z);
so the kernel of f : l?(G, L*) + L1[ s’(G”, E*) is isomorphic to the cokernel of g1 : u it’3(Go, Z) + l?‘3(G, Z). It is easy to see that the map g1 is given by gl(l+) =~c0r2zo. Using the fundamental duality theorem in the cohomology of finite groups, which states that the cup product pairing &(G, Z) x &‘(G, Z) + l?‘(G, Z) z Z/nZ is a perfect duality of finite groups, one sees that the cokernel of g1 is the dual of the kernel of the map h : I?‘‘(G, Z) + n fi3 ‘(GO, Z) which is defined by (h(z)), = res g.(z) four all v E 9Xx. Case r = 0. ala E K*, a is a local norm everywhere Kerf = ala E K*, a is a global norm >’ and coker g is dual to ker (H3(G, Z) E, n H’(G”, Z)). For example, if I)
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199
G = G” for some a, then this is an injection, so local norms are global norms. If G is cyclic, then H3(G, Z) z H’(G, Z) = 0, so local norms are global norms and we recover Hasse’s theorem, 9.6. On the other hand, if for instance G is the Vierergruppe, it is possible that G’ is always one of the subgroups of order 2, so fi3(G”, Z) = 0 but @(G, Z) = Z/22. Explicitly, we can consider Q(,/13,
,/17)/Q;
here ($(A!)=
1, and the extension
is unramified at 2 because 13 = 17 = 1 (mod 4), so all the decomposition groups are cyclic. Thus the set of elements of K* which are local norms everywhere is not the same as the set of elements of K* which are global norms (see exercise 5). Case r = 3. H3(G, L*) is cyclic of order n/n,,, the global degree divided by the lowest common multiple of the local degrees, generated by 6u,,, (6 : H’(C,) + H3(L*)), the “Teichmtiller 3class”. This can be killed by inflation (replace L by a bigger L’ so that the no for L’ is divisible by n); so H3(x/K,
K*)
= 0.
For a more precise description of the situation, announced at the Amsterdam Congress (Proc. 11,6667), see Tate: “The cohomology groups of tori in finite galois extensions of number fields”, Nagoya Math. J. 27 (1966), 709719. Group Extensions. Consider extensions M/L/K, where L/K is Galois with group G, and M/K is Galois with group E and M is a class field over L with abelian Galois group A. So 1 + A + E + G + 1 is exact. By the Artin isomorphism A N CL/NM,LCM (see Theorem A of 10.2 and Consequence 9.4). We want to know about E. 11.5. THEOREM. (i) Let a E E have image Z E G. Let x E C,; then y(iix)
= ay(x)al,
where y: C,+A is the Artin map. A) be the class of the group extension E; then v = $*(uLIK), where Ic/* is the map: H*(G, Cd + H*(G, A) induced by @ : C, + A, and where u,,, is the fundamental class for L/K.
(ii) Let v E H*(G,
It is straightforward to see (i). As usual in such cases the situation becomes clearer if we consider an arbitrary field isomorphism (T: M + M’ rather than an automorphism. Denoting aL by L’ and the restriction of d to L by t?, we have the picture M&M’
and by transporting the structure of M/L and Y E M then (Y’GW>(~Y) = 4.44(~)>.
to M’/L’
we see that, if x E CL
200
J. T. TATE
(ii) is nontrivial; SafareviE did the local case (“On Galois Groups of Padic Fields”, Dok&‘y, 1946), but it is really a general theorem about class formations (see ArtinTate notes, p. 246). We do not prove it here, and will not make any use of the result in these notes. 11.6. Before the structure of H’(G, CL) was known, Weil (“Sur la 2’ ‘. du Corps de Classes”, J. Math. Sot. Japan, 3 (19X), l35) looked Z,S’l*;*c. situation from the opposite point of view. Taking M to be Lab, the rnz$,, : .,, abelian extension of K (so that now A = G(Lab/L) is a profinite ab&‘ar” group), Weil asked himself whether there was a group extension G = G(L/K) by CL which fits into a commutative diagram of the sort 1 +
c, ,d
1 +A
I*L 1 EG
(14)
,G I 1
.l id 1
,
1,
where E = G(Lab/K), and if so, to what extent was it unique? In the function field case, lc/L is for all practical purposes an isomorphism, and the existence of such a diagram is obvious. Moreover, in that case, the grouptheoretical transfer map V (Verlagerung) from d to CL (which has its image t in C,) gives a commutative diagram
as follows from the commutativity of the lefthand diagram (13). Inspired by the case of function fields, Weil proved that also in the number field case a diagram. (14) did indeed exist, and was essentially uniquely determined by the condition that (15) (together with its analogues when K is replaced by an arbitrary intermediate field K’ between K and L) should be commutative. In particular, the class u E H’(G, CL) of such an extension 8 was unique, and that is the way the fundamental class was discovered. Nowadays one can proceed more directly, simply constructing B as a group extension of G by CL corresponding to the fundamental class uL,x, and interpreting the unicity as reflecting the fact that H’(G, CL) = 0 (cf. ArtinTate, Ch. 14). The kernel of the map IV: 8 + E is the connected component DL of CL. As Weil remarks, the search for a Galoislike interpretation of d (or even a “natural” construction, without recourse to factor systems, of a group 8 furnished with a “natural” map IV: 8 + E) seems to be one of the fundamental problems of number theory.
GLOBAL
CLASS
FIELD
THEORY
201
In support of the idea that 8 behaves like a Galois group, Weil also describes how to attach L.series to characters x of unitary representations of 8. These Lseries of Weil generalize simultaneously Hecke’s Lseries “mit Griissencharakteren” (which are obtained from representations which +?rough C, via the arrow V in (15)) and Artin’s “nonabelian” , (which are obtained from representations which factor through E, 2: %ticular through G N E/A, via the arrow Win (15)). (The intersection ,I; ft ,$ke’s and Artin’s Lseries are those of Weber obtained from repre“I/$ons of Eab N C,lD,, i.e. from ordinary congruence characters.) “‘!:g Brauer’s theorem on group characters, Weil shows that his Lseries can be expressed as products of (positive or negative) integral powers of Hecke Lseries, and are therefore meromorphic. 12. Proof of the Existence Theorem We still have to prove the Existence Theorem (D) of 5.1. Our proof, more traditional than that used by Serre in Chapter VI, works just as well in the local case. If His an open subgroup of C, of finite index [C, : H], we say temporarily ;hat H is normic if and only if there is an abelian extension L/K such that H = NLIKCL. The existence theorem asserts that every open subgroup H of finite index in C, is normic. (We have already shown that if L/K is abelian, then NLIKCL is an open subgroup of C, of finite index; in fact the normic subgroups are just the inverse images of the open subgroups of G(Kab/K) under the Artin map $k : C, + G(Kab/K).) First, two obvious remarks: If Hi 3 H, and H is normic, then HI is normic (the field L corresponding to H has a subfield L1 corresponding to H,). If HI, H, are normic, so is HI n H, (take as the field the compositum Ll w Next, we go to 9.5 in order to prove KEY LEMMA. Let n be a prime, and K ajield not of characteristic n containing the nth roots of unity. Then every open subgroup H of index n in C, is normic.
Proof. In fact, suppose H is open in C, with [Cx: H] = n. Let H’ be the inverse image of H in Jx. Then H’ is open in JK, so there is a finite set S c ‘3Xx such that H’ 3 n (1) x n U, = Us. Furthermore, H is of VES VCS index n in C,, so H’ =I Ji. Therefore H’ 3 n K*” x n U,, = E, say. VS.7 VLS Thus H = HI/K* I EK*JK*, and from consequence 9.5, it follows that H is normic. If L is an extension of K, there is a norm map N: CL + C,; conversely, if we start with H c C, we get a subgroup N‘(H) c C,.
202
J. T. TATE
LEMMA. If L/K is cyclic and H c C.,, and if N$(H) c C, is normic for L, then H is nor&c for K. Proof. Write H’ for N,ii(H) and let M/L be the class field of H’. We claim that M is abelian over K, and NIKIRCM c H, so H is normic. That N,,,C, c H is clear, since N is transitive; the difficulty is to show that M/K is abelian. (In point of fact, if something is the norm group from a nonabelian extension, then it is already the norm group from the maximal abelian subextension (see exercise 8). But this has not been proved hereif it had, we would not need L/K to be cyclic, and we would be already finished with the proof of the lemma.) M/K is a Galois extension since H’ is invariant under G(L/K). The Galois group E of M/K is a group extension, 0 + A + E + G + 1; since E/A N G is cyclic, it is enough to show that A = G(M/L) is in the centre of E. We can use the first part of Theorem 11.5. Let I,+be the Artin map C, + A. To show that A is in the centre, it is enough to check that
l)(x) = aljqx>a l = $(0x) for all x E C, and ts E E. Now @: C, + A has kernel H’, so we want to check that OX/~ E H’, which is clear since N(cx/x) = 1. Proof of the Theorem. (In the function field case we can only treat the case in which the index is prime to the characteristic; for the general case, see ArtinTate, p. 78.) We use induction on the index of H. If the index is 1 everything is clear. Now let n be a prime dividing the index. Adjoin the nth roots of unity to K to get K’, and replace H by H’ = N&(H). By the last lemma, it suffices to consider H’. The index of H’ divides the index of H; we can assume (C,. : H’) = (C, : H), otherwise H’ is normic by induction hypothesis. So n divides (Cx, : H’). Take Hi so that Hi XI H and (Cx, : H’) = n. By the above Key Lemma, Hi is normic. Let L be its class field, i.e. Hi = NLIK, C,. Put H” = N& (H’). Then [C, : H” ] < cc,* : H’]  [C, : H]. (For CJH” N4& C,#JH’ is an injection, whose image is Hi/H’, properly contained in C&/H’.) Hence H” is normic by induction hypothesis; L/K’ is cyclic, so we can apply the above lemma again; so H’ is normic.
203
GLOBAL CLASS FIELD THEORY LIST OF SYMBOLS
The numbering refers to the section of this Chapter where the symbol was first used. Note (i) [A] is the cardinality of the set A. (ii) “ 3 ” denotes inclusion with the possibility of equality. 1.1. K, K* (nonzero elements of K) G(L/K),
4.
JR, J; w
J K,Ss
s
=
mK
JLS, S = 9G
G
; w, WI,. . .
7.3. u
4.1. *
XL VP, Ku, L
No
ET,, us
8.1. h(G, A), h(A) Sunits = KS
h/K
9.1. Conorm = Con
GIV 4.2.
1.2. m2, 2.1. k(v) NV 2.2.
Im$ 5.1. Norm group 5.5. 2 5.7. Q”“, R*,
&K(U)
S 3.1. IS 3.2. NK*/K wS
6.1. I+?, 7.1. w/v, n 40 7.2. G” L”
9.7. inv, inv, (infl) inv, (res) inv, (car) Br (9 10.1. 8,, 0 Artin maps. 10.3. ., cupproduct Gab
CHAPTER VIII
ZetaFunctions
and LFunctions
H. HEILBRONN 1. Characters . . . . . 2. Dirichlet Lseries and Density Theorems 3. Lfunctions for Nonabelian Extensions References . . . . . .
. . . .
. . . .
. . . .
. . . .
. . . .
204 209 218 230
The object of this Chapter is to study the distribution of prime ideals in various algebraic number fields, the principal results being embodied in several socalled “density theorems” such as the Prime Ideal Theorem (Theorem 3) and Cebotarev’s Theorem (Section 3). As in the study of the distribution of rational primes, Lseries formed with certain group characters play an important part in such investigations. Much of what we have to say goes back to the early decades of this century and is described in Hasse’s “Bericht tiber neuere Untersuchungen und Probleme aus der Theorie der algebraischen Zahlenkiirper” (Hasse 1926, 1927 and 1930), henceforward to be referred to as Hasse’s Bericht. 1. Characters Throughout, let k be a finite extension of Q, of degree v. Let S denote a set consisting of the infinite primes (archimedean valuations) of k together with a finite number of other primes (nonarchimedean valuations) of k. S is sometimes referred to as the exceptional set. In slight modification of earlier notation, an id&le x of k will be written as x =&p. * “xpo;xp.+l,. * ‘,xpb;xpb+i,. * .I, where pl,. . . , p,, are the infinite primes, pa+r,. . ., pb are the finite primes of S, and the remaining pi are the primes not in S. Let J denote the idele group. By a character $ of the idele class group we understand a homomorphism from J into the unit circle of the complex plane satisfying (1) $(x) = 1 if x E k* (i.e. xp = x for all p); (2) $(x) is continuous on J (in the idele topology); (3) $(x) = 1 if xp = 1 for p E S and Ix,&, = 1 for p 4 S. As was pointed out in Chapter VII, “Global Class Field Theory”, $4, I/ generates a character of the ideal group Is (the free abelian group on the set of all p # S) in the following way: 204
ZETAFUNCTIONS
AND LFUNCTIONS
205
Let
a = JJ p:’
E Z)
(ai
i>b
be a general element of Zs (so that almost all the exponents ai are 0). For each i > b, let n, be an element of k with IXilp, < 1 and maximal. With a we associate the idele x” having (xyv, = {ii,,
j 1;;‘;
 .’ bs ,**,
then we define x(4 = It/(x?. We observe that although x” is not unique, x(a) is, nevertheless, welldefined function in the in view of property (3) above. Clearly x is a multiplicative sense that for all ideals a, b coprime with S, xW = xWx@). Any ideal character x defined in’ this way is called a Grossencharacter, and such characters were first studied by Hecke (1920) and later, in the idMe setting, by Chevalley (1940). Let S, S’ be two exceptional sets and x, x’ characters of Is, Is’ respectively. We say that x and 1’ are cotrained if x(a) = x’(a) whenever both x(a), x’(a) are defined. This definition determines an equivalence relation? among characters. In the equivalence class of x there is a unique x’ corresponding to the least possible exceptional set s’ (s’ being the intersection of all exceptional sets corresponding to the characters cotrained with x); we refer to this character 2’ as the primitive character cotrained with x. The principal character x0 of Is is defined by x0(a) = 1 for all a E Z’. The principal characters form a cotrained equivalence class, and the primitive member of this class is 1 for all nonzero finite ideals. We shall now look more closely into the structure of a character of the ideal group Z”, with a view to describing the important class of Hilbert and Dirichlet characters. We can write each idtle x in the form x=nx
where each factor x(p) is the idele defined by
P= Pi xw,, = {71’ , P f
Pi*
t To prove the transitivity of the relation it suffices to show that if v(y) = 1 for all idtles y for which yqi = 1, i = 1,2,. . . , m, then I&) = 1 for all idkles x. However, for any E > 0 there is an a E k* such that i= I,2 ,..., m, la  qh < 4 by the Chinese Remainder theorem and thus V(X) = &alx) tends to 1 as I + 0.
.
206
H. HEILBRONN
Then QKd = F $pW, where $, is a character of J defined by II/,(4 = W(Pji and is referred to as a local component of $ (Chevalley, 1940). We note that I,++,is continuous and satisfies (3). We first follow up some consequences of the continuity of JI. Let JV be a neighbourhood of 1 containing no subgroup of J/(J) except 1. By continuity of +, there exists a neighbourhood N’ of 1 in J, say Ixpllp
<sp,
PEE,
(1.1) U2)
lxplp = 1, P 4 E, where E is some finite set of valuations, such that l&v’)
c N.
We choose N’ first to make the set E minimal, and then to make the E~‘S maximal; so that for the finite p in E we have sP < 1. For finite p, (1.1) is now equivalent to saying that xP E 1 +p”+’ where p,, is a positive integer. But since 1 +pPp is a group, so is $,(l +p”p), whence ti+,(l +p@) = 1, and this holds for no smaller integer than pP. We write
f.Y=pPGE LLP”’ and we refer to f, as the conductor of the character x derived from JI. If p is finite and belongs to E, (1.1) implies, by virtue of (3), that p E S. Moreover, if p is finite and not in E, (1.2) shows that it is not necessary, in view of (3), to include p in S. Thus the finite p in E are just those nonarchimedean primes in the exceptional set of the primitive character cotrained with x. Let m be a given integral ideal of k and let x be any character such that f&n. Then if a = (a) where CIE 1 (mod m), x(a)=ti(l,..., = *(al,.
i;l,...,
l;cr,a ,...)
. .,a‘;a‘,.
. .,a1; l,l,.
by (11, =+(a‘,...,
a‘;1 ,...,
l;l,l,...
since al E 1 +p’P for finite p E S. Hence x(4 = ,JT,JlpW where Se is the set of archimedean
primes.
‘1
..
ZETAFUNCTIONS
AND
LFUNCTIONS
207
We now restrict attention to those characters x for which $,(J) is a discrete subset of the unit circle for all p E S,,. Consider a particular p E S,. There exists an integer m such that for all x E J, 9,(x”) = 1 and, in particular, for all x E k*. The completion k, is R or C according as p is real or complex, with each x of k* mapped onto the pconjugate x, of x. However, a homomorphic mapping of C* into the unit circle and having finite order must map C* onto 1 and similarly, a homomorphic map of R*, of finite order, into the unit circle either maps R* onto 1 or maps R* onto & 1 by x + sgn X. Thus, if p is complex, I& is identically 1, and if p is real I,$, is either identically 1 or is f 1; in the latter case, for x E k* we have
If X(~) > 0 for all real p E Se, x is said to be totally positive and we write x $0. Thus, if $ has discrete infinite components, x is a character determined by the subgroup of totally positive principal ideals = 1 (mod m). Such a character is called a Dirichlet character moduIo m; if m = 1, x is called a Hilbert character. Conversely, we shall now show that any character x of the ideal group Is which is 1 on the subgroup HCm) of totally positive principal ideals congruent 1 modulo m (where S consists of the archimedean primes together with the primes dividing m) arises in this way from an idele class character JI with exceptional set S. Let x be such a character, and let x*, xi be the restrictions of x to the group AS of principal ideals coprime with m, and to the group B(‘“) of principal ideals 3 l(mod m) respectively. Then xi(~) is determined by the signs of the real conjugates of a. We extend this definition to all elements tc of k* in the obvious way. Then we write xz = x*x;’ ; clearly xz is a character of AS equal to 1 on B(“‘). We define $ corresponding to x by stages. For any idele x, and each p not in S, define where? @‘P]]x~. Il/,w = X(P”P) (1.3) This part of the definition ensures that $ satisfies (3). We next consider together all the finite primes of S, that is, all p E SS,,. We choose y E k* such that y = xp (mod p’p) for each p E SS,, where p’pllm (this is possible by the Chinese Remainder Theorem). We then define p JIso~pb~
= X2(Y)
for those ideles x having 1x,1, = 1 for all p E SS,,. t pvjlxp means that xp E p”  @“+I.
(1.4)
208
H. HEILBRONN
To complete the definition of II/ on this subset of J, we need to define tip(x) for P E So. For real p E S,, let x”‘) be the character on k* defined by fp) (a) = sgn a p. Then x1 is the product of some subset of these characters xcp), say (1.5) Xl =pil$p) CT = So) and we define tip(x) for p E SO by (1.6)
provided xp E k*, and we extend this definition to k: by continuity. We postpone defining $ for those ideles x for which [x,1, # 1 for some PCSSO. It is clear from our construction that $ is multiplicative. We proceed to verify that $ satisfies (1) and (2), (3) having been satisfied by construction. Consider (1) first. Then ,g,Jlpw
= x; %>
p JIsotipw
= x34
by (1.5) and (1.6),
by (1.4), and pIjJshc4
= x(4 = x*(4
by (1.3). Since x* = x1 xz, $ is seen to be 1 on k*. It remains to consider (2). But I&) = 1 if P E so, Ixpllp < 1, xpE1+p”p, P ESS,, lslp = 1, P 4 s, and this is an open set in the idele topology. Finally, we need to extend the definition of $ to ideles x for which, for some p E SS,, lx,j, # 1. Let x be such an idele and suppose that p E SS,,. If p”pllx, (where up is, of course, not necessarily positive), choose a E k* such that for each p E S SO. V”pIIa We now define W = #(ax), noting that the righthand side has already been defined. Hence Ic/ is well defined; for, if jI is another element of k* such that p“~1 ID for each p E SS,,, ILO
= WW~@x)
= 4X4,
ZETAFUNCTIONS
AND
209
LFUNCTIONS
since j?/u E k* and $ is multiplicative. This completes the definition of JI as an idHe class character. We recall that Is is the group of all ideals of k relatively prime to S, AS denotes the subgroup of principal ideals in I’, B(“) the subgroup of AS consisting of those principal ideals (a) where ct E 1 (mod m), and HCm) the subgroup of B(“‘) consisting of those principal ideals (a) which satisfy a g 0. CL= 1 (mod m), We note that H(“) is the intersection of the kernels of all the distinct Dirichlet characters modulo m, and the number h, of such characters is thus the index of H+) in P. The index of AS in Is is equal to the socalled absolute class number h. The index of BCm) in AS is the number of units in the residue class ring mod m, denoted by d(m); and the index of H’“’ in B”“’ is the number of totally positive units G 1 (mod m), equal to H/(h4,(m)) where 41(m) is the number of residue classes mod tn containing totally positive units and H is the number of ideal classes in the “narrow” sense, i.e. relative to the subgroup of 1’ consisting of all totally positive principal ideals of k. Thus the number h, of distinct Dirichlet characters mod m is given by
2. Dirichlet Lseries and Density Theorems In this section we shall take “character” to mean “Dirichlet character”. If x is a character, we define x on S by x(p) = 0 if p E S. Let a denote a general integral ideal, with the prime decomposition of a in k given by a = JJ P”P, vP 2 0, vP = 0 for almost all p. By N(a) we shall understand the absolute norm of a, i.e. N&a). We define the Dedekind zetafunction &(s) by &)
=&
N&s =!(Ij&q.)’
(s=d+it),
and with each character x we associate a socalled Lseries
Since each rational prime p is the product of at most [k : Q] primes p, the convergence of both sum and product in each case, and the equality between them, for CJ> 1 all follow from the absolute convergence of sum and product fora> 1.
210
H. HEILBRONN
We observe that L(s, x0) and c,(s) differ only by the factors corresponding to the ramified primes, and that &Js) is the Lseries corresponding to the primitive character cotrained with x0. As in classical rational number theory, Lfunctions are introduced as a means of proving various density theorems (such as Theorem 3 below). An important property of these functions is that their range of definition can be extended analytically to the left of the line G = 1. The continuation into the region d > 1  A can be effected in an elementary
way (using only Abel
summation and ihe fact that c(s) is regular apart from a simple pole at s = 1) by virtue of the following THEOREM
1
where K depends on the degree, class number, discriminant and units of k. We indicate briefly the proof of this result. Let C denote a typical coset of I?“‘) in I’. Then the sum considered in the theorem is equal to
and since
it suffices to estimate the inner sum. Let 6 be a fixed ideal in C‘. Then a6 = (a) where a is a totally positive element of k such that a E 1 (mod m). As a varies, a is a variable integer in the ideal 6. Also, IW>I = [N(a))1 = WOJ@), so that the inner sum may be written in the form ,N(&TXN(b)l aeb
where the asterisk indicates that a 3 1 (mod m), a 9 0, and that in each set of associates only one element is to be counted. The integers a of the ideal 6 may be represented by the points of a certain ndimensional lattice, and the problem is essentially the classical one of estimating the number of these points in a certain simplex (Dedekind, 187194; Weber, 1896 and Hecke, 1954). The estimate takes the form?
t
A
better error term can be derived as in Landau
(1927),
Satz 210.
ZETAFUNCTIONS
AND
211
LFUNCTIONS
A result of this type may also be derived for Grossencharacters, but the proof is complicated (Hecke, 1920). As in classical theory, deeper methods lead to analytic continuation of Lfunctions into the whole complex plane, and thence to a functional equation for such functions. In the case of the ordinary Cfunction, one obtains? co
sqY; i(s)Ll 0
72
;+
1 s
(x *“~+x~sl)~(e(x)
1)dx = Q(s),
1
where O(x) =
g enm2x, WI=03 and one derives at once the functional equation Q(s) = ar(l s). Hecke (1920) extended this classical result in a farreaching way to Lseries formed with Grossencharacters; and, more recently, Tate (Chapter XV) generalized Hecke’s result to general spaces. For Dirichlet characters, Hecke’s result is summarized below (Hasse’s Bericht, 1926, 1927 and 1930). Let
then Q(s,x) is meromorphic
and satisfies the functional
equation
w, x> = w’(xw  s, 2) where W is a constant of absolute value 1, and d denotes the discriminant; riFrz are the numbers of real and complex valuations respectively; a4 = 0 or 1 according as the value of x in the domain of all principal ideals (N) with a E 1 (mod fJ does or does not depend on the sign of the qth real conjugate of a. An explicit expression for W is given, e.g. in Hasse’s Bericht (Section 9, Satz 15); its presence in the functional equation leads to interesting information of an algebraic nature, for instance, about generalized Gaussian sums (Bericht, Section 9.2 et seq). It follows from the proof of the functional equation that L&x) is an integral function, except for a simple pole at s = 1 when x = x0. We shall see that information about the zeros of Lfunctions plays an important part in applications to density results. The distribution of zeros in the “critical” strip 0 < u < 1 is particularly important. According to the t Titchmarsh
(1951).
212
H. HEILBRONN
generalized Riemann hypothesis, L(s, x) # 0 if u > 3; but this conjecture is far
from being settled. Nevertheless, significant weaker THEOREM
information
can be extracted from the much
2
if aal. us, xl # 0 Proof. In view of the product representation of L for u > 1 we restrict ourselves to the case cr = 1. The proof falls into two parts, the due to Hadamard, the second to Landau. We suppose on the contrary 1 + it is a zero of L(s, x). (i) (Hadamard) Assume that x2 #x0 if t = 0. If CJ> 1, we have, from product representation, that
may first that the
No L(s, x) = exp Fmzl ~N(p)muei’miog . 1 > When p is prime to fx, we write x(p) = e’“p, & = t log N(p) + cp and consider the function IL3(a, xo)L4(a+
it, x)L(a+2it,
= exp
x2>/
1 1 pprimetofxm=l
f
! N(p)““(3
+4 cos m/3, + cos 2mfl,)
m
1
since 3 +4 cos o +cos 20 > 0 for all real w. Keeping t fixed, we now let CJ+ 1 +O. Of the terms on the left, the first is O((a 1)3), the second is @(a 1)4) by hypothesis and the third is O(1) because if t = 0, x2 # x0. Hence the expression on the left tends to 0 as 0 + 1 +0, and we arrive at a contradiction. (ii) (Landau) It remains to consider the case when x2 = x0 (so that x is real) and, if possible, L(1, x) = 0. We consider the product &(s)L(s,
x) = T N(a)“l(a)
= 5 a,. u’ u=l
where a, = C 44 N(a)=”
and
%a) = C x(b) = n (1 +X(P) + .   x(P”)) 610 PTb where p”lla signifies that m = mp is the highest exponent to which p divides a. Moreover, if mp is even for all p dividing a, n(a) > 1.
ZETAFUNCTIONS
AND
213
LFUNCTIONS
Hence, if CJis real and both series converge, c N(a)“ql) 2 T N(a)% a We now utilize the following simple result from the theory of Dirichlet
(2.1)
series :
A Dirichlet series of type
f(s)=“~l%.c
a,20
(24=1,2,...),
has a halJlplane Re s > Q,, as its domain of convergence, and ij c,, is finite, then f(s) is nonregular at s = ran. (Titchmarsh, 1939, see the theorem of Section 9.2.) By hypothesis, f(s) = Ck(s)L(s, x) is everywhere regular (the hypothetical zero of L(s, x) at s = 1 counteracting the pole of &(s)); hence the series on the left of (2.1) converges for all real rr. However, the series on the right of (2.1) is equal to &J20) which tends to + co as c + 4 + 0. Hence we arrive at a contradiction. The Landau proof is purely existential, whereas Hadamard’s argument can be used to yield quantitative results about zerofree regions and orders of magnitude of Lfuncti0ns.t We are now in a position to prove THEOREM
3 (Prime Ideal Theorem) x = x09 NJ< F)
= x f x0*
Proof.
For Re s > 1, form the logarithmic
derivative of L(s, x), namely
 ;’, =;m~lNP)mslog N(p). x(P”)
where g is a function regular for Re s > 3. Our first, and major, step will be to estimate the coefficient sum &
xx(~> log N(p).
One way of doing this is to express the sum as a contour integral of 2 L’(s, xl ~ s J&x) t See Estennann
(1952)
and Landau
(1907).
214
H.HEILBRONN
along the line (cice, c+ ico) with some c > 1, and then to shift the line of integration to the parallel line Re s = 1, with an indent round s = 1. This we now know to be the only point at which a pole can occur (and does occur precisely when x = x0, the residue in this case then leading to the dominant term). For an account of this treatment, see Landau (1927). It is clear from even this sketch that the nonvanishing of L(s, x) on Re s = 1 is vital. An alternative approach is via the WienerIkehara tauberian theorem : Suppose that
are DirichIet series, convergent for Re s > 1, regular on Re s = 1 with simple poles at s = 1, of residue 1 in the case off, and q in the case of g, where 1 may be 0. Assume that there exists a constant c such that lb,,,] =Sca,. Then m&h
 v6
as x + 00.
We apply this result with
f(s)=  c’(s> g(s)=  us,  xl C(s) ’ L(s,x>’
c=Fk:
Clearly q = 1 when x = x0, and otherwise q = 0. Finally, it is an easy matter to show that
e.g. by partial summation; and this proves the theorem. (The tauberian approach is described in Lang (1964).) It is worth remarking that the analytic method can be made to give sharper estimates of the error terms (Hecke, 1920). Both methods extend to Grtissencharacters. In close analogy to Dirichlet’s famous proof of the infinitude of rational primes in arithmetic progressions, one can show that every Dirichlet ideal class C contains infinitely many primes p of k. Indeed, by means of the prime ideal theorem we can show that the primes p are equally distributed among the classes C; we have, in particular, that THEOREM 4
N(g  h’m lg? VGC Proof: We have only to remark that
; j(C)&) =
x+00.
ZETAFUNCTIONS
By virtue of this orthogonality
AND LFUNCTIONS
215
formula, we have
on the other hand, the sum on the right is asymptotic by the prime ideal theorem. If B is any set of ideals in k, and
to x/log x as x + co
lim 10gx card (p E BIN(p) < x> = x+co x exists, then I is called the density of primes in B. Thus the density of the set of all primes of k is 1. To investigate the density of primes in a given set, it suffices to consider only primes p of absolute first degree, that is, those p for which N&p) is a rational prime; for all primes p whose norms N(p) are powers of rational primes greater than the first, and satisfy N(p) < x, are 0(x*) in number. For the same reason we may clearly disregard the finite number of ramified primes. We shall use this remark in proving the next result, known in classical parlance as the first fundamental inequality of classJield theory (see Chapter VII, “Global Class Field Theory”, $9, and Chapter XI, “The History of Class Field Theory”, $1). THEOREM 5. Let K be a finite normal extension of k, and denote [K : k] by n. Let S be the exceptional set in k. K determines the subgroup Hs of I’, offinite index hs in I’, composedof those cosetsof H” which contain norms (relatiue to k) of ideals of K coprime with S. Then
h, < n. Proof. We identify l/n and l/h, as the densities of two sets of primes in k. First of all, if C is the set of all primes p (of k) in Hs, then, by the pre
ceeding theorem, the density of C is l/h,. Next, since K/k is normal, the prime decomposition of any p in K is of the form P=(r)1 ..rpr) where all the primes ‘$3, have the same degree f relative to y, and efr = n.
Since the number of ramified p is finite, we need consider only primes p for which e = 1. Now consider the set B of all primes p whose prime decomposition in K is characterized by e =f = 1, that is, by r = n. On the one hand, by the prime ideal theorem for K (noting that the primes I are all the primes of K of absolute first degree) the density of B is clearly l/n; on the other
216
H. HEILBRONN
hand, if p c B and ‘Qplp, then N,&J) = p, so that B c H,. The result follows at once. Note. It is worth remarking that Theorem 5 can be proved without appeal to the relatively deep Theorems 3 and 4 and, in particular, to the nonvanishing of L(s, x) on Re s = 1. A more elementary argument, using only real variable theory, runs as follows: Let s be real and > 1. By Theorem 1 and partial summation we have lim (s  l)L(s, x0) = 1c s1
(a> and
lim L(s, x) exists and is finite if x # x0.
@I
s1
From the product representation
of L(s, x) it follows that
logus,x>= c g$ + d%x> where g(s, x) is a Dirichlet series absolutely convergent for s > 3. Using the orthogonality properties of the characters x, we conclude that
where M(s) = x;xoIlog Vs, xl  ds, xl> + log (s  WXs, xo) ids, x01. By (a) and (b), limsupf(s)
is not + co and, indeed, is finite unless one of the
Sr1
L(s, x) (x # x0) vanishes at s = 1. Now let K be as described in the statement of Theorem 5. We have, analogously to (a), that liiy Cx(s).(s 1) exists and is finite. Taking logarithms
of the product representation
where lim G(s) exists and is finite. s1 B c H,, we obtain
of &(s), gives
Subtracting
(d) from (c), and using
+f(s)G(s)>0 for all s > 1. Letting s + 1 +0, the result follows. For the next theorem we require some results from classfield theory.
ZETAFUNCTIONS THEOREM
AND
LFUNCTIONS
217
6. Zf K is a finite abelian extension of k, then MS) = lJ m, x, k) x
(2.2)
where, in the notation of the preceding theorem, the product on the right extends over the primitive characters cotrained with the characters of the class group Is/H,. Proof. The proof is carried out in terms of local factors, and we consider separately nonramified and ramified primes p of k. (i) Let p be a nonramified prime of k, so that P=%***rPr where Ipi,. . ., Yj3,are distinct primes of K. From class field theory, NK/Q(rZi)
where If= [K : k] = n. Thus the corresponding whilst the corresponding
=
Nk/Q(P>f
local factor on the left is (1 N(p)fS)“‘f, local factor on the right is IJ (1 xtPvw9“)‘.
Since f is the least positive integer such that x(p/) = 1 for all x, we have the easily verifiable identity (take logs of both sides and use hs = n) (lJ+“” = n (lx(p).y)‘; x from this, with y = N(p)“, equality of the local factors follows at once. (ii) The proof for ramified primes is more difficult, and depends on the functional equations satisfied by the various Lfunctions. We begin by writing M)
= g(s) n us, x9 42 x and prove that g(s) is identically 1. From above, g(s) is equal to the finite product over ramified p of the expressions n: (1 XtPYw“)
“IJ(1NW”1 * If this product is not constant, then it has a pole or zero at a pure imaginary point it,, t,, # 0. In view of the functional equations, g( 1  s)/g(s) is a quotient of gamma functions and so can have only real poles and zeros. Thus 1 it,, is also a pole or zero of g. But we know that 1 it,, is not a singularity or zero of any of the Lseries or of &(s). Hence g is constant, and so equal to 1.
218
H.
Example
unity.
HEILBRONN
1.7 Let k = Q, K = Q(o) where o is a primitive
mth root of
Then
IAS) = C(s) n us, xl x where the product on the right extends over the primitive cotrained with all nonprincipal rational characters mod m. Proof. See Chapter III, $1. Example 2. Let k = Q, K = Q(Jd) w h ere d is the discriminant
characters
of K. Then
g denotes the Kronecker symbol (Hecke, 1954, Chapter VII). 0m We touch upon another consequence of this theorem. We recall that the functional equation of an Lfunction contains a factor where
{pp(f,v’2. Applying the functional equation to (x(s) on the left of (2.2) and to each of the Lseries on the right, one can derive the relation
where D is the discriminant of K (Hasse’s Bericht, Section 9.3). If we assume for the moment that k = Q, we obtain n fx = Disc (K/k). (2.3) This inference is not as ea,“, to justify in the general case when k # Q, because two distinct ideals in k can have equal norms. However, it can be proved (Hasse’s Bericht, Section 9.3, formula (12) and Note 44) that (2.3) is valid in general. 3. Lfunctions for Nonabelian Extensions Suppose now that K is a finite, normal but not necessarily abelian extension of k of degree n. The problem is to develop in this case an analogue of the above theory of Lseries formed with abelian group characters. As usual, let G be the Galois group of K over k. Let {Mh)},, o be a representation of G into matrices over the complex field. Thus to each element p of G corresponds a matrix Mb). The character ~(,a) of p is defined to be t By (2.2), since &Jr) and L(s, x0, k) = C*(S) have simple poles at s = 1, and the remaining L(s, x, k) on the right are regular there, it follows that these L(s, x, k) do not vanish at s = 1. In particular, in the situation of Example 1, it follows that ail the Lfunctions formed with nonprincipal Dirichlet characters mod m do not vanish at s = 1. This is an alternative proof of the key step in the proof of Dirichlet’s theorem on the infinitude of primes in arithmetic progressions mod m.
ZETAFUNCTIONS
AND
LFUNCTIONS
219
the trace of il4@) and depends only on the conjugacy class (p) in which /J lies. Two representationst {M&)},,c, (Nb)},, o are said to be equivaht if there exists a nonsingular matrix P such that PMQP1
= N(p)
for all p E G.
A representation {Mb)} is said to be reducible if and only if it is equivalent to a representation {AQ)} such that for all p E G, where (N(‘)Q}, {N(2~Q) are themselves representations of G. The character of an irreducible representation is said to be simple. From the general theory of group representations (for an account of this theory see e.g. Hall, 1959) the number g of simple characters of G is equal to the number of conjugacy classes of G, and the following orthogonality relations are known to hold: (3.1) in particular &xQ
= {;*
;;;;z;, , , where the principal character is the character of the representation M(P) = 1 for all p E G, to be denoted henceforward by x0. Also, if el, $2,. . . , $,, are the simple characters of G, then (3.2) where I,, is the number of elements in the conjugacy class
In (3.3)
where n, is the degree of JI, (that is, $i is the character of a representation
by
nl x ni matrices).
If G is abelian, then each I,, = 1, so that g = n and each n1 = 1. Thus each character er is a homomorphism of G into the unit circle, and so an abelian character. Let ‘$3be a nonramified prime of K, and let p be the prime of k lying under ‘p. We shall denote the Frobenius automorphism of K/k relative to ‘$ (for t It should be clear from the context that N is not used here to denote a norm1
220
H.
definition
HEILBRONN
see Chapter VII $2) by
is the characteristic polynomial of M(p)so that I denotes the unit matrixwe take as local factor of our Lfunction the expression
We note that this definition cannot be extended to ramified ‘$3 since there is no corresponding Frobenius automorphism. It is important to note that this local factor depends only on the character x of the representation, and not on the explicit representation matrix. For any similarity transformation of Mb) leaves its characteristic polynomial invariant and, since M@) has finite order, the Jordan canonical form of M(p) must therefore be a diagonal matrix all of whose diagonal elements are roots of unity. Thus, without loss of generality,
may be taken to be
The local factor is then equal to
b
since c E;” is equal to the trace of { M r[]
)“which,
by definition,
is equal
We collect up the local factors corresponding to nonramified define the socalled Artin Gfunctiont essentially by
‘$3 and
i=l
tax
K/k m . ([ v I>
L(s, x) = L(s, x, K/k) = P we shall introduce later a factor derived from the ramified primes. t See also Artin (1930).
ZETAFUNCTIONS
AND
221
LFUNCTIONS
We now list a number of observations about L(s, x): (I) L(s, x) is regular for u > 1, since the product is absolutely and uniformly convergent in every closed subset of the halfplane Re s > 1. (II) If K/k is abelian, and if x is simple then, apart from the factor corresponding to the ramified primes, this definition coincides with that given in Section 2 above. (III) Suppose that n is a field intermediate between K and k, normal over k. Let H = Gal(K/Q), so that H is a normal subgroup of G and G/H = Gal (O/k). Then if x is a character of G/H, it can be regarded in an obvious way as a character of G, and L(s, x, K/k) Proof.
= L(s, x, n/k).
Take the character over G defined by the representation iw4=
MPm
We have ~I%
E xN(P) (mod p)
for all integers X E K, and in particular normal extension of k,
(3.5)
for all integers X E 0.
Since Q is a
for all p E G.
XEa*x~ER
Hence (3.5) is a congruence in a, and since ‘$ is unramified, in R we have
if ‘$ lies over q
Hence also
i.e.
[Klk 1 
cp
,c~l”
H is the Frobenius
E XNcp) (mod q),
automorphism
in R.
Thus
jlM’(
F])N(p)“1
= IIM( = pvf
(IV) Suppose that x is a nonsimple
F]
H)N(&‘1
(F])
N($yl.
character of G, say x = xi +x2. Then
us, xl = us9 x&o, since log L is linear in x by (3.4).
x2),
222
H. HEILBRONN
(V) Suppose again that R is a field intermediate between K and k, this time not necessarily normal over k. Let H = Gal (K/Q, and suppose that is the partition of G into right cosets of H. To each character 2 of H there corresponds an induced character x* of G, given by (3.6) C XC% PC ‘13 PEG; x*w = i lt,pa, 1Ea then J% x*, K/k) = Us, x, W3. Proof.? Let !JJ be a nonramified prime of K, and p the prime of k under ‘$3. Suppose that, in R, p has the prime decomposition P =ifJqi9
N*/Q(Si)
=
CNk/QP)
ft.
9
and suppose that r, is an element of G such that q1 lies under oils. (Hasse’s Bericht, Section 23, I) G can be decomposed in the form
where
From the definition
since (z~~~z;‘)~ The logarithm
(3.7) Then
po= .rp1 [Klk
of induced character given above, we have
E H if and only iff,lm. of the pcomponent (p nonramified)
t For an alternative proof see Hasse’s Bericht, Section 27, VII.
of L(s, x*, K/k) is
ZETAFUNCTIONS
AND
LFUNCTIONS
223
by (3.7), on writing Ott =f;t in the inner summation; and this is nothing but the sum of the logarithms of those qcomponents of L(s, x, K/Q) which, by (3.7), correspond to p. We proceed to consider some special cases of(V). V(i) n = K. Then H = Gal (K/Cl) = (1) and we have only one character, the principal character x0. In this case, then, L(s,x,,,K/Q reduces to [k(s). By (3.6) the corresponding induced character, x0*, of G is given by n if p is the unit element of G, xx4 = 0 otherwise. (3.8) Let $1, 11/2,. . . ,1,4#be all the simple characters of G. By (3.2), with p’ = 1, we have I1 = 1 and arrive, by (3.8), at j$l$iQh(l)
= XZPh
We observe that, of course, $i(l) must be a positive integer. application of (IV), it follows that CKts) = if! Us,
$is
KlkY”“s
(3.9) By repeated (3.10)
V(ii) R = k. By (III) (with R = k and therefore G = H), Us, xo, K/k) = Us, xo, k/k) = tk(S). We shall now deduce from the preceding theorems the remarkable result that a general Artin Lfunction L&x, K/k) can be expressed as a product of rational powers of abelian Lfunctions L(s, $, K/S2), where the R’s are fields intermediate between k and K with K/Q abelian. With the notation used in (i), each character x of G can be written in the form (3.11) Hence, by repeated where the r,‘s are nonnegative rational integers. application of (IV), it suffices to consider the simple Artin Gfunctions Us, Icli, K/k). Let H be any subgroup of G, and let l, run through the simple characters of H. Each r, induces a character c$’ of G, given, let us say, by for all p E G, (3.12) tjYPl= T rji$ioL> in accordance with (3.11). The restriction of $, to H is, itself, a character of H and therefore has an expression of type (3.11) in terms ofthe rj’S. Moreover, by the theory of
224
H. HEILBRONN
induced characters, this representation Iclitz> = T
‘;*
actually takes the form all r E H,
tjCz)9
(3.13)
where the coefficients rjl are the same as in (3.12). Now take H to be a cyclic subgroup of G. Then if R is the subfield of K consisting of elements invariant under H, K/Cl is abelian and so, by (V), Us, 579 Klk) = Us, tj, K/a), which is an abelian Lfunction. To prove our result, it therefore suffices to express each ei as a linear combination of characters of type $. Each element y of G generates a cyclic subgroup H,, of G and so, by (3.12), denoting the simple characters of H,, by &,, we have Cr*;jQ =i$l~y;ji~ioc),
all CCE G
(3.14)
The system of equations (3.14) is described by a matrix with each fixed pair (r,j) determining a row and each i a column and we shall prove that its rank is equal to g. Suppose on the contrary that the rank is less than g. Then the columns are linearly dependent, i.e. there exist integers cl, c2,. . . , c8 not all zero, such that = O
$Try;ji
Thus, by (3.13), interpreted
for all y E G and all j.
in relation to a particular
7 ci$i(r) = T ct T ru;ji &j(r),
In particular,
= 0, taking r = y, we arrive at T C,$i(y) = 0
HY,
all 7 E H,, all r E H,.
for all y E G,
(3.15)
contradicting the linear independence of the simple characters $r, Jlz, . . . , $#. Following on from (3.1!9, multiply this relation by ij,Jr) and sum over all the elements y of G. From (3.1) it follows that nc, = 0 (k = 1,2,. . .,g) (3.16) whence c1 = c2 = . . . = ce = 0 (so that, incidentally, we have here another way of arriving at a contradiction). Either way, we have now that the system of equations (3.14) can be solved for pi, giving @i =,TGT where the coefficients uy; ji E Q.
u7;ji$j
ZETAFUNCTIONS
AND
LFUNCTIONS
225
The argument centering on (3.16) can be carried out modulo any prime p which does not divide n, whence it follows that the coefficients u,:,~ have denominators composed of prime factors of n. We have thus proved the following THEOREM 7. For any character x of G, the Artin Lfunction L(s, x, K/k) is given by L(S9 X9Klk) = JJ JJ L(s, 51j9 K/QIY’ where each Gal (K/a,) is cyclic, the &,‘s are (abelian) characters of Gal (K/R,) and each qj is rational with denominator composed of prime factors of n. It has been proved by Brauer (1947) that the exponents in the above theorem can be taken to be rational integers, and it follows, in particular, that the Artin Lfunctions are meromorphic. Furthermore, if x is nonprincipal, then the r’s in the above product representation of L(s, x, K/k) can also be chosen nonprincipal. Hence Artin Lfunctions formed with nonprincipal characters are, in addition, regular and nonzero for 0 > 1. If, on the other hand, x = x0 is the principal character, then L(s, x0, K/k) has a simple pole at s = 1. Artin has conjectured that, apart from this simple pole, in the case x = x0, the L(s, x, K/k) are integral functions. This conjecture would imply that &(s)/&.(s) is an integral function whenever k c K. If K/k is normal, this is true by (3.10), (V(ii)) and a fundamental theorem on group characters due to Brauer (see Lang, 1964 p. 139). Artin’s conjecture has been verified when G is one of the following special groups: S3 and, more generally, any group of squarefree order; any group of prime power order; any group whose commutator subgroup is abelian (Speiser, 1927); S, (Artin, 1924). The conjecture has not been confirmed for G=A5. By Theorem 7, each Artin’s Lfunction L(s, x) is seen to satisfy a functional equation by virtue of the fact that each abelian Lfunction occuring on the right satisfies its own functional equation. This equation will be of the form w, xl = wx>w  s, Z) where W is a constant of absolute value 1, and Q, is of the form (9(s,x) = A(X)SI (;)‘or(f+)L(s,x) with a, b in Q and A a positive constant. At this point we recall that, in the definition of Artin’s Lfunction, we omitted to include a factor corresponding to the ramified primes. Now we introduce on the righthand side of (3.17) the ramified local factors corresponding to the abelian Lfunctions (see beginning of Section 2), and take the new product as the definition of L. Then the above functional equation is satisfied automatically by the redefined Lfunction.
226
H. HEILBRONN
What is lacking in the theory of Artin Gfunctions is a nonlocal approach (provided, in the abelian case, by a series definition of Lfunctions). This lack causes the difficulties in extending these Gfunctions to the complex plane. Example:
The case G = S,.
The elements of S, fall into three conjugacy classes Cl : (1); c, : U,2,3), (3,2,1>; c, : &2), (2,3), (391). Hence there are three simple characters. Let 11/l be the principal character and 1,4~the other character determined by the subgroup C, v C,. These are both of first degree and so, by (3.3), $3 is of degree 2. Thus, by (3.2) with p’ = 1, we have that
Hence the table of simple characters of S, can be set out as follows: +1 $2 $3 1 1 2 1 1 1 c2 1 1 0. c3 Working from (3.6), we obtain the induced characters x* corresponding the characters x of (i) H= A,: Cl
(ii) H=
(iii) H=
xt 2
c2
2 2
c3
0
0
Cl
xx 3
c2
0
C3
1
xs 3 0 1;
Cl
x2 6
Cl
1
c2 c3
From these tables we see that x:
=*I+$29
x:=IcI1+J13*
0 0.
2
1
{1,(1,2)}:
(1):
x: 0;
to
ZETAFUNCTIONS
AND
LFUNCTIONS
227
Ss is the Galois group of any nonabelian normal extension K of k of degree 6. Let Q2,, sZzbe the two intermediate fields tied under the subgroups A,,{(1),(1,2)} respectively. Thus R, is a quadratic extension and Qz a cubic extension of k, and K is therefore an abelian extension of each of them. Also, Sz, is an abelian extension of k. Thus, by (V(ii)), and (IV), MS) 1 ~(s’~xo$/I<) llrl C,,(s)
= Us, = 41
L,<s>
$1
= Us, +I+ tiz + %
K/k)
JI3’
xo, K/W
= Us,
$I+
$2,
= Us,
@I +@a
K/k)
L*,,
z ;G,,, 91
WM
K/k)
$3’
and C,(s) = Us, xo, K/k) = L,,. We remark that L,, = Us, $2, K/k) = Us, x5, Wk) by (III) and that L,, = Us, Jls, K/W = Us, xz, W4) by (V). Hence L,, and Le3 are integral functions. Incidentally, this example verifies Attin’s conjecture for Gal (K/k) = S,. We conclude this brief survey of the nonabelian case by quoting cebotarev’s density theorem: Let K/k be normal and let G = Gal (K/k). Let C denote a given conjugacy class in G. Then the class of all nonramified primes p, each having the property that equal to
[Klk 1 
E C for some? prime ‘$3 of K lying over p, has density
rp
card C card Using arguments of the kind described above (see Section 2), this result follows from the fact than an Artin Lfunction has no zeros on the line Res= 1. Example. As an illustration of the theorem, let us consider the case of a nonnormal cubic extension KS/k. A (nonramified) prime p in k factorizes in K3 in one of the following three ways: (1) P = rp,cp,% (2) p = !Q, !& where the deg !JJJp = 1, deg ‘pz/p = 2 (3) P = % t In fact, C consists of the Frobenius $3, lying over 4.
automorphisms
c
Klk 5
1
corresponding
to those
228
H.H&LBRO~
We wish to determine the density of primes p falling into each of these categories. Let K6 denote the minimal extension of K3 that is normal over k. We study the factorization of p in K6, bearing in mind that in any decomposition of p into primes of KS, these all have equal order. Thus in (1) p decomposes either in six primes (in KS) of degree 1, or !&, ‘pz, !& are themselves primes of KS, each of degree 2. The latter possibility is ruled out because KS/k is not normal. [Suppose, on the contrary that, each ‘$, remains a prime in Ke, and hence is of degree 2 in K8. Write oi=
[
Klk . Pi
I
Then each crl is of order 2, and hence is a 2cycle (note that Gal (K,Jk) = S,). Further, the u,‘s span a conjugacy class, and hence are distinct transpositions. Now K3 is the set of elements of KS invariant under some 2cycle, say ul. Thus d’;Pi n KS) = ‘% n KS. But since there are three distinct prime ideals in K3, and since each has a unique extension to Ke (for each remains a prime in K8),
(i = 1, 2, 3). a1’Pi = rp, Let the subfields invariant under uz and a3 be Kj and Ki. we have (since Kj and Kg are the conjugates of K3) But the transpositions
(i,j, OjVP = Pi generate S,. Hence
Thus there is no automorphism
Then, in a similar way
= 1, 2, 3).
s,%Ji = IPrz such that 4%
=
%
and this is impossible.]
In (2), we must have p = q1 q, ‘$J1 as the prime factorization over KS, each factor being of degree 2, with q, qz = !&. In (3), either p remains prime, or p decomposes into two primes of degree 3. The former would imply that G/k generates S3, which is impossible. Hence the latter decomposition F1P must hold. We now apply the fact (Chapter VII,
92) that the order of
K,lk [ q1 
equals the degree of q. From this it follows for each prime q of K6 lying over p that &lk is an icycle in case (i) (i = 1,2,3). Hence, by Cebotarev’s [ 9 I
ZETAFUNCTIONS
AND
LFUNCTIONS
229
theorem, the required densities in cases (I), (2) and (3) are 6, 3 and 3 respectively. The following result is a direct consequence of Kummer’s theorem (see Chapter III, Appendix, or Weiss (1963), 4.9 and in particular 4.9.2). “Supposefg k[x], is irreducible, and 8 is a zero off. Let p denote a prime of.k. Then, for almost all primes p of k, p splits in k(8) asf(x) splits over the residue class field of k module p.” Now let n(p,j’) denote the number of solutions of the congruence f(x) = 0 (mod p); then, for almost all p, n(p,f) may be regarded, alternatively, as the number of prime factors from k(B) of p having degree 1 relative to k. Applying the prime ideal theorem (Theorem 3) to primes of k(B), we arrive therefore at the following THEOREM8. Using the notation defined above, &/(PYf) Corollary I. factors, then
 log ? x
More generally, if
asx,co.
f is the product of I distinct irreducible
Corollary 2. Ift for almost all p, f splits completely mod p, then f factorizes completely over k[x]. Proof. If n’ denotes the degree off, then we are given that n(p,f) = n’ for almost all p. Hence, by the prime ideal theorem for k, I = n’.
It would seem plausible that if f has at least one linear factor mod p for almost all p, then f has at least one linear factor in k[x]. However, the following counterexamples show this to be false. f(x) = (x2a)(x’b)(x’c) (0 where abc is a perfect square in Z, with none of a, b, c a perfect square. For, if a, b are quadratic nonresidues mod p, then c is a quadratic residue modp and hence f has two linear factors modp. (ii) f(x) = (x2+3)(x3+2).
For if p s 1 (mod 3), x2 +3 factorizes mod p, and if p = 2 (mod 3), x3 + 2 factorizes mod p. However, we do have the following, THEOREM 9. If f is a nonlinear polynomial over k which has at least one linear factor modulo p for almost all p, then f is reducible in k[x]. Proof. Assume that f is irreducible. Then it follows from Theorem 8 and
230
H. HEILBRONN
the prime ideal theorem for k (Theorem 3) that N(&xMP9f’
1) = +&).
Hence, since n(p,f) > 1 almost everywhere, n(p,f) = 1 almost everywhere. Let K be the root field off over k, and q a prime ideal in k which splits totally in K. Then q splits totally in k(8) and n(q,f) is equal to the degree of f, with a finite number of exceptions. These q have positive density by the prime ideal theorem (Theorem 3) in K. Hence f is linear. REFERENCES
Artin, E. (1923). uber eine neue Art von LReihen. Abh. math. Semin. Univ. Hamburg, 3, 89108. (Collected Papers (1965), 105124. AddisonWesley.) Artin, E. (1930). Zur Theorie der LReihen mit allgemeinen Gruppencharakteren. Abh. math. Semin. Univ. Hamburg, 8,292306. (Collected Papers (1965), 165179. AddisonWesley.) Brauer, R. (1947). On Artin’s Lseries with general group characters. Ann. Math. (2) 48, 502514. Chevalley, C. (1940). La thCorie du corps de classes. Ann. Math. (2) 41, 394418; and other papers referred to in the above. Dedekind, R. Vorlesungen i.lber Zahlentheorie von P. G. Lejeune Dirichlet. Braunschweig, 18711894. Supplement 11. Estermann, T. (1952). Introduction to modem prime number theory. Camb. Tracts in Math. 41. Hall, M., Jr. (1959). “The Theory of Groups”. Macmillan, New York. Hasse, H. Bericht tiber neuere Untersuchungen und Probleme aus der Theorie der algebraischen Zahlkdrper. Jber. dt. Mat Verein., 35 (1926), 36 (1927) and 39 (1930). Hecke, E. (1920). Eine neue Art von Zetafunktionen und ihre Beziehungen zur Verteilung der Primzahlen (Zweite Mitteilung). Math. Z. 6,1151 (Mathematische Werke (1959), 249289. Vandenhoeck und Ruprecht). Hecke, E. (1954). Vorlesungen tiber die Theorie der algebraischen Zahlen, 2. Unverlnderte Aufl., Leipzig. Landau, E. (1907). Uber die Verteilung der Primideale in den Idealklassen eines algebraischen Zahlkijrpers. Math. Annln, 63, 145204. Landau, E. (1927). Einftihrung in die elementare und analytische Theorie der algebraischen Zahlen und der Ideale, 2. Aufl. Leipzig. Lang, S. (1964). “Algebraic Numbers”. AddisonWesley. Speiser, A. (1927). “Die Theorie der Gruppen von endlicher Ordnung”, 2. Aufl. Berlin. Tit&marsh, E. C. (1939). “The Theory of Functions”, 2nd edition. Oxford University Press, London. Titchmarsh, E. C. (1951). “The Theory of the Riemann Zetafunction”. Oxford University Press, London. Weber, H. (1896). Uber einen in der Zahlentheorie angewandten Satz der Integralrechnung. Giittinger Nachrichten, 275281. Weiss, E. (1963). “Algebraic Number Theory”. McGrawHill, New York.
CHAPTER IX
On Class Field Towers PETER R~QUETTE
1. Introduction . . . . . . 2. ProofofTheorem3 . . . . . 3. Proof of Theorem 5 for Galois Extensions . References . . . . . . .
. . . .
. . . .
. . . .
. 231 : Z?? . 248
1. Introduction Let k be an algebraic number field of finite degree and hk its class number, i.e. the order of the finite group Cl, of divisor classes of k. One of the most striking features of algebraic number theoryas compared to rational number theoryis the existence of fields k with class number ht > 1, i.e. whose ring of integers is not a principal ideal ring. There arises the question as to whether every such k can be imbedded into an algebraic number field K of finite degree with hK = 1. We refer to this problem as the imbedding problem for k, and to K as a solution of the imbedding problem. Let k, be the Hilbert class field of k. It can be defined as the maximal unramified abelian field extension of k. The Galois group of kJk is isomorphic to Cl, via the reciprocity isomorphism of class field theory. In particular, (k, : k) = hk. The principal divisor theorem states that every divisor of k becomes a principal divisor in k,. But there may be divisors of kl which are not principal; so let k2 be the Hilbert class field of kl. Continuing, we obtain a tower of fields k c kl c k2 c k, c . . . in which each field is the Hilbert class field of its predecessor. This is called the Hilbert classfield tower of k. We denote by k, the union of the fields ki. This is an algebraic number field whose degree may be finite or infinite. PROPOSITION 1. If the imbedding problem k c K with hK = 1 has a solution K, then k, c K. In particular, the degree of k, is then finite. Conversely, if the degree of k, is Jinite, then hk, = 1 and hence k, is then the smallest solution of the imbedding problem for k. Proof. (i) First we show k, c K for all i. Using induction we may assume i = 1. The field extension k,/k is unramified and has abelian Galois group. 231
232
PETER
ROQUJZTTE
Both these properties carry over to the composite field extension k, K/K. Hence k, K is contained in the Hilbert class field Kl of K. By assumption, (K,:K)=h,= 1. It follows ki c K. (ii) Now we assume k, of finite degree. Then k, = k, if i is large. Hence hk, = hk, = (ki+i : ki) = 1. QED. By Proposition 1 we see that the imbedding problem for k is equivalent to the problem of whether the Hilbert class field tower of k is finite. This problem has been posed by Furtwlingler and it is mentioned in Hasse’s Klassenkbrperbericht 1926, in connection with the (then yet unsolved) principal divisor theorem for k. Although the principal divisor theorem was proved already in 1930 (Furtwangler), it was not until 1964 that the finiteness problem of the Hilbert class field tower could be solved. The solution has been given by SafareviE (together with Golod) who showed that the answer in general is negative, i.e. the Hilbert class field tower can be in&rite. The purpose of this course is to give a report about SafareviE’s results, together with the related work of Brumer. Let p be a prime number. A field extension K/k is called a pextension if it is Galois and if its Galois group is a pgroup. (Warning: if K/k is not Galois it is not called a pextension, even if its degree is a ppower.) Let ky) be the maximal pextension of k contained in k, ; this is called the Hilbert pclass field of k. Let kip) be the Hilbert pclass field of kp’. We obtain a field tower k c k’p’ c kp’ c k$“’ c . . .
in which each field is the Hilbert pclass field of its predecessor. This is called the Hilbert pclass field tower of k. The union of the kp) is called k’,P). It is easy to see that klp) c k, and in fact kjp) is the maximalpextension of k which is contained in k,. It follows k’,P’ c k,. In particular, if k’,P’ is of infinite degree we see that k, is of infinite degree too. Hence we ask: Under what conditions is k’,P’ of finite degree, ifp is a fixed prime? The fact that we shall deal with k’,P’ rather than with k, is due only to the fact that pgroups are easier to handle than arbitrary solvable groups. The following proposition is the analogue to Proposition 1 for the pclass field tower and is proved similarly, using the fact that (ky) : k) = hip) is the ppower part of the class number h,. The proof is left to the reader. PROPOSITION 2. Let p be a prime number. If the imbedding problem k c K with p ,/‘h, has a solution K, then k’,P) c K. In particular, the degree of k’,P’ is then finite. Conversely, if the degree of k’,P’ is finite, then its class number is not divisible by p and hence k’,P) is then the smallest solution of the imbedding problem for k with respect to p.
Now we state our main result.
First a definition:
If G is any group, we
ON CLASS
FIELD
233
TOWERS
denote by G/p the maximal abelian factor group of G of exponent p, regarded as a vector space over the field of p elements. We define the prank d@‘G to be the dimension of G/p: dcP)G = dim G/p. If G is finite abelian, then d@)G is the number of factors of ppower order which occur in a direct decomposition of G into cyclic primary components. THEOREM 3 (GolodSafareviE). There exists u function r(n) such that dcp) Cl, < y(n) for any algebraic number field k of degree n whose pclass field Remark
tower is finite.
4. As we shall see in the proof of Theorem 3, we have
dcp) Cl, < 2 + 2,/(rk + SF)) (1) where rk denotes the number of infinite primes of k and Sp) = 1 or 0 according to whether the pth roots of unity are contained in k or not. Since rk 6 n and @‘I 6 1 we see that we can take r(n) = 2+2J(n + 1). In order to formulate our next theorem, we introduce the following notation. Let q be a finite prime in the rational number field Q, and let D be any extension of q to k with the corresponding ramification degree e(Q). Let us put ek(d = tg$ e(Q)
where D ranges over the extensions of q to k (gcd means “greatest common divisor”). We shall call q completely ramified in k if e,(q) > 1. Let tP be the number of completely ramified q such that p divides e,(q). THEOREM 5 (Brumer). There exists a function c(n) such that dcP) Clk 2 tip’  c(n) for every algebraic number field k of degree n. Remark
6. One can show that dcp) Cl, 2 tip’  r,n
where tk has the same meaning as in Remark 4. Since rk 5 n we see that we can take c(n) = n2.
Actually, following Brumer, we are going to prove here Theorem 5 only in a somewhat weaker version, insofar as we shall consider only Galois extensions k of Q. We shall exhibit a function c’(n) such that for every Galois extension k/Q of degree n the following inequality holds: dfP) 51, 2 tp’ 
C’(n).
Our proof of Theorem 5 in the general case requires the use of the Amitsur
234
PJTI'ERROQUEXTE
cohomology in number fields which we do not want to assume known in this Chapter. For Galois extensions, we shall show that d(P) Cl, zi: tP’ 
F; + w,(n).bp L > where wp(n) denotes the exponent of p occurring in n. Since wp(n> s n 1 we see that we can take c’(n) = (n1)+(n1) = 2(n1). Combining Theorems 3 and 5 we obtain the COROLLARY
7. If k is an algebraic numberfield of degreen and if tip’ 2 y(n) + c(n)
then the pclassfield tower of k is infinite. In particular, for any given degree n > 1 and any prime p dividing n, there exist injbtitely many algebraic number fields k of degree n with an infinite pclassjield tower, for instance the Jields
k=Q(;/(q,...
a~)) with N 5 r(n) + c(n) where the q, > 0 are different prime numbers in Q. Numerical examples: We consider the case n =p = 2 and use the
estimates given by formulae (1) and (2). We have Sp)= 1, w,(2) = 1. Furthermore, ti2) is just the number of finite primes of Q which ramify in k. It follows: A quadraticfield k has an infinite 2classfield tower if the number of finite primes of Q which are ramified in k is 2 2 + 2J(rk + 1) + rk. If k iS iIIXigiXUUy, we have r& = 1. Since 2+2&+ 1 < 6 it follows that any imaginary quadratic field with at least six ramified finite primes has an infinite 2class field tower. A small numerical example is k = Q(,/(2.3.5.7.11.13)) = Q(,/(30030)). If k is real, we have rk = 2. Since 2 +2&+2 < 8 we see that in the real case we must have at least eight finite ramified primes in order to deduce the infinity of the 2class field tower. A small numerical example is k=Q(,/(2.3.5.7.11.13.17.19)) = Q&/(9699690)). 2. Proof of Theorem 3 We consider the following situation: k an algebraic number field of finite degree; Cl& its divisor class group; C& its idkle class group; u& the group Of idele U!litS ; = u& n k the group of units in k; Ek
ON
CLASS
FIELD
235
TOWERS
the group of roots of unity in k; a fixed prime number = kcp) as explained in Section 1. G = GTK/k) the corresponding Gaiois group. Under the assumption that (K: k) is finite we are going to prove the inequality (1) of Section 1. The proof will be divided into two parts. In the first part, we shall reduce Theorem 3 to a purely group theoretical statement about finite pgroups. In the second part, we shall prove this group theoretical statement. The unit theorem says that Ek is a direct product of the finite cyclic group W, and a free abelian group with r, 1 free generators. (See Chapter II, $18.) This gives dtp)Ek = (rk 1) + dtp) W, = (r,  1) + 69). Therefore the inequality (1) to be proved can also be written as d(P) Cl, < 2 + 2 J(d’P’ Ek + 1) or equivalently, in rational form: *(d(P) Cld2  dcp)Cl, < d’p’Ek. (3) First we observe that dtP) Cl, = dtp) G. Namely, we have W, P K
d(P)
G
=
d(P)(@)
where Gab is the maximal abelian factor group of G. This follows directly from the definition of the prank of G as given in Section 1. Now, Gab is the Galois group of the maximal subfield of K which is abelian over k; this subfield is the Hilbert pclass field kl(p). By the reciprocity isomorphism of class field theory, we have therefore Gcab= c~$P’ where Cl,$‘) is the pSylow group of Clk. By definition of the prank, d(P) C$‘) = d(P) Cl,. This proves our contention. Therefore the inequality (3) to be proved may be written as $(dcP) G)’  dcp) G < dcp’E,. (4) Any factor group of Ek has prank 5 d(P)&. In particular, the norm factor group:
W&&K) Hence it suffices to show that (5)
= fi”G
E3.
+(dfP’ G)2  dcp) G < dcp’fio(G, I&).
this is true for
236
PETER ROQUETTE
To determine the righthand side, we use the exact sequence l+EK+UK‘UK/ER+l. Since K/k is unramified, UK is cohomologically trivial as a Gmodule. (For the proof, decompose UK into its local components and prove the similar fact for a local Galois extension and its unit group, cf. Chapter VI, 0 2.5,)t It follows that .tl’(G, EK) = H ‘(G, UK/&).
Now we use the exact sequence 1+U~/E~+C~+Cl~+1. Here, the group Cl, is of order prime to p. This follows from the fact that K = k’,P) is the maximal field in the Hilbert pclass tower; see Proposition 2 in Section 1. Since G is a pgroup, we conclude that Cl, is cohomologically trivial as a Gmodule. Therefore H‘(G,
UK/EK) = H‘(G,
C,) = H3(G,Z).
using Tate’s fundamental theorem of cohomology in class field theory (Chapter VII, 0 11.3). It follows that the inequality (5) to be proved may be written as *(d(p) G)2  dcp)G < d’P’H2(G, Z). (6) using the fact that H,(G, Z) = H3(G,
Z)
by definition of the cohomology groups with negative dimensions. The inequality (6) is now a purely group theoretical statement; we shaN see that (6) is true for anyjinite
pgroup G.
According to our notation introduced in Section 1, let Z/p denote the cyclic group with p elements. The homology groups Hi(G, Z/p) are annihilated by p and may therefore be regarded as vector spaces over the field with p elements. Let us put d@) G = dim H.(G, Z/p). LEMMA 8. For any group k, there is a na;ural isomorphism H,(G, Z/p) = G/p. In particular, dip’ G = dcp) G. LEMMA 9. For anyjinite group G, we have d(P) H,(G, Z) = d$” G  d’P’ G. Proofs.
Consider the exact sequence o,z+z+z/p+o P
where 7 denotes the map “multiplication t Recall that the absence of ramification local extension is trivial, i.e. of degree 1.
byp”. Consider the corresponding
for an hfinite
prime means the corresponding
ON CLASS
exact homology
FIELD
sequence H,(Z) ) Hi(Z) ) Hi(Z/p)
TOWERS
237
j Hi l(Z) , Hi,(Z).
(For brevity, we havz written H,(Z) instead of H,~~, Z).) In general, if A is any abelian group, let A, be the kernel of the map A 7 A; its cokernel is A/p. With this notation, we obtain the exact sequence 0 ) Hi(Z)/p + Hi(Z/p) * Hi l(Z), + 0. (7) First, take i = 1. We have H,(Z) = Z which has no ptorsion; it follows Km, = 0 and therefore (8)
HIWP
= H,(Z/P).
Here, H,(Z) = H,(G, Z) = Gab is the maximal abelian factor group of G. (See Chapter IV.) By definition of G/p, we have G/p = Gab/p. This proves Lemma 8. Secondly, take i = 2 in formula (7). Since G is finite, all groups occurring there are finite; hence they are finite dimensional vector spaces over the field with p elements. Comparing dimensions : dy) G = dP)Hz(Z) + dim H,(Z) P In general, if A is any finite abelian group, we have dim A, = dim A/p which is proved by decomposing A into cyclic direct factors. Applying this to A = HI(Z) and using formula (8) we obtain dimHi = dip’ G. P = dimH,(Z/p) This proves Lemma 9. Using Lemmas 8 and 9, we see that the inequality (6) to be proved is equivalent to the following. THEOREM 10. Let G be a finite pgroup, p a prime number. Then d%’ G > *(dip’ G)’ , where dp’G = dim H.(G 1 3Z/p).
We use the following notations: A = Z(G) is the integral group ring of G; I is the augmentation ideal of A, defined to be the kernel of the augmentation map A + Z; A/p is the group ring of G over the field of p elements. First we prove two preparatory lemmas. LEMMA 11. Let G be a jinite pgroup and A a Gmodule with pA = 0. Then the minimal number of generators of A as Gmodule equals dim H,(G, A) = dim A/IA, the dimension to be understood over theJield with p elements.
238
PETER
ROQUETTE
More precisely: Let a, E A. Then the a, generate A as a Gmodule if and only if their images a, in A/IA generate AJIA as a vector space. Proof. Assume the a, generate A/IA. Let B be the Gsubmodule of A generated by the a,. Then the natural map B/IB + A/IA is epimorphic, which is the same as the map H,(B) + H,(A). The exact sequence O+B+A,A/B+O yields the exact homology
sequence
H,(B) + H,(A) + H&W) ) 0 from which we infer that H,(A/B) = 0. Since A/B is annihilated by p and G a pgroup, it follows A/B = 0, i.e. A = B. (See Chapter IV, Q9.) QED. LEMMA 12. Let G be ajinite pgroup and A a Gmodule with pA = 0. Then there exists a resolution . ..+Y.+Y,+Y,+A+O
with the following properties: (i) Each Y, is a free module over Alp. (ii) The number of free module generators of Y, over h/p equals dim H,,(G, A). (iii) The image im( Y,,+ 1) is contained in I. Y,. Proof. Put d = dim H,(G, A). By Lemma 11 there is a free Amodule X on d generators and an epimorphism X+ A. Since pA = 0 we obtain by factorization an epimorphism X/p + A. Put Y = X/p. Then Y is a free A/pmodule on d generators. In particular, Hi(Y) = Ofor i 2 1. Let B be the kernel of Y+ A so that O+B+Y+A+O is exact. The exact homology
sequence
. . . jH,+,(Y)jHl+,(A)jH,(B)Hi(Y>j... yields (9) Hi(B) = Hi+ I(A) for i 2 1. As to the case i = 0, we have the exact sequence 0 B H,(A) + H,(B) ) H,(Y) + H,(A) + 0. By construction, Y and A have the same minimal number of generators as Gmodules; hence dim Ho(Y) = dim H,(A) by Lemma 11 which shows that H,,(Y) + H,,(A) is an isomorphism. We conclude that (9) holds also in the case i = 0.
ONCLASSFIELDTOWERS
Since I&(Y)
239
+ H,,(A) is an isomorphism, the map B/IB = H,(B) + l&,(Y) = Y,UY
is zero; hence Bc1.Y. (10) Now put Y = Y,,. Then Y,,+ A , 0 is the first step in the resolution to be constructed. The second step is obtained by applying the same procedure to B. We obtain YI + B+ 0 with a kernel C such that Hi(C)
=
Hi+l(B)
=
Hi+2(A)
for i 2 0, and c c I. Yi. YI is a free A/pmodule on dim H,,(B) = dim H,(A) generators. If we define YI + Y0 to be the composite map Y, f B + Y,, then (10) says that im( YJ c I. YO. Continuing this process, we obtain Lemma 12 by induction. QED. Let us apply Lemma 12 to the case A = Z/p. Then H,(G,Z/p) = Z/p is of dimension 1. Hence Y0 is free with one generator. By (iii), the kernel of Y,,+ Z/p is contained in I. YO. Since YJI. Y0 is of dimension 1, this kernel is precisely I. YO. Hence YI + I. Y0 f 0 is exact. Changing notation, we obtain the COROLLARY 13. Let G be afinite pgroup and put d = d’,P) G, r = dp) G. Then there is an exact sequence R+D+I.E+Owithim(R)cI.D where E, D, R denote the free Alpmodules with 1, d, r generators respectively. Proof of Theorem 10: We use the abbreviations of Corollary 13. For any finite Gmodule A with pA = 0 we introduce the PoincarC polynomial PA(t) = c C”(A). t” with c,(A) = dim Z”A/Z”+‘A. OSII
(Note that c,(A) = 0 if n is sufficiently large.) We put P(t) = PE(t). Since c,(E) = dim E/IE = 1 we have P(t)  1
Pm(t) = y
Also, since D = Ed is the dfold direct product of E, we have c,(D) = d. c,,(E) and therefore PO(t) = d. P(t). Similarly, PR(f) = r . P(t).
240
PETER
ROQUETTE
In general, if 0 c t c 1 is a real variable, then PAM 1t!
=o~~(A)t”
with s,(A) =,~&c,(A)
= dim A/I”+lA.
Also,
where we have to interpret s,(A) = 0. From Corollary 13 we obtain an epimorphism I“+ ‘D + I”+‘E. ifR,+l denotes the foreimage of I”+‘D in R then the sequence 0 + R/R,+I , DIl”+‘D ) IE/I”+‘E B 0 is exact. This gives s,,(D) = s,,(IE) + dim R/R, + i. By Corollary Therefore,
13, image(R) c ID, hence image (PR) c Z”+‘D, dim R/R,+l
4 dim RII”R
Hence,
I”R c R,, i.
= s,,(R).
This gives s,(D) 5 s,UE) + s,  I(R). The numbers occuring in these inequalities are the coefficients of power series as given above. It follows that
or equivalently P(t)
d.PW I 
t
1
if 0
+ r.t.P(t)
In other words: 1 s P(t).(rtG&+l)
if
0 < t < 1.
Since P(t) has positive coefficients, we conclude that O
t + d we get 2r
r>)d’
as contended.
The substitution
d 2r
t +  is permissible
Lemma 9 that d 5 r < 2r, hence 0 < f < 1. QED
since we know from
ON CLASS
FIELD
TOWERS
241
Remark 14. Actually, Golod and SafareviE have proved in their paper only dip)G > f(djP)G 1)2. The inequality as given here has been obtained independently by Gaschiitz and Vinberg. Remark 15. Theorem 10 is important not only for the class field tower problem but also for various other problems in the theory of pgroups. This is due to the fact that the homological invariants &“G and dp)G permit the following group theoretical interpretations if G is a finite pgroup : dip’G is the minimal number of generators of G (Bumside’s basis theorem; see Chapter V, $ 2.5) and d(ZP)is the minimal number of relations among these generators which define G as a pgroup (see Serre, “Cohomologie Galoisienne”). 3. Proof of Theorem 5 for Galois Extensions Let k/Q be a Galois extension of finite degree n, and p a prime number. As said in Section 1, we are going to prove the inequality d(P)Cl, 2 $0  rL1 + w,(n). Sp) . ( Pl > Again, the proof will be divided into two parts. First, we shall reduce the proof to a group theoretical statement about the cohomology of finite groups. Secondly, we shall prove this group theoretical statement. LetK = kl be the Hilbert class field of k; G the Galois group of KJk; G* the Galois group of K/Q ; g the Galois group of kJQ. Then g = G*/G and we have the inflationrestriction sequence of cohomology groups 1 R H’(g, &) + H’(G*,E,) + H’(G, I&) where Ek as in Section 2 denotes the group of units in k. We conclude that dP’H’(G*, EJ 6 dP’H’(G, EJ +dP’H1(g, EJ. Hence the inequality (11) to be proved is an immediate following three statements : H’(G, E3 = Cl, (12) dP’H’(G*, EK) = rp’ 03)
consequence of the
r,1 J&) 5 + w,(n). sp. P1 We are going to prove these three statements. For any algebraic number field K of finite degree we consider the commutative and exact diagram: (14)
&‘H’(g,
242
PETER
i
ROQUETTE
r
t
l+ EK +UK+U~IE~+l 1
l+K”
1
1
+ I, + 1 1
l+P,+D,+ 1
1
1
c, 1
+ 1
Cl, 1
+1
1 group of K;
is the multiplicative the idble group; IK DR the divisor group; PR the group of principal divisors ; and the other groups have the same meaning as before in Section 2. The arrows denote the natural maps. (In Section 2 we have used already the first row and the last column of this diagram.) If K is a Galois extension of a subfield k with Galois group G then all groups and maps of this diagram are Gpermissible and we obtain a corresponding commutative and exact cohomology diagram:
where KX
l+
z EK 1
,
l+
k”
*
l+
j”
+
i Uk cl
i + (WEdG c JY
) fJ’(G, Ed + H ‘(G, UK) d 1 Q
Ik cl
+
C,
+
:
C$
0;
i 1 = H'G&I + H'(G, U,> 11 (" 11
1
where we have twice used Hilbert’s Theorem 90: H’(G,KX) = 1 and also the corresponding local statement : H’(G,IJ = 1. LEMMA 14. Let K be an algebraic number$eld offinite Galois extension of a subfield k with Galois group G. Assume 0: c P,, i.e. that every invariant is a natural exact sequence
l+C1,+iY’(G,E&+H1(G,UK)+l. Q
degree which is a
divisor is principal.
Then there
ON CLASS
ProoJ We use the cohomology the natural map
FIELD
TOWERS
243
diagram as explained above. Observe that
cp : H’ (G, EK) + H’ (G, U,)
occurs twice in this diagram. We have to show that, under the assumption of Lemma 14, cp is epimorphic and its kernel is Clk. The assumption in Lemma 14 means Pg = Dg. Looking at the left lower corner of our cohomology diagram we see therefore that cp is epimorphic. On the other hand, Pg =I$ shows that a= 1. Hence tlOr=jIOq= 1. Since q is epimorphic, /I = 1. Hence y is an isomorphism. Looking at the right upper corner of our diagram we see that kernel (cp) = image (6) = (UK/,??K)G/image (E); the isomorphism y shows that this equals C,Jimage (y 0 s) = C,/image (rl 0 c) = IJk. lJ, = Cl,. QED. Proof of (12): We use Lemma 14 in the case where (as in (12)) K is the Hilbert class field of k. Since K/k is unramified, we have H’(G, UK) = 1, as already observed in Section 2. In view of Lemma 14 it suffices therefore to show that the assumption of Lemma 14 is satisfied if K is the Hilbert class field of k. Since H’(G, UK) = 1 we infer from our cohomology diagram above that c : I, + 0: is epimorphic. On the other hand, its image is I,JU, = Dk. Hence D, = DE. Now, the principal divisor theorem for the Hilbert class field says that Dk c PK. QED. Proof of(13): We want to apply Lemma 14 to the Galois extension K/Q, where K (as in (13)) is the Hilbert class field of k. First we have to show that the assumption of Lemma 14 is satisfied for K/Q. G* is the group of K/Q and G the group of K/k. We have therefore G c G* and Dr c 0:. The foregoing proof shows 0: c PK. Hence Dr c P,. Since Cl, = 1 we infer therefore from Lemma 14 an isomorphism H’(G*, EK) = H ‘(G*, U,). For any finite prime q of Q let e,(q) denote the ramification index of some extension 61 of q to K. (Since K/Q is Galois, this does not depend on the choice of fllq.) Let Z/e,(q) denote the cyclic group of order e,(q). Then 11 ‘(G*, UK> = n Z/e,(q). (For the proof, decompose U, into its loci1 components and prove the similar fact for a local Galois extension and its unit group.) Now K/k is unramified and therefore e&)
= ekd
244
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ROQUETTE
for every q. Hence we obtain H’(G*,
J%) = n Z/ek(d. 4 The prank of the direct product on the righthand side is equal to the number of q with pie,(q), i.e. to tp’. QED. Proof of (14): The torsion group of Ek is the group W, of roots of unity in k which is cyclic of prank Sip). Moreover, the unit theory says that EJ W, is a free abelian group on rk  1 generators. Hence (14) is an immediate consequence of the following group theoretical statement : LEMMA 15. Let G be a finite group of order n and p a prime number. Let A be a finitely generated Gmodule with torsion group tA, and let p(A) be the number of free generators of AJtA as an abelian group. Then ~(4 dP’H ‘(G , A) 5 
Proof.
+ w (n). d(P)tA.
pl Consider the restriction map
p
H ‘(G, A) + H’(GfP’,
A)
of G to its pSylow subgroup Gcp). This is a monomorphism components (Chapter IV, Prop. 8, Cor. 3). In particular, dCP’H ‘(G, A) $ d”‘H Hence we may assume in the following
of the pprimary
‘(GCp), A).
that G = Gtp) is a pgroup.
(Observe
that w (n) = w (ntp)) where n(P) is the order of Gtp’.) No; considb the exact sequence O+tA+A+A/tA+O
and the corresponding
exact cohomology
sequence
H ‘(G, tA) + H ‘(G, A) + H ‘(G, A/tA).
It follows that dtP’H ‘(G, A) 5 dCP’H ‘(G, tA) + dCP’H ‘(G, A/tA).
We shall show that dtP’H’(G
, tA) Ip’w (n) dCP)tA
and dCP’H ‘(G, A/tA) 4 p0
pl’
In other words: We shall treat the following two cases separately: torsion module; (II) A is torsion free. (I) The torsion case: We have dCP)G = dim G/p 5 w,(n)
(I) A is a
ON
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245
and therefore it suffices to show: dtP’H ‘(A) S dtP’G. dtP’A. (1% (For brevity, we omit the symbol Gin the notation of the cohomology groups.) Let Z’(A) be the module of crossed homomorphisms/z G ) A. Let g,, . . . , gd be a minimal system of generators of G; Bumside’s basis theorem forpgroups tells us that d = d@)G. Every crossed homomorphismfis uniquely determined by its values J(gJ, 1 s i 6 d. In other words : The map is an injection
f+ tf(sl),   * s&d)) of Z’(A) into the dfold direct product Ad of A. It follows d’p’Z’(A) % d”‘(Ad) = d. (dcp’A) = dtP’G. dtP’A.
Since H’(A) is a factor module of Z’(A), we obtain (15). Remark 16. In the proof just given, we have not made use of the fact that A is a torsion module. Hence the inequality (15) holds for every &itely generated Gmodule A. This means that we have proved instead of (14) the inequality d(p)H ‘(9, EC) 6 wp(n)(rk 1 + 6!p)) and therefore instead of (11) the inequality dfp’CI, ;r tp)  wp(n)(rk  1 + Sip’). This would suffice to obtain a function c’(n) with d’P’C1, ;r: tp’  c’(h) , as explained in Section 1. Namely, since w#) 5 n  1, we could take c’(n) = (n  1). n. The following proof, which permits to obtain a better estimate in the torsion free case, is needed only in order to obtain the better estimate c’(n) = 2(n 1) as stated in Remark 6 of Section 1. (II) The torsionfree case: We begin with LEMMA 17 (Chevalley). Let G be a group of order p and A a finitely generated Gmodule. Then d’P’H’(G,A)d’P’H2(G,A) Proof.
= ~(4 plP p(AG).
(i) Let us put dl ,(A) = dcp)H ‘(A)  d(p)H ‘(A).
Note that p . H’(A) = 0 since G is of order p. Hence dfp)H’(A) is the dimension of H’(A) over the field of p elements; the order h’(A) of H’(A) is therefore p to the power d(P)H’(A). If we define the Herbrand quotient h,z(A)
= h’Wh2t4
246
PETER
ROQUETTE
then it follows that h,,,(A)
= pd+2(A) .
We may therefore regard d,,(A) as an additive analogue to the Herbrand quotient. The properties of the Herbrand quotient, as stated in Chapter IV, 8 8, can therefore be applied mutatis mutandis to dI 2. In particular, we have : If 0 * A + B + C + 0 is an exact sequence of Gmodules, then dlAB) = 4A4+4,(0 We say therefore that dI _ 2 is an additive function
of Gmodules. Let us say that two finitely generated Gmodules A, B are rationally equivalent if A @ Q and B @ Q are isomorphic as Q(G)modules. (The tensor product is to be understood over Z and Q(G) = Z(G) QDQ is the group ring of G over Q.) We may regard A/tA as a submodule of A @ Q and in fact as a Ginvariant lattice of the rational representation space A @3Q of G. In view of Propositions 10 and 11, Section 8 of the cohomology course, we have:
If
A and B are rationally
equivalent, then d, 2(A) = d, _ JB). of rational equivalence classes.
We say therefore that d,,(A) is a function (ii) For brevity, we define the function
T(A) = p(A)P.P(A’) pl
*
We claim that z(A) too is an additive function of rational equivalence classes, and have to show that both p(A) and p(A’) are. It follows from the definition of p that p(A) = dime A@Q and p(AG) = dimo AG@Q. From the 8rst relation we infer that p(A) is a function of rational equivalence classes. From the second relation we infer the same for p(A’) since AG@Q=(A@Q)O.
IfOtA+B+C+Oisexactthen 0 + A@Q P B@Q+
is exact too.
This shows the additivity
C@Q+
of p(A).
0
As to the additivity
of
p(AG), we have to use the fact that
O,(A~Q)Gj(B~Q)Gt(C~Q)GjO is exact too, since H’(G, A @ Q) = 0 because A @ Q is uniquely divisible. (iii) We have now seen that dI 2 and r are both additive functions of rational equivalence classes. Furthermore, an easy computation shows d, emz(Z) = z(Z) =  1 dl&I)
= t(A) = 0
ON CLASS
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247
where we consider the integers Z and the group ring A = Z(G) as Gmodules. Hence the contention of Lemma 17 follows from the following statement: Every additive function f of rational equivalence classes of finitely generated Gmodules A is uniquely determined by its values for the modules Z and A. (iv) Proof of the statement in (iii): The exact sequence O+IFA+Z+O so that f(I) is determined by f(Z) and f(A). Hence shows f(l> =f(N f(Z), it remains to be shown that every finitely generated Gmodule A is rationally equivalent to a direct sum of Gmodules which are isomorphic either to Z or to I. In other words: A@Q=CAi@Q
where A , = Z or A, = I. Since A @ Q as a representation space of G over Q is a direct sum of irreducible representation spaces Vi, this amounts to showing that every irreducible representation space V # 0 of G over Q is either isomorphic to Z@ Q = Q or to I@ Q. Now every such V is isomorphic to a direct summand of Q(G) = A @ Q. Hence we have to determine the direct decomposition of Q(G). The exact augmentation sequence 0 + I@Q+Q(G) splits; the corresponding
+ Q+ 0
map Q + Q(G) being given by x+x.e
(xEQ)
where e is the idempotent e=l
C g. Pu=G
We have therefore the direct decomposition Q(G) = (IC3QNBQ.e
and it remains to show that 1@Q is irreducible, i.e. that I@ Q, as a Qalgebra, is a field. Let K denote the field of pth roots of unity. Let x denote an isomorphism of G on to the group of pth roots of unity in K. By linearity, x extends uniquely to an algebra homomorphism of Q(G) on to K. Its kernel contains e since the sum of all the pth roots of unity in K is 0. Hence x defines an algebra homomorphism of Q(G)/e = I@ Q on to K. Since dimoK=pl=dim,i@Q it follows that this is an isomorphism
of 18 Q on to K. QED.
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PETER ROQUJZTTE
LEMMA 18. Let G be a finite pgroup and A a finitely generated Gmodule which, as an abelian group, is torsion free. Then
d’P’H ‘(G, A) 5 p(A;;yG). In particular,
it follows that d’P’H’(G,
A) $ @ P1
which we wanted to prove. Prooj First let G be of order p. Then Lemma
17 shows
 P . P(A3 + dfP’H ‘(G, A). dfP’H ’ (G 9A) S P(A) Pl Since G is cyclic, H’(G, A) = fi’(G, A) is a certain factor group of AC and has therefore prank 5 dCp)AG= p(A’), the latter equality holding since AC is torsion free. It follows d’P’H’(G,A)
< 
p(A)p’p(Ac_)
+
p(~G)
Pl which proves our assertion if G has order p. Now let G be of order 2 p2. Let U be a proper normal subgroup. The inflationrestriction 1 + H ‘(G/U, AU) + H’(G, A) + H’(U, A) shows dtP’H ‘(G, A) S d’P’H’(G/U, A”) + dtP’H ‘(U, A). Using induction, we may assume that
sequence
dCp’li ’ (G/U, A”) 5 PC&  dAG> Pl dCP’H ‘(U, A) 5 ~(4  PU”) Pl Adding, we obtain our contention. QED.
.
REFERENCES
Brumer, A. (1965). Ramification and class towers of number fields. A&h. moth. J. 12,129131. Brumer, A. and Rosen, M. (1963). Class number and ramification in number fields. Nagova math. J. 23,97101. Friihlich, A. (1954). On fields of class two. Proc. Land. math. Sot. (3) 4, 235256. Frohlich, A. (1954). On the absolute class group of abelian fields. J. Lond. math. Sot. 29,211217.
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Frohlich, A. (1954). A note on the class field tower. Q. JI. Math. (2) 5, 141144. Friihlich, A. (1962). On nonramified extensions with prescribed Galois group. Mathematika, 9,133l 34. Golod, E. S. and SafareviE, I. R. (1964). On class field towers (Russian). Zzv. Akud. Nauk. SSSR, 28, 261272. English translation in Am. math. Sot. Transl. (2) 48, 91102. Hasse, H. (1926). Rericht tiber neuere Untersuchungen und Probleme der Theorie der algebraischen Zahlkorper, Teil 1. Jber. dt. Matverein. 35, in particular p. 46. Iwasawa, K. (1956). A note on the group of units of an algebraic number field. J. Math. pures appl. 35, 189192. Koch, H. (1964). Uber den 2Klassenkorperturm eines quadratischen Zahlkorpers. J. reine angew. Math. 2141215, 201206. SafarevE, I. R. (1962). Algebraic number fields (Russian). Proc. Znt. Congr. Mufh. Stockholm, 163176. English translation by H. Alderson in Am. math. Sot. Transl. (2) 31, 2539. SafareviE, I. R. (1963). Extensions with prescribed ramification points (Russian with French summary). Inst. Hautes Etudes Sci. Publ. Math. 18, 7195. Serre, J.P. (1963). “Cohomologie galoisienne”. (Lecture notes, Paris 1963, reprinted by Springer, Berlin.) Scholz, A. (1929). Zwei Remerkungen zum Klassenkorperturmproblem. J. reine angew. Math. 161, 201207. Scholz, A. and Taussky, 0. (1934). Die Hauptideale der kubischen Klassenkorper imaginiirquadratischer Zahlkorper usw. J. reine angew. Math. 171, 1941.
CHAPTER X
SemiSimple
Algebraic
Groups
M. KNESER Introduction . . . . . . . . . 1. Algebraic Theory 1. Algebraic groups ‘over an algebraically ciosed field : 2. Semisimple groups over an algebraically closed field 3. Semisimple groups over perfect fields . . . 2. Galois Cohomology . . . . . . . 1. Noncommutative cohomology . . . . 2. Kforms . . . . . . . . 3. Fields of dimension < 1 . . . . . 4. padic fields . . . . . . . 5. Number fields . . . . . . . 3. Tamagawa Numbers. . . . . . . Introduction . . . . . . . 1. The Tamagawa measure . . . . . 2. The Tamagawa number . . . . . 3. The theorem of Minkowski and Siegel . . . References . . . . . . . . .
. . . . . . . . . . . . . . . . .
250 250 250 252 253 253 253 254 255 255 257 258 258 258 262 262 264
Introduction Section 1 contains the basic definitions and statements of some fundamental Sections 2 and 3 are devoted to certain results of an algebraic nature. arithmetical questions relating to algebraic groups defined over local or global fields. Throughout, very few proofs are given, but references are given in the bibliography. For more information on recent progress in the theory of algebraic groups, the reader is referred to the proceedings of the following conferences: Colloque sur la ThCorie des Groupes Algebriques, Bruxelles, 1962. International Congress of Mathematicians, Stockholm, 1962. Summer Institute on Algebraic Groups and Discontinuous Subgroups, Boulder, 1965. References to the bibliography are at the end of this Chapter.
1. Algebraic
Theory
Basic reference (Chevalley, 195658). 1. Algebraic groups over an algebraically closed jieId (Chevalley, 195658 and Rosenlicht, 1956) Let k be an algebraically closed field. An algebraic group defined over k is 250
SEMISIMPLE
ALGEBRAIC
GROUPS
251
an algebraic variety G defined over k, together with mappings (X,JJ)HXJJ of G x G into G and x H XI of G into G which are morphisms of algebraic varieties and satisfy the usual group axioms. From now on, the words group, subgroup, etc., shall mean algebraic group, algebraic subgroup (i.e. closed subgroup with respect to the Zariski topology), etc. An algebraic group G is linear if it is alline as an algebraic variety. For example, GL,(k) is a linear algebraic group, since it can be represented as the set of points (xr,,u) in k”‘+l which satisfy the equation y . det (xi,) = 1. Hence any (closed) subgroup of GL,(k) is a linear algebraic group, and it is not difficult to show that conversely every linear algebraic group is of this form. The additive and multiplicative groups of k, considered as onedimensional algebraic groups, are denoted by GOand G,,, respectively; they are both linear (G,,, = GL,). A torus is a product of copies of G,,,. A linear algebraic group G is unipotent if every algebraic representation of G consists of unipotent matrices (i.e. matrices all of whose latent roots are 1). A connected algebraic group which is projective as an algebraic variety is an abelian variety. Every abelian variety is commutative. For example, any nonsingular projective elliptic curve carries a structure of an abelian variety. If G is any algebraic group, the connected component GO of the identity element of G is a normal subgroup of finite index. GO has a unique maximal connected linear subgroup G1, which is normal, and Go/G1 is an abelian variety (Chevalley’s theorem: proof in Rosenlicht, 1956). G1 has a unique maximal connected linear solvable normal subgroup Gz, called the radical of G1, and G1/G2 is semisimple, i.e. it is connected and linear and its radical is (1). G, has a unique maximal connected unipotent subgroup G3, which is normal in G,, and Gz/G3 is a torus. G3 is called the unipotent radical of Gz. Thus we have the following chain of subgroups of G:
linear
Y G, I G,
solvable
G,
connected
finite abelian variety semisimple torus
G3 unipotent (1) Example. G = GL,,. G is linear and connected, hence G = Gr. The radical Gz is the centre of GL,, consisting of the diagonal matrices (z G,), and G/G2 is simple.
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M. KNESER
We shall concentrate our attention
mainly on sem simple groups. 2. Semisimple groups over an algebraically closed field (Chevalley, 195658) Let k be an algebraically closed field, and G a semisimple group defined over k. A Cartan subgroup of G is a maximal torus, and a Bore1 subgroup of G is a maximal connected solvable subgroup. For example, if G is the special linear group SL,, then the group of diagonal matrices contained in G is a Cartan subgroup (it is clearly a torus, and it is maximal because it is equal to its centralizer), and the group of upper triangular matrices in G is a Bore1 subgroup. A homomorphism cp: G ) His an isogeny if G, H are connected groups of the same dimension and the kernel of cp has dimension zero (i.e: is finite). It follows that cpis surjective, since cp(G) is connected and of the same dimension as H. (For example, SL, + PGL, is an isogeny : its kernel is the group of nth roots of unity.) If the characteristic of k is zero, the kernel of rp is contained in the centre of G. In characteristic p > 0 there are unpleasant phenomena connected with the Frobenius automorphism. For example, let G = H = SL,, and let cpbe the mapping x = (xii) I+ (x;), which is an isogeny. If the xu are in the field k and q(x) = 1, then x = 1; cpis bijective but is not an isomorphism, since cp ’ is not algebraic. If we allow the xi, to be, say, dual numbers (that is, elements of the kalgebra k[e], where s2 = 0), then x = 1 +sy is in the kernel of cp for every y E G, but is not in the centre. We want to exclude this type of phenomenon, so we define a central isogeny to be an isogeny whose kernel is contained in the centre, for points with coordinates in any kalgebra. If cp: G + H is a central isogeny, G is said to be a central covering of H. Among the groups which centrally cover G there is a maximal one, G, which admits no proper central covering, and among the groups centrally covered by G there is a minimal one, G, which centrally covers no other group. 6 is called simply connected, and G is the aa’joint group of G. Example : G = SL,, G = PGL,,.
In order to classify semisimple groups, it is enough to classify the simplyconnected groups, and then to look for the possible central isogenies. Every simplyconnected group is uniquely expressible as a product of almost simple groups (i.e. groups G with finite centre C such that G/C is simple). SL, is an almost simple group. The classification of almost simple groups is the same as for simple Lie groups : 4,: %,+I G:
SP2.
B,, D,: isogenous to orthogonal groups E6, E ,,E8, F4, G, : exceptional groups.
SEMISIMPLE
ALGEBRAIC
GROUPS
253
3. Semisimple groups over perfect fields (For groups over nonperfect fields, see Demazure and Grothendieck, 196364) Let k be a perfect field, with everything defined over k: this means that the coefficients of the defining equations lie in k. A ktorus is an algebraic group defined over k which over the algebraic closure k of k becomes a torus, i.e. is isomorphic to a product of groups G,,, = GL, ; it is split over k if it is isomorphic to such a product over k. A semisimple group is said to be split over k if it has a maximal ktorus which is split over k; it is quasisplit over k if it has a Bore1 subgroup defined over k. As this terminology indicates, a split group is quasisplit. For example, SL, is split, and the group of elements of norm 1 in a quaternion division algebra over k is neither split nor quasisplit. In the decomposition of a simplyconnected semisimple group G as a product x G, of almost simple groups, the factors Gi and the isomorphism ~~~& are defined over li, not necessarily over k. The Galois group r = Gal (E/k) permutes the G,, and the product of the Gi belonging to a given transitivity class is a group defined over k. Thus we may assume that r permutes the G1 transitively. The isotropy group of G1 is a subgroup of finite index in l?, whose fixed field is a finite extension I of k, and Gr is defined over 1. G is said to be obtained from Gr by restriction of thejieid of definition from I to k : G = Rllk(G1) @Veil, 1961). This procedure allows one to reduce many questions to the case of almost simple groups, so we shall assume from now on that G is almost simple. In view of the classification of almost simple groups over E mentioned in Section 2, it is natural to ask whether, for each of the types A,, . . ., G2, there exists a group defined over k. In fact it is true that for each type there is a group G defined over k and split over k; and G is unique up to kisomorphism if we require in addition that it should be simplyconnected or adjoint (Chevalley, 1955; Demazure and Grothendieck, 196364). The classification of the various kforms of G, i.e. of groups G’ isomorphic to G over k, is a problem of Galois cohomology. 2. Galois Cohomology Basic reference (Serre, 1964). 1. Noncommutative cohomology Let G be a group and let A be a Ggroup, i.e. a (not necessarily commutative) group on which G acts. (If G is profinite we require that the action of G should be continuous with respect to the discrete topology on A.) The action of s E G on a E A is denoted by “a. We define H’(G, A) to be the group AG of Ginvariant elements of A. Next
254
M. KNESER
we define a cocycle to be a function SH a, on G with values in A which satisfies a, = as.‘a, (s, f E G). Two cocycles a,, b, are equivalent if there exists c E A such that b, = cl.as.‘c (s E G). The set of equivalence classes of cocycles is the cohomology set H’(G, A). This is a set with a distinguished element, namely the class of the identity cocycle. Both H’(G, A) and H’(G, A) are functorial in A. ~Ef 14 A + B + C + 1 is an exact sequence of Ggroups and Ghomomorphisms, then one can define a coboundary map H’(G, C) + H’(G, A), and the sequence (of sets with distinguished elements) 1 + H”(G, A) + H’(G, B) + H’(G, C) + H’(G, A) , H’(G,B) + H’(G, C) is exact. (Exactness is to be understood in the usual sense, the kernel of a map being defined as the inverse image of the distinguished element of the image set.) If A is contained in the centre of B (so that A is abelian and therefore H’(G, A) is defined) then one can define a coboundary map H’(G, C) + H’(G, A) which extends the exact sequence one stage further. Let K be a perfect field and L a Galois extension of K, with Galois group G. Let A be an algebraic group defined over K, and A, the group of points of A which are rational over L. Then AL is a Ggroup, and we write H’(L/K, A) in place of H’(G, AL). If R is the algebraic closure of K, we write H’(K, A) in place of H’(K/K, A). 2. Kforms Let V, V’ be algebraic varieties with some additional algebraic structure (for example, algebraic groups, or vector spaces with a quadratic form, or algebras) all defined over a perfect field K. Suppose that f: V+ v’ is an isomorphism for the type of structure in question, defined over a Galois extension L of K. Ifs is any element of the Galois group G of L/K, then s transforms f into an isomorphism “f: V, V’, and a, = f ’ o “f is a cocycle of G with values in Aut (V). It is easily seen that such a “Kform” v’ of V is Kisomorphic to another Kform V” of V if and only if the corresponding cocycles are equivalent, and therefore we have an injective mapping of the set of Kforms of V (modulo Kisomorphism) into the cohomology set H’(L/K, Aut (V)). In many important cases (algebraic groups, vector spaces with a finite number of tensors) this mapping is bijective. In such cases the problem of classifying Kforms is solved by (i) classification over the algebraic closure R of K, and (ii) computation of H’(K, Aut (V)). Consider for example the classification of Kforms of simplyconnected semisimple linear algebraic groups. As we have seen in Section 1, it is enough to consider almost simple groups, and then the classification over
SEMISIMPLE ALGEBRAIC
GROUPS
255
R is known. Let G be the split group of a given type; then the group of inner automorphisms of G is isomorphic to the adjoint group G rG/C where C is the centre of G, and the quotient Aut (G)/G is isomorphic to tne group Sym (G) of symmetries of the Dynkin diagram of G. So we have an exact sequence \\~ l+G+Aut(G)+Sym(G)+l and the derived cohomology sequence: H’(K, Sym(G)) + H’(K, G) + H’(K, Aut (G)) I H’(K, Sym (G)). The Galois group g of K/K operates trivially on Sym(G), so that a cocycle of g with values in Sym (G) is a homomorphism g + Sym(G), and therefore H’(K, Sym(G)) is a quotient of the set Horn (g, Sym(G)). In fact cp is surjective, and for each u EH’(K, Sym(G)) the fibre cp‘(cr) contains a quasisplit Kform G, of G, which is unique up to Kisomorphism (Steinberg, 1959); and the elements of cp‘(or) correspond to those Kforms G’ of G such that there exists an isomorphism f: G, + G’, defined over it, for which f  ’ O “f is an inner automorphism of G. for all s E g. By a twisting argument (cf. Serre, 1964, I5.5) qo‘(or) is seen to be a quotient of the cohomology set W(K, G,&. 3. Fields of dimension < 1
A field K is of dimension < 1 if the Brauer group Br(L) is zero for every algebraic extension L of K. Examples of such fields are all finite fields, and the maximal unramified extension K,, of a field K which is complete with respect to a discrete valuation. THEOREM 1 (Steinberg, 1965). If K is a perfectfield of dimension < 1, and if G is a connected linear algebraic group defined over K, then H’(K, G) = 1. In the case where K is finite, this theorem was proved earlier by Lang. From Steinberg’s theorem and the results stated in Section 2, it follows that cp‘(a) contains exactly one element, so that every Kform of G is quasisplit, and the Kforms are classified by H’(K, Sym (G)). 4. padic fields
If K is a padic field (i.e. a completion of a global field with respect to a discrete absolute value) it is no longer true that H’(K, G) = 1 for every connected group G. However, there is the following result: THEOREM 2 (Kneser, 1965). If K is a padicjield and G is a simplyconnected semisimplelinear algebraic group defined over K, then H’(K, G) = 1. The proof in Kneser (1965) proceeds by reduction to the case of almost simple groups and then deals with each type A,, . . . , Gz separately. For the classical groups the theorem is closely related to known results on the
256
M.
KNESER
classification of quadratic, hermitian, etc., forms over a padic field. More recently Bruhat and Tits (1965) have indicated a uniform proof, based on results of Iwahori and Matsumoto (1965). If G is the simplyconnected covering group of an orthogonal group, Theorem 2 is essentially equivalent to the classification of quadratic forms over a padic field. Let q be a nondegenerate quadratic form in n > 3 variables over a padic field K of characteristic +2, let 0 be the orthogonal group of q, and SO the special orthogonal group of q. All nondegenerate quadratic forms in n variables are equivalent over the algebraic closure R of K, so the classification over K of nondegenerate quadratic forms in n variables is .equivalent to the determination of H’(K, 0). We have an exact sequence 1+so+o+~~+1 where p2 = (k 1); now the map H’(K, 0) + ZZ’(K,&, surjective, hence we have an exact sequence (of sets)
i.e. Oxt p2, is
13 H’(K, SO) : If’@, 0) : H’(K, /.Q. By a twisting technique it may be shown that CI is injective. H’(K,,u,) we use the exact sequence
(1) To compute
2
1+/l,+K*+K*+1 2 where R* + fz* is the map XI+X~. This exact sequence gives rise to a cohomology exact sequence (of groups) K* : K* + H’(K, p2) + 1 + 1 + H2(K, p2) P Br (K) t Br (K) (since H’(K, R*) = 1 by Hilbert’s Satz 90). We deduce that H’(K, p2) E K*/K*2, H2W, p2)= Br WI2 where Br(K), denotes the group of elements of order 2 in the Brauer group Br(K). (Since BrQ = Q/Z, we have Br(K), = p2.) The exact sequence (1) now becomes 1 +H1(K,SO):H’(K,0):K*/K*2. Here d is essentially the discriminant: more precisely, if r E H’(K, 0), d(t) is the discriminant (modulo squares) of the quadratic form defined by r divided by that of q (and therefore d is surjective). The quadratic forms defined over K which have the same discriminant (modulo squares) as q are classified by H’(K, SO). The simplyconnected covering of SO is the spin group: Spin. We have an exact sequence 1+~2+Spin+SO+1
257
SEMISIMPLE ALGEBRAIC GROUPS
and hence a cohomology
exact sequence (of sets)
+ H’(K, Spin) + H’(K, SO): Br(K)2. Spin, + Son: K*/K*’ Here 6 is the spinor norm andd is closely related to the Witt invariant of a quadratic form. From Theorem 2 we have H’(K, Spin) = 1; hence (i) the spinor norm 6 : SOx , K*/K*’ is surjective; (ii) every quadratic form with the same discriminant d and the same Witt invariant as q is Kisomorphic to q. Conversely, (i) and (ii) imply that H’(K, Spin) = 1. 5. Number Jields Let K be an algebraic number field, G a linear algebraic group defined over K. If v is any prime of K and if K, denotes the completion of K at v, we have mappings H’(K, G) + H’(K,, G) obtained mapping
by restriction
together with the embedding
Gx + GKV, hence a
8: H’(K,
G) + n H’(K,, G). (1) ” The theorem of MinkowskiHasse states that two quadratic forms defined over K are isomorphic over K if and only if they are isomorphic over K, for all v (discrete and archimedean). In terms of Galois cohomology, this theorem is equivalent to the injectivity of the map 6:H’(K,0)+~H1(K,,0) ” where 0 is the orthogonal group of a quadratic form. The map 1’3defined by (1) is not always injective, even if G is connected and semisimple (Serre, 1964). However, if G is simplyconnected, the following theorem seems to hold: THEOREM 3. Let K be an algebraic number field, G a simplyconnected semisimple linear algebraic group defined over K. Then 0: H’(K, G) + n H1(KU, G) ” is bijective. In fact, H’(K,, G) is trivial for all discrete v (Theorem 2), so that Theorem 3 is equivalent to THEOREM 3’. The mapping
~‘KG)+)‘jHIUG,G) “EC0 is bijective, where 03 denotes the set of archimedean primes of K.
258
M. KNEWR
The proof of Theorem 3 proceeds by reduction to the case of almostsimple groups, and then by examination of each type A,, . . . , Gz separately. For the classical groups it is related to known results on quadratic, hermitian, etc., forms. For the exceptional groups other than Es see Harder (1965). The case E, is still oben. 3. Tamagawa N’umbers Basic reference (Weil, 1961). Introduction
Let K be an algebraic number field, A the addle ring of K. The letter u will denote a prime (discrete or archimedean) of K; K, denotes the completion of K at v, and o, the ring of integers of K, if v is discrete. Let G be a connected linear algebraic group defined over K. G is isomorphic to a subgroup of a general linear group GL,, and we may therefore regard G as a closed subset of affine mspace (m = nz + 1, see Chapter 1, Section 1) when convenient. The addle group GA of G is the set of all points in A” which satisfy the equations of G. We give GA the topology induced by the product topology on A”, and GA is then a locally compact topological group. GA may also be defined as the restricted direct product of the groups GKU with respect to their compact subgroups G,“, where GKY (resp. G,) is the set of points of G with coordinates in K, (resp. 0,). At first sight, GA appears to depend on the embedding of G in affine space. But it is not difficult to show that if we change the embedding we change G,” at only a finite set of primes v, and hence GA is unaltered. Examples. If G = G,, then GA = A. If G = G,, then G, is the idele group of K. Since K is a discrete subgroup of A, it follows that G, is a discrete subgroup of GA. Since G, is locally compact, it has a leftinvariant Haar measure, unique up to a constant factor. The product formula shows that right multiplication by an element of GK does not change the measure on GA, and therefore we have an induced leftinvariant measure on the homogeneous space GA/GK. There is a canonical choice of the Haar measure on GA, called the Tamagawa measure z (Section 1). Once this is defined there is the problem of computing the Tamagawa number z(G) = r(GA/GK) (Section 2). This, applied to the case where G is an orthogonal group, is equivalent to the classical arithmetical results of Minkowski and Siegel on quadratic forms (Section 3). 1. The Tamagawa measure
Let V be an algebraic variety of dimension n, defined over K. Let x0 be a simple point of V and let x1,. . . , x,, be local coordinates on V at x0 (not
SEMISIMPLE
ALGEBRAIC
necessarily zero at x0). A differential hood of x0 in V by an expression
GROUPS
259
nform on V is defined in a neighbour
co = f(x) dxl . . . dx,
where f is a‘ rational function on V which is defined at x0. The form o is said to be defined over K if f and the coordinate functions xi are defined over K. The rule for transforming o under change of coordinates is the usual one. If cp: W+ V is a morphism of algebraic varieties, a differential form o on V pulls back to a differential form q*(o) on W. Suppose now that V is a connected linear algebraic group G. The left translation 1, : x H ax (a E G) is a morphism V + V and therefore transforms a differential form w on G into a differential form n:(w). Thus we can define leftinvariant differential forms on G, and the basic fact is that there exists a nonzero leftinvariant differential nform o on G, defined over K, and o is unique up to a constant factor c E K*. Examples. G=G,,
o=dx;
G = G,,,
o = dx/x;
o = (n dxij)/(det xij>“. We shall use o to construct measures o, on the local groups GKY. For this purpose we need to fix Haar measures K on the additive groups K,‘. When K = Q we fix pP (p a fi@te prime) by pJZ,> = 1, and p, we take to be ordinary Lebesgue measure on R. When K is an arbitrary number field, there are various possible normalizations: all we require is that (i) p(,(o”) = 1 for almost all discrete v, and (ii) if p = n pL, is the product measure on A, then ,u(A,/K) = 1. G = GL,,
(One such normalization is to take ~~(0”) = 1 for all discrete v, and pV to be c, times Lebesgue measure for archimedean u, where the c, are positive real numbers such that lI c, = 2’2/dKI*, "EC0
where 00 denotes the set of infinite primes of K, r2 is the number of complex infinite primes, and dK is the discriminant of K.) To define oU, we proceed as follows. The rational functionfcan be written as a formal power series in tj = xix: with coefficients in K. If the xp are in K,, thenfis a power series in the xi with coefficients in K,, which converges in some neighbourhood of the origin in K,“. Hence there is a neighbourhood U of x0 in Gk, such that cp :xH(tl,. . . , t,,) is a homeomorphism of U onto a neighbourhood U’ of the origin in K,“, and such that the above power series converges in 27’. In U’ we have the positive measure If(t)l,dtl . . .dr, (where dtl . . . dt, is the product measure pV x . . . x pL, on K,“); pull this back
260
M.
KNESER
to U by means of cp and we have a positive measure w” on U. Explicitly, if g is a continuous realvalued function on Gx, with compact support, then dt1 * * * et. 9% = Lp‘(0)lf(0l” Iu s The measure o” is in fact independent of the choice of local coordinates xi. In the archimedean case this follows from the Jacobian formula for change of variables in a multiple integral, and in the discrete case from its padic analogue. If the product “IJ%W converges absolutely, we define the Tamagawa measure by 7= Y
a”
Explicitly, if S is a finite set of primes such that co E S and if, for each u E S, U” is an open set in GK” with compact closure, then 7 is the unique Haar measure on GA for which 7 “4u”x (
lh, “6s
= )
“US =
TV
x vIjlsdGoJ
If the product
does not converge absolutely, we have to introduce factors to make it converge. A family (A”) of strictly positive real numbers, indexed by the primes v of K, is a set of convergencefactors if the product n G ‘c&L) “Cm is absolutely convergent. The Tamagawa measure
7
(relative to the 2”) is then
7=lJ43D,. Y
In either case, 7 does not depend on the choice of the form w. For if we replace o by cw (c E K*) then (cw)” = Jcl”o”, and n ICI” = 1 by the product ” formula. Let k(v) denote the residue field at v (v discrete) and let G(“) denote the algebraic group defined over the finite field k(v) obtained by reducing the equations of G modulo the maximal ideal p” of 0”. Then it can be shown by a generalization of Hensel’s lemma that for almost all v we have o”(G,“) = (NV)” card (G(“) (1) kc”)) ’ where NV is the number of elements in k(v), and G$& is the group of points of G(“) rational over k(v) .
SEMISIMPLE ALGEBRAIC GROUPS
261
Examples. If G = G,, o,(GJ
= 1.
If G = G,,,,
~v(G,,.) = 1 &; if G = GL,,, then
%(G.S=(lk)...(l&); if G = SL,,, then
Since
rI(1 ”  &)
= MS)1
is convergent for Re (s) > 1 but not for s = 1, it follows that the product n o,(G,,$ converges when G = SL, but not when G = GL,. In the latter &se we may take
as convergencefactors. PROPOSITION.
If G is semisimple, the product n w,(GJ
is absolutely con
vergent (and so convergencefactors are not needed).
In outline, the proof is as follows (for details, see Ono (1965)). From (1) we have to prove that the product n {WV)” card (G&,)1 (2) ” is absolutely convergent. Using the theorem (due to Lang) that isogenous groups over a finite field have the same number of rational points we can reduce to the case where G is simple over the algebraic closure R of K i.e. has no nontrivial algebraic normal subgroups defined over K. For such a group the order of Gg!, can be explicitly determined and is of the form (Nvy{l+O((Nv)2)~. The convergence of (2) now follows from the convergence of the zetafunction t&(s) for Re (s) > 1.
262
M. KNESER
2. The Tamagawa number The Tamagawa number z(G) is defined to be r(GA/GK). THEOREM 1 (Bore1 and HarishChandra, 1962). If G is semisimple, z(G) is jinite. For another proof of this theorem see Godement (196263). THEOREM 2 (Ono, 1965). Let G be a semisimple group, e its universal (i.e. simplyconnected) covering, C the (finite) kernel of the covering map G + G, and let c denote the character group Horn (C, G,,,). Then z(G) ho(c) =i’(e)’ a where ho(e) = Card H’(K, t?) and i’(e) is the number of elements in the kernel of the map H’(K, e) + n H’(K,, c). Hence it is enough to com;ute one Tamagawa number in each isogeny class, for example r(c). In this direction, there is Weil’s CONJECTURE.T(G)= 1.
This conjecture is proved for many of the classical groups [(Weil, 1961, 1964, 1965) for all groups of types B, and C,,, and some of types A, and D,; and (Mars, 196?; Weil, 1961) for some exceptional groups]. Langlands gave a nonenumerative proof for all split semisimple groups (BruhatTits (1965)). Example. G = SO,, (n > 3). Here C has order 2, and it can be shown that h’(c)/i’(e) = 2. So in this case z(G) = 1 is equivalent to z(G) = 2. As we shall see (Section 3), z(G) = 2 is equivalent to the theorems of MinkowskiSiegel on quadratic forms. 3. The theorem of Minkowski and Siegel Let K be an algebraic number field, I/ an ndimensional vector space over K (n > 3), q a nondegenerate quadratic form on V defined over K, G = SO(q) the group of all automorphisms of the vector space V which preserve q and have determinant 1. We regard G as acting on V on the right. We shall identify V with K” by choosing a fixed basis of V; then the elements of G are n x n matrices. A lattice M in V is a finitely generated osubmodule of V of rank n, where o is the ring of integers of K. In particular L = o” is a lattice, called the standard lattice (with respect to the chosen basis of V). Two lattices M, M’ are isomorphic if M’ = Mx for some x E GK. For each finite prime v, let M, = o, @ M, which is a lattice in K,“. It can be shown that M is uniquely determined by the M, [namely M = n (M, n V)]; that M, = 0; for ” almost all v; and that, conversely, if M, is a lattice in K,” for each finite u,
SEMISIMPLE
ALGEBRAIC
GROUPS
263
such that M, = 0: for almost all U, then there is a unique lattice A4 in K” whose local components are the M,. The addle group G,, operates on the set of lattices in K” as follows: if M is a lattice and x E GA, then (MA), = i&x, (observe that, since M, = 0: for almost all u, and X, E G,,” for almost all t), we have (MC), = 0: for almost all v). The orbit of M under GA is the genus of iki, which therefore consists of the lattices in V which are locally isomorphic to M at all finite primes. The class of M is the orbit of M under Gx, i.e. the set of all lattices globally isomorphic to M. The stabilizer of the standard lattice L in GA is the open subgroup
where co denotes the set of archimedean primes of K. Hence the lattices in the genus of L are in oneone correspondence with the cosets GAx in GA, and the classes in the genus with the double cosets G,,xGK in GA. For any semisimple group, the number of double cosets G,,xGx in GA is finite (Bore1 and HarishChandra, 1962). In our special case G = SO(q), this is the wellknown finiteness of the number of classes in a genus. Let h denote this number. We have r(G) = ~GA/&c) = i+L, = i$l7K
xi G&L) %., xi WW
(1)
since 7 is leftinvariant. The group xi ’ GA,xI is the stabilizer of LXi in GA. Hence we need consider only one term in this sum, say r(GA, Gx/Gx). We have GA Gx/Gx z GA/G,, (since GA n Gx = GO). Hence G/W = ~GJGo) = 70 where F is a fundamental domain for GO in GA, i.e. a Bore1 set in GA, which meets each coset xG, exactly once. The projection GA, = G, x G, + G,, restricted to G,, embeds G, injectively in G,, and we may take F to be a set of the form G,x F,, where Fa is a fundamental domain for G, in G,. Hence we have ~(GA,
7(G,m
WW
= 7(G, x F1 = vIjTmdG,J x wdG,/Go)
where “co =“lJp. If we replace L by Lx,, G,,” is replaced by XI,: G,“xi, V and therefore the term
264
M. KNESER
o,(G,J is unaltered (since COis rightinvariant as well as leftinvariant, any semisimple group) ; and GO= Gx n GA, is replaced by G,(x,) = GK n x; ’ GA, xi. Hence ‘5(X; lG~,xi
GKIGK) =vSI_wu(Go,,l X ~m(Gce/Go(Xi))
for
(2)
and therefore, from (1) and (2),
Suppose now that the form q is totally definite, i.e. that each archimedean v is real and that q is definite at each archimedean completion K,. Then G, = n G, 06 00 is a product of real orthogonal groups and is therefore compact, and G&K,) is the group of units E(Lxi) of the lattice Lx*, i.e. the group of all integer matrices which transform LXi onto itself, and is finite. (If S, is the matrix of q with respect to a basis of Lxi, then E(LXi) is the group of all integer matrices X such that xl&X = S,; since S, is definite, E(LXi) is clearly finite.) Hence (3) now takes the form E
I= I Card
l
WW)
=Ll+L=L. %A%)
u I m dG,J
7GJ
(4)
When K= Q, this is equivalent to Minkowski’s formula for the weight of a genus of definite quadratic forms. Similarly, we can recover Siegel’s theorem (193537) on the number of representations of one quadratic form (in n variables, say) by another (in m variables), by considering an mdimensional vector space V endowed with a nondegenerate quadratic form q, an ndimensional subspace W and the Tamagawa number of the group G of all qorthogonal transformations which induce the identity on IV. For details see Kneser (1961) and Weil (1962). REFERENCES
Bore], A. and HarishChandra
(1962). Arithmetic
subgroups of algebraic groups.
AM. Math. 75, 485535. Bruhat, F. and Tits, J. (1965). A.M.S. Summer Institute on Algebraic Groups and Discontinuous Subgroups, Boulder. (To appear.) Chevalley, C. (1955). Sur certains groupes simples. Tohoku math. J. 7, 1466.
SEMISIMPLE
ALGEBRAIC
GROUPS
265
Chevalley, C. (195658). Classification des groupes de Lie algebriques. SPminuire E. N.S. Demazure, M. and Grothendieck, A. (196364). Schemes en groupes. Sbminaire de Giometrie Algibrique I.H.E.S. Godement, R. (196263). Domaines fondamentaux des groupes arithmetiques. S&ninaire Bourbaki, No. 257. Harder, G. (1965), l%er die Galoiskohomologie halbeinfacher Matrizengruppen, I. Math. 2. 90, 404428; and II, (1966), 92, 396415. Iwahori, N. and Matsumoto, H. (1965). On some Bruhat decomposition and the structure of the Hecke rings of padic Chevalley groups. Publs. math. ht. Etud. scient., No. 25. Kneser, M. (1961). DarstellungsmaBe indefiniter quadratischer Formen. Math. Z. 77, 188194. Kneser, M. (1965). GaloisKohomologie halbeinfacher algebraischer Gruppen tiber padischen Korpern. Math. 2. 88,4047; and 89, 250272. Mars, J. M. G. (196?). Les nombres de Tamagawa de certains groupes exceptionnels. (To appear.) Ono, T. (1965). On the relative theory of Tamagawa numbers. Ann. Math. 82, 88111. Rosenlicht, M. (1956). Some basic theorems on algebraic groups. Am. J. Math. 78, 401443. Serre, J.P. (1964). “Cohomologie Galoisienne”, Lecture notes, College de France, 196263, 2nd edition, SpringerVerlag. Siegel, C. L. Uber die analytische Theorie der quadratischen Formen. Ann. Math. (1935), 36, 527606; (1936), 37,230263; (1937), 38,212291. Steinberg, R. (1959). Variations on a theme of Chevalley. Pucif. J. Math. 9,875891. Steinberg, R. (1965). Regular elements of semisimple algebraic groups. Publs. math. Inst. ht. Etud. scient., No. 25. Weil, A. (1961). “Adeles and Algebraic Groups”, Lecture notes, Princeton. Weil, A. (1962). Sur la theorie des formes quadratiques. Colloque sur la ThCorie des Groupes Algebriques, Bruxelles, 922. Weil, A. (1964). Sur certains groupes d’operateurs unitaires. Acta Math. Stockh. 111, 143211. Weil, A. (1965). Sur la formula de Siege1dansla theorie des groupes classiques. Actu Math. Stockh. 113, l87.
CHAPTER XI
History
of Class Field Theory HELMUT
HASSE
1. The notion of cZassfieZdis generally attributed to Hilbert. In truth this notion was already present in the mind of Kronecker and the term was coined by Weber, before Hilbert’s fundamental papers appeared. Kronecker, in his large treatise “Grundziige einer arithmetischen Theorie der algebra&hen Griissen” of 1882, discusses at length the “zu assoziierenden Gattungen” (species to be associated). What he means by this is, in modern terminology, an algebraic extension K of a given algebraic number field k such that all divisors? in k become principal divisors in K. By his investigations he had found that for imaginary quadratic fields k the “singular moduli” generate such extensions K. By this notion Kronecker anticipates the principal divisor theorem of class field theory, stated and in special cases proved by Hilbert. Weber (1891, 1897a, b, 1898, 1908), however, did not define the notion of class field on this basis, a basis which, as we know today, is unsuitable for building the theory. What he postulated is part of the law of decomposition. Whereas Hilbert in his later definition considered only the case of absolute divisor classes, Weber gave his definition in full generality, viz. Let k be an algebraic numberfield and A/H a congruencedivisor classgroup in k. An algebraic extension K/k is called cIassJield to A/H, if exactly those prime divisors in k of first degree which belong to the principaI ‘classH split completely in K. In order to define Weber’s notion of a congruence divisor classgroup AJH in k, consider integer divisor moduli m in k, i.e. formal finite products composed of finite prime spots (prime divisors) of k with positive integer exponents and real infinite prime spots of k with exponents 1. Call a number or a divisor in k prime to m if it is prime to all prime divisors contained in nt,
and understand congruence mod m as congruence modulo all prime divisor powers contained in m and equality of sign for all real prime spots contained t I use throughout the term “divisor” instead of the classical term “ideal”, because valuation theory, derived from the KroneckerHensel notion of divisor, has been nowadays adopted widely as a more suitable foundation of algebraic number theory than Dedekind’s ideal theory. 266
HISTORY
OF CLASS
FIELD
THEORY
267
in nt. Further consider quotient groups A,,/H,,, where A,,, is the group of all divisors in k prime to m for a given integer divisor modulus m and H,,, is a subgroup containing all numbers a = 1 mod nt in k (considered as principal divisors in k). Call two such quotient groups A,,,,/H,,,, and A,,/H,,,, “equal” to each other if for the least common multiple m = [m,, nt2] (and hence for every common multiple of ml and ntJ there holds the equality = H,,,, n A,,,, which implies the isomorphy A,,,/H,,,, LZ A,,/H,,,,. Hm, n Am Each set of quotient groups A,,/H,,, A,,/H,,,,,. . . that are all “equal” to each other defines a congruence divisor class group A/H in Weber’s sense. The single quotient groups A,,/H,,, A,,/H,,,,,. . . are called the interpretations (Erklirungen) mod m,, mod m2,. . . of A/H. The lowest possible interpretation modulus (Erklirungsmodul) f, which turns out to be the greatest common divisor of all possible m,, m,,. . . is called the conductor (Fiihrer) of A/H. Occasionally the single interpretations are also called congruence divisor class groups. Besides the theorems on class fields proved by Weber which will be adduced presently, in the cases he treated he further observed also the fundamental isomorphy theorem : The Gulois group @(K/k) is sureIy ubeliun.
is isomorphic
to the CIUSSgroup A/H,
and hence
Already Kronecker (1853, 1877) knew that the cyclotomic fields are class fields in the above sense. He stated his famous completeness theorem: Every ubeliun field over the rational number field is a cyclotomic field, and hence is a ClussJield.
This was first proved completely by Weber (1886, 1887) and later more simply by Hilbert (1896, 1897); further proofs were given by Weber (1909), Speiser (1919) and Delaunay (1923). Further Kronecker (18831890), by his investigations on modular functions and elliptic functions with “singular moduli”, had ascertained that their transformation and division equations generate relatively abelian fields K over imaginary quadratic fields k. It was his “liebster Jugendtruum” (1880) (dearest dream of his youth) to prove also in this case the completeness theorem, that every relatively ubeliun jield K over an imaginary field k is obtained by such transformation and division equations.
quadratic
This was proved later, first partially by Weber (1908) and Fueter (1914), then completely by Takagi (1920) and once more by Fueter (with Gut) (1927). Weber (1897, 1898) conceived his notion of class field from those examples. By making use of Dirichlet’s analytic means (Lseries) he deduced from his class field definition thefirst? fundamental inequality of class field theory: [A:H]=h
t
In
modern terfninology: second.
268
HELMUT
HASSE
further the uniqueness theorem: H = H’cs
K = K’,
the ordering theorem: Hz
and the following presupposition
H’csKEK’,
for the isomorphy
theorem:
KJk is normal.
From the introduction to Weber (1897, 1898) it is safe to conclude that he was convinced of the validity of the general existence theorem: To every congruence field K/k.
divisor
cIass group A/H
in k there exists
a class
He points out that from the existence of a class field K/k follows the existence of an itinity of prime divisors in the single classes of A/H, i.e. a farreaching generalization of Dirichlet’s famous theorem on primes in prime residue classes. He did not know, however, other examples for the existence of class fields than those from the theory of cyclotomic, modular and elliptic functions. 2. The preceding references to Weber’s share in the origin of class field theory are not meant to detract from the great merit of Hilbert in the development of this theory, but only to put it in the right light. Hilbert himself had a high opinion of Weber; he has repeatedly quoted and duly acknowledged his ideas and results on class fields. The publications of Hilbert (1898, 1899a, b, 19OOa, b, 1902) on class field theory are apparently concerned only with the special case of absolute divisor classes, i.e. where the principal class H contains all principal divisors (without or with sign conditions). Moreover, he had carried through the proofs only for the reZativeIy quadratic number fieldsi.e. n = 2with class number h = 2. Before his mind’s eye, however, he had throughout the most general case. For instance, in his lecture (1899a) before the DMV (German Mathematical Association) meeting at Braunschweig (Brunswick) in 1897 he declared (in free translation): “In this lecture we have restricted ourselves to the investigation of relatively abelian fields of second degree. This restriction however is only a provisional one, and since all conclusions in the proofs of the theorems are capable of generalization, it is to be hoped that the difficulties of establishing a theory of relatively abelian fields will not be insurmountable.” In this sense he had in mind in full generality the main theorems of class field theory already mentioned (existence theorem, uniqueness theorem, ordering theorem, isomorphy theorem and decomposition law). For Hilbert class field theory was not only, as it was for Kronecker and
HISTORY
OF CLASS
FIELD
THEORY
269
Weber, a means towards proving the completeness theorem and towards generalizing Dirichlet’s prime number theorem. As clearly stated in utterances like that quoted above and in the headline to his note (1898, 1902) in the G&finger Nachrichten, he rather regarded it as a “Theory of relatively abelian fields”. A further aim of his was the problem of the higher reciprocity laws, handed down from Gauss, Jacobi, Eisenstein and Kummer, and appearing in his famous Paris lecture of 1900 as Problem 9. Indeed, one of Hilbert’s most profound achievements is the conception of the reciprocity law as a product formula for his norm residue symbol a,b 1. n( P 7 > “= Here k is assumed to contain the nth roots of uinty and p runs through all prime divisors of k, including what we call today the infinite prime spots, introduced by him as symbols 1, l’, l”, . . . . He pointed out the analogy of this formula (given by him only in special cases) to the residue theorem for algebraic functionsthe prime spots p with norm residue symbol # 1 corresponding to the ramification points of the Riemann surface.? This analogy was later splendidly justified by the subsummation of the product formula in algebraic function fields with finite constant field under the residue theorem by Schmid (1936) and Witt (1937). Moreover, Hilbert stated (again only in special cases) the norm theorem: = 1 for all p c> a is relative norm from k(Jb), so important for the further development of the theory. This theorem was proved later in full generality by me (193Oc). Due to the restriction to the absolute divisor class group A/H, Hilbert’s list of class field theorems contains, in addition to those already mentioned, the discriminant theorem: K/k has relative discriminant b = 1. He further stated the principal divisor theorem, already touched upon earlier in this Chapter: AN divisors from k become principal divisors in K. Moreover, he asserted : The class number of K is not divisible by 2, and this not only under his original restriction, that k has class number h = 2, but also for h = 4. t From today’s standpoint this is not quite correct. In number theory the norm residue symbol may be different from 1 also without ramification, namely by inertia. In algebraic function fields over the complex number field this does not occur, so that for them only ramification comes into question.
270
HELMUT
HA&SE
Excepting only this latter assertion, which is true for h = 2 but not necessarily for h = 4, all Hilbert’s assertions on class fields have proved true in the general case. For the principal divisor theorem, however, things turned out to be much more complicated than Hilbert apparently thought. One cannot help feeling the greatest admiration for his keenmindedness and penetration, which enabled him to conceive with such accuracy general laws from rather special cases. 3. As to the reciprocity law, in 1899 the “Kiinigliche Gesellschuft der Wissenschaften zu Giittingen”, presumably on Hilbert’s suggestion, had set as prizesubject for 1901 the detailed treatment of this law for prime exponents Z # 2. This problem was solved by Furtwgngler. In the prizetreatise (1902, 1904) itself he gave the solution only for fields k with class number not divisible by Z, and without distinguishing between the Zl different nonresidue classes. In further papers (1909, 1912, 1913, 1928), however, he proved the law of reciprocity for prime exponents 1 (including 2) in full generality. Not only did he prove it in the classical shape: a i7,=
b a,
0 0
if a and b are primary
(and in case I= 2, totally
positive),
but also in the shape of Hilbert’s product formula for the norm residue symbol. According to a very beautiful idea of his, the law in the classical shapeby passing to the extension field li = k(y(ab‘)) over which Ii = E(vb) is unramified and hence contained in the absolute class field Kcomes back to the decomposition law in ZQli. In other words, that
(ii=0,
over E depends only on the absolute
class in E to which p
belongs. This decomposition law was at his disposal since he had previously proved the existence theorem for the absolute class field (1907). 4. A decisive step forward was taken by Takagi (1920, 1922) in two highly important papers on class field theory and the law of reciprocity. Takagi had been studying Hilbert’s papers on the theory of numbers at the turn of the century and had investigated the relatively abelian fields over the Gaussian number field. This resulted in his solution (1903) of Kronecker’s famous completeness problem over this field by means of the division theory of elliptic functions in the lemniscatic case (i.e. where the period parallelogram is a square). In his big class field paper of 1920 Takagi gave class field theory the decisive new turn by starting from a new definition of the notion of “class field”. His new definition was more suitable than Weber’s because it allows one to envisage the important completeness theorem right from the beginning.
HISTORY
OF CLASS
FIELD
THEORY
271
For an arbitrary relative field K/k he makes correspond to each integer divisor modulus m in k a congruence divisor class group interpreluted (erkhart) mod m, viz. the smallest quotient group A,,,/&,, whose principal class I& contains all relative norms N(N) of divisors ‘3 in K prime to m and hence consists of all divisors a in k prime to m with a N i’V(‘%) mod m, i.e., $&
E a = 1 mod m (a E k).
For m sufficiently high, more precisely for all multiples m of a lowest f (the conductor of K/k) the class groups A,,,/& turn out to be “equal” and hence constitute a congruence divisor class group A/H in Weber’s sense with conductor f. According to Weber there holds the already mentioned tist fundamental inequality [A:Hj=h
?+Pjf, an assertion which was later supplemented
by me (1926, 1927, 193Oa, 1930d)
to
b = v fp
f = M fp x
272
HELMUT
HASSE
(conductordiscriminant theorem) where x runs through the characters of and fx denotes their conductors, while M denotes the least common x multiple taken over all x. Whereas the existence theorem is obtained from the uniqueness theorem and completeness theorem by a suitable enumeration, the proof of the completeness theorem comes back to the second? fundamental inequality: A/H
h 2 n.
This is obtained by a farreaching generalization of the classical theory of genera from Gauss’ Disquisitiones Arithmeticae. Nowadays the theory of cohomology has permitted one to systematize the rather complicated chain of conclusions that leads to this inequality. 5. As to the reciprocity law, the above mentioned further paper of Takagi’s (1922) already gave considerable simplifications of Furtwangler’s rather complicated argument (only 40 instead of 80 pages!). These simplifications were possible because of the availability of the full class field theory instead of that only for the absoZute class field. It was not before Artin, however, that an entirely new idea of fundamental importance was brought to bear on this law. He found the key to its conceptual (as opposed to formal) significance in explicitly giving a canonical isomorphism of the class group A/H onto the Galois group 0(K/k).
Artin (1927) showed that such an isomorphism is obtained by making correspond to each prime divisor p,#, or rather to its class in A/H, the socalled Frobenius automorphism FP of K/k with respect to p. This automorphism is defined by AF’ 3 ABlu’) mod p, for all integers A E K, where a(p) is the absolute norm of p. Today one writes
Klk a as a multiplicative function ( > a in k prime to b or, what is the same, to f.
and one defines from this the Artin symbol on the group A, of all divisors The isomorphy between A/H and B(K/k) reciprocity
may then be expressed by Artin’s
law:
=loa~H,. Artin (1924) had conceived this reciprocity law four years earlier, and had proved it in simple special cases. However, it was not until he came to know t In modern terminology: first.
HISTORY
OF CLASS
FIELD
273
THEORY
Tchebotarev’s method of crossing the classes of A/H with the congruence classes corresponding to a relatively cyclotomic field, that he succeeded in giving a general proof. Tchebotarev (1926) had invented this method in order to improve on a theorem of Frobenius (1896), viz. : The prime divisors ‘$ of a normal K/k with Frobenius automorphism F, in a given division (Abteilung) S  ‘F; S (v prime to the order of FV, S running through @(K/k)) have a density, namely the relative frequency of the division in the whole group (li(K/k). The method in question enabled Tchebotarev to generalize this theorem to a single conjugacy class S  ‘F+i’. For a Kummer field K = k(qb) ( w h ere k contains the nth roots of unity) Klk the Frobenius automorphism p changes J/b into i ;lb, as is clear ( > 0 from the definition
of the power residue symbol
b

0 Pn
’
by Euler’s criterion.
Hence Artin’s reciprocity law yields the assertion: 0
b p
depends only on the class to which p belongs in the congruence class
group kod fb corresponding
to k(qb).
Here fb denotes the conductor of k(Jfb). On the other hand, from the definition
i depends only on the residue 0 Pn class to which b belongs mod p. From this it is easy to deduce the reciprocity in its classical shape and also in the shape of Hilbert’s product formula. Thus it becomes understandable and justified that Artin called the above isomorphy assertion the “general law of reciprocity”. With the help of this law, Artin (1930) could also reduce the principal divisor theorem, enunciated by Hilbert and not yet proved by Takagi, to a pure grouptheoretical proposition, which then was proved by Furtwangler (1930). Further proofs of this proposition were given by Magnus (1934), Iyanaga (1934), Witt (1936, 1954) and Schumann and Franz (1938), whereas Taussky and Scholz (1932, 1934) investigated more closely the process of “capitulation”, i.e. of becoming a principal divisor in the subfields of the absolute class field. Exhaustive results concerning this latter problem, however, have not been achieved up to now. 6. Analogously
to the process of basing the power residue symbol
the more general Artin symbol basing the Hilbert
b
 on 0 0”
, I succeeded (1926,1927 and 1930a, b) in
norm residue symbol
contains the nth
274
HELMUT
HASSE
a, Klk
over an arbitrary k (which need not P > contain the nth roots of unity). At first, for all prime spots p of k, my definition followed the roundabout way Hilbert was forced to take for the prime divisors pin. As a consequence of this, the connection of the symbol with the norm residues became apparent only on the strength of Artin’s reciprocity law. Shortly afterwards, however, I was able to give (1933) a new definition from which this connection was immediately clear. My new definition proceeded on a conceptual basis from the theory of algebras. I had shown previously (193lb) that every central simple algebra of degree n over a local field k, with prime element rt possesses an unramified (and hence cyclic) splitting field Z,. Consequently such an algebra has a canonical generation un = xv* %‘zvup = z,“p roots of unity) on a symbol

(
(F, the Frobenius automirphism).
On the strength of this, the residue class
y_pmod+ 1 is invariantly associated with the algebra. n cyclic field K/k I put then
For a global relatively
when the cyclic algebra over k, generated by 24”= a, after extension to the completion ticular, for a Kummer
= KS,
k,, has the invariant
field K = k(db)
unity) the automorphism From this definition
u‘Ku
: mod+ 1. In par
(w h ere k contains the nth roots of
changes vb . into one infers immediately
the local property:
= 1 G a norm from KP/k,, where KP denotes the isomorphy
type of the completions
divisors Cplp. Hence the symbol
K, for the prime
is now called simply the norm
symbol. Globally I could prove (1926, 1927, 1930a, 193Oc) the norm theorem: = 1 for all pea
anticipated
by Hilbert
norm from K/k,
for his symbol, as mentioned
before.
So Hilbert’s
HISTORY
product formula
OF CLASS
FIELD
275
THEORY
for his symbol became the product formula
for
the norm
symbol:
l$9)=1. This is equivalent
to my sum theorem for central simple aIgebras (1933): F:zO
mod+
1.
Whereas the definition of the norm symbol generalizes from relatively cyclic to arbitrary relatively abelian fields K/k by formal composition, I showed (1931a) that the norm theorem does not remain true in this generality. In connection with the local property of the norm symbol I was led (193Oc) to establish the main theorems of a classfield theory over local fields kP. Here one has a oneone correspondence between all relatively abelian fields KP/kP and all congruence number class groups A,/H, in k,, such that a, Kplkp gives a canonical isomorphism of Ap/Hp onto Q(Kp/k,). The ( P > connection with class field theory over global fields k is given by the relations: (i) KP/kP represents the isomorphy type of the completion of K/k for the ‘PIP; (ii) B(K+‘/k,) is the decomposition group of K/k for the !Q[p. Subsequently, Schmidt (1930) and Chevalley (1933a) gave a systematic development of local class field theory without making reference, as I had done, to that connection with global class field theory. Here I should also mention the essential contributions made by Herbrand (1931 and 1932) to the group theoretical mechanism of certain proofs in local as well as global class field theory, and to the higher ramification theory of normal extensions. 7. In Takagi’s class field theory the characterization of the relatively abelian fields K/k by means of the corresponding congruence divisor class groups A/H in k has a disadvantageous fault of beauty. This fault arises from the approximations A,,,/H, to A/H by a kind of limit process with ascending divisor moduli tn. After padic concepts and methods had been brought to bear on class field theory in the manner described before, Chevalley (1933b) had further the happy idea of replacing also the WeberTakagi characterization in terms of the congruence divisor class groups A/H by a smoother padic characterization. In this he succeeded by introducing his ideal elements, or briefly idsles, namely vectors a = (. . .,uy,. . .) with components
ap from the single completions
k, satisfying a suitable
276 finiteness condition,
HELMUT
HASSE
viz.,
up r 1 for almost all p. He replaced the WeberTakagi congruence divisor class groups A/H by factor groups A/H of the absolute id2le class group A/k*, where k* denotes the group of principal idbles (. . ., a,. . .) corresponding to the numbers a # 0 from k. He proved that the symbol (y)
= jj (%y>,
named after him the Chevalley symbol, gives an isomorphism between an idele class group A/H and the Galois group B(K/k). Here, the principal class H consists of all those absolute idble classes in k that contain idele norms from K. In Chevalley’s class field theory this idble class group takes the role of Takagi’s congruence divisor class group A/H. To the set of all relatively abelian fields K/k corresponds in this way the set of all idele class groups A/H with open principal class H in a suitable topology of A, namely that in which the idele units s 1 mod m for all divisor moduli m form a complete system of neighbourhoods of 1. Thus one can say that by Chevalley’s ideas (not to say: idrYes) the localglobal principle has taken root in class field theory. There is a further fault of beauty in Takagi’s class field theory that was obviated by Chevalley (1940). This is the recourse to analytic means (Dirichlet’s Lseries) for the proof of the first fundamental inequality h < n. Chevalley succeeded in proving this inequality in a purely arithmetical manner. 8. There still remains much to be said about further developments arising from the class field theory delineated up to this point. For instance, what lies particularly close to my heart, the explicit reciprocity formulae (determination of the norm symbol a, p b for prime divisors pin); further inclusion of ( >n infinite algebraic extensions K/k, class field theory over congruence function fields (algebraic function fields with finite constant field), Artin’s Lseries and conductors, and so on. I must, however, refrain from talking about those
subjects here, because that would exceed the frame of this lecture. I cannot touch either on the further development of class field theory after the war, but suppose I must conclude my survey of the historical development at this point. If I have understood rightly, it was my task here to delineate for the mathematicians of the postwar generation a vivid and lively picture of the great and beautiful edifice of class field theory erected by the prewar generations. For the sharply profiled lines and individual features of this
HISTORY
OF CLASS
FIELD
THEORY
277
magnificent edifice seem to me to have lost somewhat of their original splendour and plasticity by the penetration of class field theory with cohomological concepts and methods, which set in so powerfully after the war. I should like to think that I have succeededto some extent in this task. REFERENCES?
Artin, E. (1924). I%er eine neue Art von LReihen. Abh. Math. Semin. Univ. Hamburg, 3, 89108. (Collected Papers(1965), 105124.Addison Wesley.) Artin, E. (1927). BeweisdesallgemeinenReziprozitltsgesetzes.Abh. Math. Semin. Univ. Hamburg, 5, 353363.(Collected Papers(1965), 131141.Addison Wesley.) Artin, E. (1930). Idealklassenin Oberkorpem und allgemeinesReziprozitatsgesetz. Abh. Math. &mitt. Univ. Hamburg, 7, 4651. (Collected Papers(1965), 159164. Addison Wesley.) Chevalley, C. (1933a). La theorie du symbole de restesnormiques.J. reine angew. Math.
169, 140157.
Chevalley, C. (19336). Sur la theorie du corps de classesdans lescorps finis et les corps locaux. J. Fat. Sci. Tokyo Univ. 2, 365476. Chevalley, C. (1940). La theorie du corps de classes.Ann. Math. 41, 394417. Delaunay, B. (1923). Zur BestimmungalgebraischerZahlkijrper durch Kongruenzen; eine Anwendung auf die AbelschenGleichungen.J. reine angew. Math. 152, 120123. Frobenius, G. (1896). Uber die Beziehungen zwischen den Primidealen eines algebraischenKorpers und den Substitutionen seinerGruppe. Sber. preuss. Akad. Wiss., 689703.
Fueter, R. (1914). Abel’sche Gleichungen in quadratischimaginlren ZahlkGrpern. Math. Annln. 75, 177255. Fueter, R. (1927). “Vorlesungen tiber die singularen Moduln und die komplexe Multiplikation der elliptischen Funktionen II” (unter Mitwirkung von M. Gut). LeipzigBerlin. Furtwangler, Ph. (1902). Uber die Reziprozitatsgesetzezwischen lten Potenzresten in algebraischenZablkiirpem, wenn 1eine ungerade Primzahl bedeutet. Abh. K. Ges. Wiss. Giittingen. (Neue Folge), 2, Nr. 3, l82.=Math. Annbt. 58 (1904), l50. Furtwlngler, Ph. (1907). Allgemeiner Existenzbeweisfur den Klassenkorpereines beliebigenalgebraischenZahlkdrpers. Math. Annln. 63, l37. Furtwlngler, Ph. Reziprozitltsgesetze ftir Potenzreste mit Primzahlexponenten in algebraischenZahlkijrpern I, II, III. Math. An&. (1909), 67, 131; (1912), 72, 346386;
(1913), 74,413429.
Furtwangler, Ph. (1928). I%er die Reziprozitltsgesetze ftir ungerade Primzahlexponenten. Math. Ant&t. 98, 539543. Furtwangler, Ph. (1930). Beweis des Hauptidealsatzesftir Klassenkijrper algebraischer Zahlkiirper. Abh. Math. Semin. Univ. Hamburg, 7, 1436. t This list of publications from the years 18531940 (with one exception from 1954) does not pretendto be a completelist of publicationsaboutclassfield theory and related subjects for this period. It contains only those publications which appeared important as references for what has been said in the lecture.
278
HELMUT
HASSE
Hasse, H. Bericht tiber neuere Untersuchungen und Probleme aus der Theorie der algebraischen Zahlkiirper I, Ia, II. Jber. dt. MutVerein. (1926), 35, l55; (1927), 36,233311; Exg. Bd. (193Ou),6, l204. Hasse, H. (19306). Neue Begrtindung und Verallgemeinerung der Theorie des Normenrestsymbols.J. reine angew.Math. 162, 134144. Hasse,H. (193Oc).Die NormenresttheorierelativAbelscher Zahlkorper alsKlassenkorpertheorie im Kleinen. J. reine ungew.Math. 162, 145154. Hasse,H. (193Od).Ftihrer, Diskriminante und Verzweigungskiirper relativAbelscher Zahlkorper. J. reine angew.Math. 162, 169184. Hasse,H. (1931u).BeweiseinesSatzesund Widerlegung einer Vermutung tiber das allgemeineNormenrestsymbol. Nuchr. Ges. Wiss.Giittingen, 6469. Hasse,H. (1931b). Uber padischeSchiefkorper und ihre Bedeutung ftir die Arithmetik hyperkomplexer Zahlsysteme.Math. An&. 104,495534. Hasse, H. (1933). Die Struktur der R. BrauerschenAlgebrenklassengruppetiber einem algebraischenZahlkorper. Math. Annln. 107, 731760. Herbrand, J. (1931). Sur la theorie des groupes de decomposition, d’inertie et de ramification. J. Math. putesuppl. 10,481498. Herbrand, J. (1932). Sur lestheortmes du genreprincipal et desideaux principaux. Abh. Math. Semin. Univ. Hamburg, 9, 8492. Hilbert, D. (1896). Neuer Beweis des Kronecker’schen Fundamentalsatzestiber Abel’sche Zahlkorper. Nuchr. Ges. Wiss. Giittingen, 2939. Hilbert, D. (1897). Bericht: Die Theorie der algebraischenZahlkijrper. Jber. dt. MutVerein. 4, 175546. Hilbert, D. (1898). Uber die Theorie der relativAbel’schen Zahlkorper. Nuchr. Ges. Wiss.Giittingen, 377399= Actu math. Stockh. (1902), 26, 99132. Hilbert, D. (1899u).Uber die Theorie der relativquadratischenZahlkorper. Jber. dt. Mat Verein. 6, 8894. Hilbert, D. (18996). Ubcr die Theorie desrelativquadratischen Zahlkiirpers. Math. Annln. 51, 1127. Hilbert, D. (19OOu).Theorie der algebraischenZahlkiirper. Enzykl. math. Wiss. I C 4a, 675698.
Hilbert, D. (19OOb).Theorie desKreiskorpers. Enzykl. math. Wiss.I C 4b, 699732. Hilbert, D. (19OOc).MathematischeProbleme. Vortrag auf internat. Math. Kongr. Paris 1900. Nuchr. Ges. Wiss. Giittingen, 253297 (reprinted in “A Collection of Modern Mathematical Classics,Analysis” (ed. R. Bellman), Dover, 1961). Iyanaga, S. (1934). Zum Beweis des Hauptidealsatzes.Abh. Math. Semin. Univ. Hamburg, 10,349357. Kronecker, L. (1853).Uber die algebraischauflijsbarenGleichungenI. Sber.preuss. Akud. Wiss.365374= Werke IV, l1 1. Kronecker, L. (1877). Uber Abel’sche Gleichungen.Sber. preuss.Akud. Wiss.845851= Werke IV, 6372. Kronecker, L. (1882). Grundziige einer arithmetischenTheorie der algebraischen Grossen. J. reine ungew.Math. 92, l122=Werke II, 237388. Seethere 0 19, 6568 = 321324.
Kronecker, L. (188390). Zur Theorie der elliptischen Funktionen IXXII. Sber. preuss.Akud. Wiss.=Werke IV, 345496= V, 1132. Kronecker, L. Auszug aus Brief an R. Dedekind vom 15 M&z 1880. Werke V, 453458. Seealso my detailed “Zusutz”, ibid., 510515. Magnus, W. (1934). Uber den BeweisdesHauptidealsatzes.J. reine ungew.Math. 170,235240.
HISTORY
OF CLASS
FIELD
THEORY
279
S&mid, H. L. (1936). Zyklische algebraische Funktionenkorper vom Grade pm fiber endlichem Konstantenkorper der Charakteristik p. J. reine angew. Marh. 175, 108123. Schmidt, F. K. (1930).Zur Klassenkorpertheorieim Kleinen. J. reine angew.Math. 162, 155168.
Schumann,H. G. (1938). Zum BeweisdesHauptidealsatzes(unter Mitwirkung von W. Franz). Abh. Math. Semin. Univ. Hamburg, 12, 4247. Speiser,A. (1919). Die Zerlegungsgruppe.J. reine angew.Math. 149, 174188. Takagi, T. (1903).Ober die im Bereich der rationalen komplexen Zahlen Abel’schen Zahlkorper. J. COILSci. imp. Univ. Tokyo, 19, Nr. 5, 142. Takagi, T. (1920). Uber eine Theorie des relativAbel’schen Zahlkorpers. J. CON. Sci. imp. Univ. Tokyo, 41, Nr. 9, 1133. Takagi, T. (1922). Uber das Reziprozitltsgesetz in einem beliebigenalgebraischen Zahlkorper. J. COIL Sci. imp. Univ. Tokyo, 44, Nr. 5, l50. Taussky, 0. (1932). Uber eine VerschPrfung des Hauptidealsatzesfti algebraische Zahlkiirper. J. reine angew.Math. 168, 193210. Taussky, 0. and Scholz, A. (1934). Die Hauptideale der kubischenKlassenkijrper imagintiquadratischer Zahlkiirper : ihre rechnerischeBestimmungund ihr EinfluD auf den Klassenkorperturm.J. reineangew.Math. 171,1941. Tchebotarev, N. (1926). Bestimmungder Dichtigkeit einer Menge von Primzahlen, welchezu einer gegebenenSubstitutionsklassegehoren.Math. Annln. 95,191228. Weber, H. Theorie der Abel’schen Zahlkijrper I, II. Acfu math. Sfockh. (1886), 8, 193263; (1887), 9, 105130. Weber, H. uber Zahlengruppen in algebraischenK&pen I, II, III. Math. Ann/n. (1897), 48,433473; (1897), 49, 83100; (1898), 50, l26. Weber, H. (1891).“Elliptische Funktionen und algebraischezahlen”.Braunschweig. Weber, H. (1908). “Lehrbuch der Algebra III”.? Braunschweig. Weber, H. (1909). Zur Theorie der zyklischen Zahlkorper. Math. An&n. 67, 3260. Witt, E. (1936). Bemerkungenzum Beweisdes Hauptidealsatzesvon S. Iyanaga. Abh. Math. Semin. Univ. Hamburg, 11, 221. Witt, E. (1937).Zyklische K&per und Algebren der Charakteristikp vom Gradep”. J. reine angew.Math. 176, 126140. Witt, E. (1954).Verlagerungvon Gruppen und Hauptidealsatz. Proc. Internat. Math. CongressII, Amsterdam 1954,7173.
t Secondedition of “ElliptischeFunktionenund Algebra&he Zahlen”.
CHAPTER XII
An Application
of Computing
to Class Field Theory
H. P. F. SWINNERTONDYER Let G be a connected algebraic group. Chevalley has shown that G contains a normal subgroup R such that R is a connected linear group and GJR is an Abelian variety?; and these properties determine R uniquely. The general theory of algebraic groups therefore depends on theories for linear groups and for Abelian varieties. In Chapter X of this book Kneser has described the known numbertheoretical properties of linear groups, and the same thing will be done here for Abelian varieties. But there is a fundamental difference between these two topics. Kneser has expounded a theory which is reasonably complete and satisfactorythe major results have been or are on the point of being proved, and there is no reason to think that there are important theorems as yet undiscovered. For Abelian varieties, on the other hand, very few numbertheoretical theorems have yet been proved. The most interesting results are conjectures, based on numerical computation of special cases, and the fact that one has no idea how to attack these conjectures suggests that there must be important theorems which have not yet even been stated. The original calculations are largely due to Birch and SwinnertonDyer ; and their results can be expressed in respectable language through the efforts of Cassels and of Tate. One object of this chapter is to describe the numerical evidence, to show how far the conjectures are directly based on it, and to explain why it seems right to state the conjectures in very general terms although all the evidence refers to cases of one particular kind. With the advent of electronic computers, calculations which would previously have been out of the question are now quite practicable, provided that they are sufficiently repetitive. There must be many topics in number theory in which intelligently directed calculation would yield valuable results. (At the moment, most numbertheoretical calculations merely pile up lists of integers in the manner of a magpie; they are neither designed to produce valuable results nor capable of doing so.) The second object of this chapter 7 An Abelian variety is a connected algebraic group which is also a complere algebraic variety; the group operation on an Abelian variety must necessarily be commutative. The simplest example of an Abelian variety is an nonsingular cubic curve in the projective plane, together with a point on it which is the identity for the group operation. 280
AN
APPLICATION
OF COMPUTING
TO
CLASS
FIELD
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is to show what can reasonably be done on a computer, and what the numbertheorist must consider before he can ask someone to do his calculations for him. When a modern algebraist defines something, he usually begins by taking the quotient of one very infinite object by another. A process of this kind can seldom be carried out on a computer even in theory; and it never can in practice. Thus an object which can only be defined in this sort of way is not effectively computable. Conversely, if something has to be computed it must be defined in a way which is more downtoearth, even if less canonical. This applies not only to the final result of the calculation, but to anything which turns up in the course of it. This may be difficult or even impossible: for example the TateSafareviE group (defined below) has at the moment no constructive definition. Once a numbertheorist has phrased his problem in computable terms, he can estimate how much machine time it will take. This depends on the number of operations involvedan operation for this purpose being an addition, subtraction, multiplication or division. (This is a crude method of estimation, and ignores for example the time spent in organizational parts of the program; but it will give the right order of magnitude.) A calculation which takes lo6 operations is trivial, and is worth doing even if its results will probably be useless. A calculation which takes lo9 operations is substantial but not unreasonable. It is worth doing in pursuit of any serious idea, but not just in the hope that something may turn up; moreover, the method of calculation should now be reasonably efficient, whereas for a smaller problem one chooses the simplest possible method in order to minimize the Finally, a calculation which takes 1Ol2 effort of writing the program. operations is close to the limit of what is physically possible; it can only be justified by a major scientific advance such as landing a man on the moon. When a calculation is complete, one must still ask whether the results are reliable. A typical program, in machinecoded form, is a string of several thousand symbols; and a slip of the pen in any one of them will change the calculation being carried out by the computer. Most errors have disastrous results, and can therefore be detected and removed; but some errors are analogous to mistakes in a formula, and will merely cause the production of wrong answers. Some calculations are selfchecking: one example is given below, in which an integer is obtained as the value of a complicated analytic expressionif this had been wrongly evaluated, the result would not always have come out to be approximately an integer. There are other calculations in which once the answer has been found it can be checked with relatively little effort: for example in searching for a solution of a diophantine equation. But in general it is desirable either to work out some typical results by hand, for comparison with those obtained from the machine, or to have the whole
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calculation repeated ab initio by a different programmer on a different machine. If I is a curve defined over Q, the field of rationals, under what conditions can we say that r contains rational points? This problem is completely solved for curves of genus 0, for I? is then birationally equivalent over Q to a straight line or a conic; so it is natural to consider the case when r has genus 1. This is equivalent to saying that if PI, . . , P., Q,, . ., Qm, are any points on r then there is a unique Q, such that the P, are the poles and the Qi the zeros of some function on I’. It follows that for any given point 0 on I’ we can give I? the structure of a group with 0 as the identity; for if PI, Pz are any points on I? we define Q = PI + Pz by the property that PI, Pz are the poles and 0, Q the zeros of some function on r. Moreover if 0 and I are defined over Q, then so is the group law; thus the set of rational points on r form a group, which we call To. We can associate with l? another curve J, the Jacobian of r, as follows. On I? x l? we define an equivalence relation by writing Pi x Q, h) Pz x Q, if and only if PI, Q, are the poles and Pz, Q, the zeros of a function on I’. J is defined (up to birational equivalence) as the curve, each point of which corresponds to an equivalence class on I x I. J is defined over Q and certainly contains a rational point, corresponding to the class of points P x P on I? x I’; and we can write the equation of J in the form y2z = x3  Axz2  Bz3.
(1)
We say that r is a principal homogeneousspacefor J. Clearly r is birationally equivalent to J over Q if and only if it contains a rational point; and because J has a canonical rational point it has a canonical group structure. The problem of rational points on r now breaks up into two parts: (i) is there a rational point on I, and (ii) what is the structure of the group of rational points on J? Of these, the second is more attractive because there is more structure associated with it. Mordell in 1922 proved the following theorem. MORDELL'S THEOREM. The group of rational points on J is finitely generated.
It is easy to find the elements of finite order in this group, in any particular case. Thus it is natural to ask for a means of finding g, the number of independent generators of infinite order, and if possible for an actual set of generators. Mordell’s argument is semiconstructive, in that it gives in any numerical case an upper bound for g which is not absurdly large; and by using the ideas underlying his proof one can usually find g and an actual set of generators. But the process can be very laborious and there is no guarantee that it will always work. Moreover g is not connected with anything else in the theory. What should it be connected with?
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We can regard a solution of the equation (1) as the result of fitting together a compatible set of solutions of the congruences y*z G x3Axz2Bz3
(modp”).
(2)
One can hope that if these congruences have a lot of solutions it will be relatively easy to find compatible sets, and so (1) will have a lot of solutions and g will be large. Hence we look for a measure of the density of solutions of the congruences (2). Let NP” be the number of essentially distinct primitive solutions of (2). Except at finitely many bad primesp it follows from Hensel’s Lemma that Npm = p”  ‘Np and we are therefore led to consider the infinite product (3)
fl (N,lp).
We can also define N,, as the number of points on (1) considered as a curve over the finite field of p elements. For finitely many primes we expect that the factor in (3) is wrong, but it is not yet clear what to put in its place. There is a more respectable reason for considering the product (3), though in the end it will turn out to be misleading. We have Np = pcr,z*+1 where laPI = p112,* and the local zetafunction
of l? is defined as
~r,,W= (l(1cr,P?uB,P? p3(1 piy The numerator
of the righthand Ws)
*
side has the value NJp at s = 1; if we write
= n Kl ~~Pw
spi,p31’
(4)
then
and formally
Lu)1’
= fl (N,/P).
The product (4) is only known to converge in Ws > 312, though it is conjectured (and in special cases known) that L,(s) can be analytically continued over the whole plane. Moreover one can at best hope that the product (3) is finitely oscillatory, because one expects that L,(s) will have complex zeros on WS = 1. However, it is easy to calculate the finite products 0’)
= I1 (N,lp)
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taken over all p 5 P for values of P up to several thousandt. To a sympathetic eye, the results suggested thatf(P) oscillates finitely when g = 0, and tends to infinity when g > 0; and this led to the linked pair of conjectures f(P) lies between constant multiples of (log P)g, h(s) has a zero of order g at s = 1.
(Of course the constants in the first conjecture depend on I.) By looking at the behaviour of f(P), Birch and SwinnertonDyer were able to predict the value of g for individual curves, and the predictions were right in about 90% of the cases tried; these were curves for which g = 0, 1, 2 or 3. But there seems to be no objective numerical method for estimating g accurately in this way. To do better than this, we have to know more about L,(s). It is therefore natural to consider curves J which admit complex multiplication, since there is then an explicit formula for ZVpand b(s) is a product of Hecke Lseries. The simplest such curves have the form y2z = x3  Dxz”,
(5)
in which we can assume that D is an integer and is fourthpowerfree; and if we write (. .)4 for the biquadratic residue symbol in Q(i) we have the following formulae. THEOREM (DAVENPORTHASSE).
(p+l
For the curve y2z = x3  Dxz2, forp
G 3 mod 4,
forp
E 1 mod 4,
where in the secondcasep = n?i in Q(i) with TC,?~: E 1 mod (2 t 2i).
There is no easy proof of the full theorem. That ap/n is a power of i goes back to Jacobsthal, who gave an elementary proof which makes it look like a piece of good luck. The reason why such formulae exist in this case is that Q(i) is here the ring of endomorphisms of J in characteristic 0. Thus if p G 1 mod 4, Q(i) must be the whole ring of endomorphisms of J in characteristic p ; for the known structure theorems rule out any strictly larger ring. Hence ap, being the image of the Frobenius endomorphism, must lie in Q(i); and since CC,&, = p we find that up/z is a unit in Q(i). t The naive way of calculating N,, is to set z = 1 in (1) and test all possible pairs x, y; this would have taken O(pa) operations, and used more machine time than was justifiable. But because (1) can be written as y’ = w = x=  Ax  B it is only necessary to tabulate for each value of w the number of ways in which it is a square; for each value of x one can then read off the number of solutions. In this way, N, can be found after only O(p) operations.
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Using the values above for the NP we have L,(s) = L,(s) =
n p=
(1 +p’23l
3mod4
where the product is taken over all Gaussian primes R E 1 mod (2 + 29, the sum is over all Gaussian integers u t 1 mod (2 + 2i) and N is the norm for Q(i)/Q. Writing c = 1601 + p, where 1 runs over all Gaussian integers and p over a suitably chosen finite set of values, we have
L,(s) = ; $ 4; 0
Lf(Nu)“.
We cannot yet write s = 1, for the inner series would not converge; thus we write E(l S) .L+ c Il(cc,dIa12”
v+.
if+q+
1


[
1q+
1
I> 9
which converges for 9% > 3, and after some reduction we obtain
Here we can at last let s tend to 1; and since $ (a, 1) = e(a), the Weierstrass zeta function, we obtain
Jw)= i&~(~)4t(i5)  &=@;
@)
This is an explicit finite formula from which the value of L,(l) can be computed. In the interests of simplicity, I have derived a relatively clumsy formula for L,(l). In fact, if A is the product of the distinct odd primes dividing D, then L,(l) can be expressed as a sum of O(A’) terms instead of the O(L)‘) given above, and can therefore be calculated in 0(A2) operations. The constant implied in this formula is quite large because each term is complicated; but it was possible to evaluate L,(l) for all 1A 1 < 108, which gave some very large values of D. Moreover, provided that 1 A 1 > 1 the second term in (6) vanishes for a suitable choice of the p and L,(l) can be written as a sum of terms involving Weierstrass pfunctions. In this way it can be shown that D”‘LD(l)/o
is a rational integer if D > 0
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where CDis the real period of the Weierstrass pfunction
defined by
P’2 =4p34p. The proof is in two parts: 23’4A &,(1)/o is an algebraic integer, because of the addition formula for p ; and D ‘I4 &,(1)/w is rational, because of the Kronecker Jugendtraum. There is a similar result for D < 0. From this, and the approximate calculations of L,(l), which are accurate to one part in lo’, we can obtain the exact values of L,(l) for a large number of particular curves; and all these support the conjecture that L,(l) = 0 ifand only ifg > 0. However, the numerical results still leave something to be explained away or fitted into the theory, to wit, the rational integer D”4LD(l)/w when g = 0. The possible nontrivial factors from which this can be built up are (i) q(D), the number of rational points of finite order on J; (ii) m (D), the order of the TateSafareviE group of J; (iii) “fudge factors” corresponding to the finitely many “bad” primes. For the curve (5), q(D) is 4 if D is a square and 2 otherwise. “Bad” primes are those for which J has a bad reduction modulo p; that is, those which divide 20. One has also the right to deem “infinity” to be “bad” if this is expedient. We have next to define the TateSafareviE group. There is an equivalence relation among the principal homogeneous spaces r for a fixed Jacobian J, defined by birational equivalence over Q. The equivalence classes under this relation form the WeilChGtelet group,t which is a commutative torsion group of infinite order. Those r which contain points defined over each padic field and over the reals, fill a certain number of equivalence classes; and these classes form the TateSafareviE group. It is conjectured that this group is always finite, though there is no case in which this has yet been proved; however Cassels has shown that when it is finite its order is a perfect square. The whole group is not constructively defined, and is not computable; but it is theoretically possible to compute those of its elements which have any given order (of which there can only be finitely many), and for the curve (5) it is practicable to find the elements of orders 2 and 4 at least. Fudge factors must be purely local, and there are now two plausible alternatives: (i) fill in the missing factors of G(S) by means of the functional equationthis is possible whenever there is complex multiplication; (ii) supply the missing factors of the product n(N,/p) by considering the Tamagawa measure of the curve J. t This can also be defined as the cohomology group W(Q,J)see Chapter X for the details of a parallel case. The group law can be defined geometrically as follows. There is a map I’ x l’ + J which can be written P x Q + (P  Q) in an obvious notation. On rl x I?, we define an equivalence relation by writing PI x Pa N Q1 x Ql if and only if PI  Q1 = QP P,. Now r1 + I?, is defined as I’,. the curve whosepoints are in oneone correspondencewith the equivalence classeson lY1 x r,.
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It must be emphasized that these two alternatives do lead to numerical different results, and that the second one appears to work while the first definitely does not. To define the Tamagawa measure, let o be a differential of the first kind on J; this induces a measure op on J over each padic field, and the Tamagawa measure of J is defined to be IISJ%
taken over all primes including infinity. Here o is defined up to multiplication by an arbitrary constant, and the contributions of this constant to the infinite product evidently cancel out. LEMMA. Zf p is a ‘good”finite
prime then SJ o,, = N,/p.
Here “good” means that both J and w have a good reduction mod p. We can take w to be dx
dy 3x2D 2y=
possibly multiplied
by a padic unit.
(8)
Let (xO,yO) be any solution of
y2 E x3Dxmodp
with y0 + 0 mod p. By Hensel’s Lemma, to each padic x = x0 mod p there corresponds just one y 3 y0 mod p satisfying y2 = x3  Dx; all these solutions together contribute to J op an amount equal to the padic measure of the set of x t x0 mod p, which is p ‘. A similar argument works for the solutions withy,, = 0 or co. This proves the Lemma. If we define o to be (a), which is in fact the canonical form for the differential on J, then the bad primes are just those which divide 20 03. It can be shown that =2oD*ifD>O I @co with a similar result for D < 0. Thus the Tamagawa measure of J, defined by (7), is formally equal to 2~00~* [L,(l)]’ fll op = z(D), say, (9)
where the product is taken over the Unite bad primes p. This is a rational number. The contributions from the bad primes can be found by arguments similar to those of the Lemma, though more tedius; and it can be shown that s e+, for a bad prime p is just the number of components into which J decomposes under reduction modulo p in the sense of N&on (Modtles minimaux . . , Publ. IHES, 21). The numerical results lead to the conjecture that the Tamagawa measure (9) is equal to
MD>12/dD)
(10)
They cannot be said to support this conjecture directly since m(D) cannot
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be calculated ; but if we write
Y(D) = MQl”/@>~ the conjectural value of m(D), it has the following properties in every case in which it has been calculated. (i) y(D) is a perfect square, and usually contains no prime factor other than 2; in most cases j(D) = 1. (ii) Whenever the exact power of 2 which divides m(D) has been calculated, this is also the exact power of 2 which divides y(D). In the outstanding cases, m(D) must be and y(D) is divisible by a substantial power of 2. Moreover, a comparison of the conjectures for the isogenous curves y2 = x3  Dx and Y2 = x3 + 4Dx yields an explicit formula for m( 40)/1u(D); this has subsequently been proved by Cassels. The surprising part of this conjecture is the presence of a square in (10). We expect to measure a union of representatives of the elements of the TateSafareviE group modulo the group of rational points on this union; and the measure turns out to be q(D) rather than a constant. To describe what happens for g > 0, or for more general curves I, it is convenient to define the modified Lseries LX4 = hM/n J q# (11) in which the product is taken over all bad primes including infinity, and L&) is defined by (4) in which the product is taken over all good primes. If J admits complex multiplication, we can give explicit formulae for the coefficients of the power series expansion of L,(s) about s = 1 by arguments similar to those above; the only change is that instead of Weierstrass zeta functions the formulae involve series which converge but have no interesting theoretical properties. Nelson Stephens, a student of Birch, has carried out these calculations for curves of the form J: y3 = x3 Dz’, (12) which also admit complex multiplication. His results are consistent with the formula LX4 = wllgK %9/MJJ12 (13) in which g is the number of independent generators of JQ of infinite order, q(J) is the order of the torsion subgroup of JQ, m(J) is the order of the TateSafareviE group of J, and K measures the size of the generators of infinite order for J, in the following way. Let P = (x, y, z) be a rational point on J, where x, y, z are integers with no common factor, and write
W) = log IMax tl~~~l~l~~z~>l. Tate has defined the canonical height of P to be R*(P) = lim n2R(nP);
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he has shown that this limit exists and behaves like a quadratic form on the additive group JQ. In particular it gives rise to the bilinear form
= h.p(~
2’4;;“fP;;;y p,l
***
where L,,,(S) = n (1 Cr,jP3l. (19) Here IOlmjl = pmi2, and the product (14) has B,,, factors, where B,,, is the mth Betti number of V. Moreover, we can write LG) = II &n(4 where the product is taken over all primes for which V has a good reduction modulo p. In general we do not know how to define the factors corresponding to the bad primes, and so cannot go on to obtain L:(s); the one exceptional case is when V is an Abelian variety A and m = 1 or 2 n  1. In this case we can proceed as we did for J; the only novelty is we must now distinguish between A and its dual A, whereas for an elliptic curve J = .f. Since the proper generalization of (13) should follow from (13) when A is a product of elliptic curves, it ought to be L;nnl(s+n1) = L:(s) z 2”(s l)“m~~(A)/~(A)tj+i). (15) The form of the denominator on the right has been obtained from the role which duality plays in the proof of Cassels’ relation between m(Ji) and III(J,) for two isogenous elliptic curves J1 and J2.
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Except in this special case, we can only try to generalize the weaker conjecture that L,(S) has a zero of order g at s = 1. In these terms, there is a natural analogue of (15) for general V: L,,,(s) has a zero of order g at s = n, where g is the rank of the group of rational points on the Picard variety of V. But the extra generality is spurious, for one believes that L,,,(s) depends only on the Picard variety of V and not on V itself’. For even suffices, Tate in his Wood’s Hole talkt produced a remarkable new conjecture : L,,(s) has a pole of order v at s = r + 1, where u is the rank of the group of rcycles on V defined over Q, module algebraic equivalence. Note that Lz,(s) converges in Ws > r + 1, so that this conjecture can be given a meaning even if L,,(s) cannot be analytically continued; by contrast, all the previous conjectures have refered to a point which is a distance 3 from the region of known convergence. Tate’s conjecture is known to hold for rational surfaces, and he has proved it for other varieties of certain special types. Moreover, by applying it to the nfold product of an elliptic curve F with itself, he has been able to state the distribution of the characteristic roots a,, and 2,, of I’ as p varies; and these predictions have been verified numerically. Is there a comparable conjecture for L,,,(s)? One expects to associate L,,(s) with rcycles modulo algebraic equivalence, and L,,,(s) with (r l)cycles algebraically equivalent to zero. Moreover, any conjecture must generalize the one stated above for L znl(~), and must take into account the duality between L,,,(s) and L,,,(s). B earing these facts in mind, there is one plausible form that such a conjecture could take. Let ai and a2 be two algebraically equivalent rcycles on V; this means that there is a nonsingular variety W whose points correspond to rcycles on V, and there are points P, and P2 on W which correspond to a, and a, respectively. We shall say that ai and a2 are Abelian equivalent if it is possible to choose W,P, and P2 in such a way that the image of PI x P2 in the Albanese variety of W is the identity. It is known that this gives an equivalence relation, though it is one about which very little has been proved. The most natural conjecture now is as follows: L,,,(s) has a zero of order at least v at s = r, where v is the rank of the group of (r  1)  cycles on V defined over Q and algebraically equivalent to zero, module Abelian equivalence. t Published paradoxically in “ Arithmetical Algebraic Geometry. Proceedings of a Conference held in Purdue University, December 57, 1963.” (Ed. 0. F. G. Schilling.) Harper & Row.
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This is supported by unpublished results of Bombieri and SwinnertonDyer for the cubic threefold, and for the intersection of two quadrics in 2n + 1 dimensions. In both these cases the conjecture appears to hold with actual equality. However, numerical calculations show that for the nfold product of an elliptic curve with itself we can have either inequality or equality, depending on the particular curve.
CHAPTER XIII
Complex
Multiplication J.P. SERRE
............... Introduction 1. The Theorems .............. 2. The Proofs ............... 3. Maximal Abelian Extension ........... ............... References
292 292 293 295 296
Introduction A central problem of algebraic number theory is to give an explicit construction for the abelian extensions of a given field K. For instance, if K is Q, the field of rationals, the theorem of KroneckerWeber tells us that the maximal abelian extension Qb of Q is precisely Qcyc’, the union of all cyclotomic extensions. There is a canonical isomorphism
so we have an explicit classfield theory eve: Q (Chapter VII, 3 5.7). If K is an imaginary quadratic field, complex multiplication does essentially the same. We get the extensions Kab 3 R 3 K (g is the absolute class field the maximum unramified abelian extension) essentially by adjoining points of finite order on an elliptic curve with the right complex multiplication. 1. The Theorems Let E be an elliptic curve over C (the complex field); in normal form, its equation may be taken as y2 = 4x3 g2 xg,. We know that E z C/T where r is a lattice in C. An endomorphism of E is a multiplication by an element z E C with zF c I?. In general, End(E) = Z; if End(E) is larger, then End(E) @Q = K must be an imaginary quadratic field. Let R be the ring of integers of such a field K; then End(E) is a subring of R of finite index. Every such subring of R has the form Rf = Z +fR (f > 1 is the “conductor”); so, if E has complex multiplication, there is a complex quadratic field K with integers R, and an integer f, such that End(E) g R,. Conversely, given RI, there are corresponding elliptic curves. THEOREM. The elliptic curves with given endomorphismring Rf correspond oneone (up to isomorphism) with the classgroup Cl(R/). 292
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[Cl(R,) is the same as &RI), the group of projective modules of rank 1 over R,. Iff = 1, it is simply the group of ideal classes of R.] Sketch of Proof. Such a curve E determines a lattice F by E r C/T (so essentially r is n,(E)). r is an R/module of rank 1, whose endomorphism ring is exactly R,; this implies that r is an Rfprojective module of rank 1. Conversely, given such a F, C/l? is an elliptic curve with End(C/r) = R, COROLLARY. Up to isomorphism, there are onlyfinitely many curves E with End(E) = R/; in fact, there are preciseZy h, = Card(Cl(R,)). To each E, associate its invariant j(E); so there are h, such numbers associated with Rp For simplicity, take f = 1. THEOREM (WeberFueter).
(i) The j(E) are algebraic integers. (ii) Let a = j(E) be one of them. Thefield K(u) is the absolute classfield of K. (iii) Gal(K(a)/K) permutes the j(E)‘s associated with R transitively.
There are analogous results for f > 1; in particular, the j(E) are still algebraic integers. In fact, we can be more explicit. Call Ell(R,) the set of elliptic curves with endomorphism ring RI; if E g C/r E Ell(R,), write j(r) for its invariant; r is an inversible ideal of R/, and j(I) depends only on its class. Hasse (1927, 1931) has proved the following theorem. THEOREM. Let p be a good prime of K, with (p,f) = (1); let p, = p n R, be the corresponding ideal of R,. Then the Frobenius element F(p) acts on i(r) by Mr)>p(p) = j(rp;l). COROLLARY (f= 1). If Ette E Cl(R), and the Artin map takes c E Cl(R) to b, E Gal(K(a)/K), then a, (e) = e  c. Briefly, Cl(R) acts on EIl(R) by translation with a minus sign.
2. The Proofs We describe Deuring’s algebraic proof of these theorems (Deuring, 1949, 1952); for generalizations, see the tract of Shimura and Taniyama (1961). First, we must make sure that everything we are using is algebraically defined. So, as before, let K be a complex quadratic field with integers R. A curve E defined over an algebraic closure K by y2 = 4x3  g2 x gs has an invariant j = 1728gs/A where A = gz  27g:; and End(E) is welldefined algebraically. We consider the set Ell(RI) of (classes of) curves with endomorphism ring R/. Our correspondence Ell(R,) c* Cl(R,) was obtained using the topology of C; so it must be replaced by something algebraic. We assert that EII(RJ is an affine space over Cl(R,) (in other words, given x E El1 and y E Cl we can define xy E Ell). In fact, if E is a curve and M is a projective module of
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rank 1 over R,, we may define M*E = Hom(M, resolution 4
E).
More explicitly,
take a
Ry+R;!,M+O;
then Hom(R/,
E) = E, so Hom(M, E) is the kernel of ‘4 : E” + Em. Proof of the Theorem. Certainly j(E) is an algebraic number; for if it were
transcendental, there would be infinitely many curves with complex multiplication by R,, but Cl(R,) is finite. In fact, thej(E) are algebraic integers, but we will not prove this (one needs other methods; for instance, show there is a model of E over a finite extension of K with a “good reduction” everywhere). G = Gal(R/K) operates on the set ElI(R/) and preserves its structure of Cl(R/)affine space; hence G acts by translations. This means that there is a homomorphism #J: G + CI(R/) such that the action of d E G on Ell(R,) is translation by 4(a). We make the following assertions. C$is onto. (9 If p is a good prime, let Fp E CI(R/) be the image of the Frobenius element by 4. (ii) Fv = WI. Clearly, (ii) implies (i); for, given (ii), Cl(p) E 4(G) for all good p and trivially Cl(R,) is generated by the Cl(p)‘s. The assertion (i) tells us that the Galois group permutes the j(E) transitively, and (i) and (ii) taken together tell us that @j(E)) is the absolute class field (for f = 1). Note that it is actually enough to know (ii) for almost all primes p of first degree; class field theory takes care of the others. Proof of (ii). Let E be an elliptic curve defined over L, where L/K is abelian. If ‘$ is a prime of L, not in a finite set S’ of bad primes, E has a good reduction J&, modulo ‘$. Suppose that ‘$Ip, p a prime of K. If Np is the absolute norm of p, (J!?~)~~is got by raising all coefficients to the Npth power. We assert (iii) UQNv s p*Ep E (@&,. Now (iii) implies (ii), for by (iii), j(E)Np E j(p*E) (modulo 5@), so the Frobenius mapj(E) +tj(E)Np is the translation j(E) ~+j(p*E). Proof of (iii). The inclusion map p + RI induces p*E c R/*E = E. This map E + p*E is an isogeny of elliptic curves, and we see easily that its degree is Np. Taking reductions modulo $3, we have an isogeny E + z Case 1. p of degree 1, Np = p. The map E + z has degree p, and may be seen to be inseparable (look at the tangent space). Hence, it can only be the map xwxP of E + EP. For our application, this is actually enough. But we had better sort out the other case:
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Case 2. p of degree 2, so Np = p2 where p is inert in K/Q. Then one can show that B has Hasse invariant zero; that is, it has no point of order p. The map Ez accordingly has trivial kernel, so again it is purely inseparable, so it is E + f?*‘. Example. There are either 13 or 14 complex quadratic RI’s with class number 1, namely Q(,/d) withf= 1 for d = 1,2,3,7,11,19,43,67,163, and possibly ?, and also Q(,/ l), Q(,/ 3) and Q(,/ 7) with f= 2 and Q(J3) with f = 3. Hence there are 13 or 14 curves with complex multiplication with i E Q. Their invariants are: (see end of chapter) j = 2633, 2653, 0,  3353, 215, 21533, 2l*3353 , 2153353113 2183353233293 m3, 2333113 , 24;553, 3353173 . j121553. 3. Maximal Abelian Extension We want the maximal abelian extension Kab of K. As a first shot, try K’, the union of the fields given by the j’s of the elliptic curves of Eli(+), for f= 1,2,...; this field is generated by the i(z>‘s where Im(@ > 0 and T E K. Now enlarge K’ by adding the roots of unity; we get a field K’? = QcyC’. K’ which is very near Kab. The following theorem states this result more precisely : THEOREM.
Gal(Kab/K “) is a product of groups of order 2.
This follows easily from class field theory and the results of Section 2. Let now k? be the absolute class field of K, and let E be an elliptic curve defined over R, with End(E) = R. Let L, be the extension of k? generated by the coordinates of the points of finite order of E. This is an abelian extension of K, whose Galois group is embedded in a natural way in the group U(K) = fl U,(K), where U,(K) means the group of units of K at the finite prime V. By class field theory, L, is then described by a homomorphism 8, : I= + U(K) where IR is the idele group of R. Let U be the group of (global) units of K, so that U = { f 1) unless K is Q(,/ 1) or Q(,/3). One can prove without much difficulty that the restriction of dE to the group U(R) is: ‘G(x) = Nr&‘1. P&) where ps is an homomorphism of U(R) into U. The extension L,, and the homomorphism pE, depend on the choice of E. To get rid of this, let X = E/U be the quotient of E by U (X is a projective line), and let L be the extension of R generated by the coordinates of the images in X of the points of finite order of E. This extension is independent of the choice of E.
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THEOREM.L is the maximal
abelian extension of K.
This follows by classfield theory from the properties of 0, given above.
Remark. If U is of order 2 (resp. 4, 6) the map E + X is given by the coordinate x (resp. by x2, x3). Hence Kab is generated by j(E) and the coordinates x (resp. x2,x3) of the points of finite order on E; this has an obvious translation in analytical terms, using thej and g functions. [For further and deeper results, analogous to those of Kummer in the cyclotomic case, seea recent paper of Ramachandra (1964).] REFERENCES
Deuring, M. (1949). Algebraische Begrtindung der komplexen Multiplikation. Abh. Math. Sem. Univ. Hamburg, 16, 3247. Deuring, M. (1952). Die Struktur der elliptischen Funktionenkorper und die Klassenkiirperder imaginiirenquadratischenZahlkorper. Math. Ann. 124,393426. Deuring, M. Die Zetafunktion einer algebraischenKurve vom GeschlechteEins. Nachr. Akad. Wiss. Giittingen, MathPhys. KI. (1953)8594; (1955)1342; (1956) 3776; (1957) 5580. Deuring, M. Die Klassenkorper der komplexen Multiplikation. Enz. Math. Wiss. Band Iz, Heft 10, Teil II (60 pp.). Fueter, R. “Vorlesungen tiher die singularen Moduhr und die komplexe Multiplikation der elliptischen Funktionen”, (1924)Vol. I; (1927)Vol. II (Teubner). Hasse,H. Neue Begrtindung der komplexen Multiplikation. J. reine angew. Math. (1927)157,115139; (1931) U&6488. Ramachandra, K. (1964). Some applications of Kronecker’s limit formulas. Ann. Math. 80, 104148. Shimura, G. and Taniyama, Y. (1961). Complex multiplication of abelian varieties Publ. Math.
Sot. Japan, 6.
Weber, H. (1908). “Algebra”, Vol. III (2nd edition). Braunschweig. Weil, A. (1955).On the theory of complex multiplication. Proc. Znt. Symp. Algebraic Number Theory, TokyoNikko, pp. 922.
STOP
PFms
H. M. Stark has shown (Proc. Nat. Acad. Sci. U.S.A. 57 (1967), 216221) that there is no tenth imaginary quadratic field with classnumber1 (the ? of the Example at the end of $2). Practically simultaneously A. Baker (Mathematika, 13 (1966), 204216) proved an important general theorem which reduces the problem of its existenceto a finite amount of computation.
CHAPTER XIV
lExtensions K. HOECHSMANN Introduction ........... 1. TwoLemmas .......... 2. Local Fields .......... 3. Global Fields .......... Appendix: Restricted Ramification References ...........
.......
297 297 298 300 302 303
Introduction Let I be a rational prime, R a group of order I (multiplicative). For a profinite group G, we shall denote by H’(G) the cohomology group H’(G, Q), G operating trivially on R. If G is the Galois group of a field extension K/k, we shall sometimes write H’(K/k). We shall be interested in these groups particularly for i = 1,2 in the case where k is a local or global field and K its maximal Iextension, i.e. the maximal normal extension whose Galois group is a proZgroup. 1. Two Lemmas LEMMA 1. Consider a family (93 of morphisms of exact sequences of prolgroups: l+R+F+G+l Q” t t t t l+R,+F,+G,+l where (a) F, F, are free? (H’(F) = H2(FV) = 1). (b) F, F, are built from a minimal system of generators. G, G, resp. (H’(G) + H’(F), etc., are surjective). Then R is generated (as a closed normal subgroup of F) by the totality of images 4pU(RV) the map H2(G) + n H2GJ 0 is injective. Proof. In analogy to Prop. 26 of Serre (1964) we have: R generated by the cp,(R,) + H’(R)’ + n H’(R,) injective. On the other hand, the transv t Cf. Serre (1964) for the meaning of freedom, etc., in this context. 297
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gressions (Tg) in the diagram H’(R)’ 2 H’(G) 1 1 H’(RU)Gv 2 If’(G,) are isomorphisms by the exactness of the HochschildSerre our hypotheses (a) and (b).
sequence and
Given field extensions as shown in the diagram (everything Galois over k), where K/k and K’/k’ are lextensions, K’ E = Gal (k/k) is finite with order prime to I, consider the I obvious map 8: Gal (K’/k’) + Gal (K/k). Suppose KAL (a) H’(K’/K) = 1 and Ik _A’ k‘f (b) Res: H’(K’/k) + H2(K’/L) to be trivial. I: Then 8 induces an isomorphism 8* : H’(K/k) + H2(K’/k’)‘. Proof. O* is the composition of Inf: H2(K/k) + H’(K’/k) and Res: H2(K’/k) + iY’(K’/k’)‘. LEMMA 2.
The bijectivity of Res follows from the fact that I is prime to the order of E. To see it, one could use dimensionshifting on finite subextensions and the HochschildSerre sequence. Hypothesis (a) yields the exactness of the pertinent InfRes sequence in dimension 2, hence the injectivity of Inf. Its surjectivity then follows from hypothesis (b) and the injectivity of the restriction from (K’/K) to (K’/L). 2. Local Fields Let k be a finite extension of the rational Zextension of k, and G = Gal (K/k). Put 6 = rank of Ztorsion part of k*,
padic field Qp, K the maximal
iflP otherwise. We are interested in the numbers d1 and dZ of generators and defining relations for G: i.e. the ranks of H’(G) and H’(G). THEOREM 1.
d’ =n+6+1, d’=b. Proof. By the Bumside Basis Theorem, it sufhces to count generators of G/G’[G, G], which by local class field theory amounts to looking at k*/k*‘.
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The 1 in the formula for d’ comes from the infinite cyclic group generated by a prime element, the n+6 from the group U of units. For U is the direct product of a finite cyclic group and a free Z,module of rank [k: Q,] (cf. Hasse (1962) II. 15. 5). So much for d ‘. To find d2, we distinguish two cases. (1) 6 = 1. Inject R into K obtaining an exact sequence I 1 l+n+K*+K*+l, where I means raising to the lth power. Because of Hilbert 90, we have an isomorphism i : H2(G) + H’(G, K*),, lower I meaning Ztorsion part. These are the Brauer classes of order I, by local class field theory a group of order 1. (2) 6 = 0. Adjoin Ith roots of unity obtaining a field k’ and, by construction of its maximal Iextension K’, a diagram as in Lemma 2. The hypotheses of Lemma 2 are readily verified. (a) H’(K’/K) = Horn (Gal @C/K), a) = 1, since a nontrivial element would yield a proper lextension of K. (b) Already the restriction from H’(K’/k’) to H2(K’/L) is trivial, since elements of these groups can, as in (l), be identified with Brauer classes of order I; these die in the vastness of L/k’. By Lemma 2, H2(G) = H2(K’/ky. E is cyclic; let E be a generator.
Our injection i:R+K’* is not an smap. Indeed, for w E fi, we have i(w) = i(0) = [si(o)]“, where m E Z such that a‘6 = cm for Zth roots [ of unity. get a commutative diagram
Accordingly,
we
If’(K’/k’) 1, Br(k’) 81 1 s.m H’(K’/k’) 1, Br (k’) i.e. for a E Hz&‘/k’) we have: inv [i(c?)] = m . inv [(ia>“] = m . inv [ia], which shows that, for a E H2(K’/k’)E, ia = 1 and hence a = 1. Remark. The invariants d’ and d2 and the exponent 1’ of the ftorsion part of k* determine G completely except in the case 6 = 1, P = 2. If
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6 = 0, we are dealing with a free group, and the remark is trivial. G has the properties: (1) H2(G) N R. (2) The pairing H’(G) is nondegenerate.
x H’(G)
P n defined by cupproduct
If 6 = 1,
and (1)
(Via the Kummer isomorphism H’(G) N k*Jk*‘, this pairing corresponds to Hilbert’s Zpower residue symbol.) ProZgroups having a finite number d of generators and satisfying (1) and (2) are called DemzGkin groups. If G is any Demu&in group such that the exponent P of the torsion part of G/[G, G] is greater than 2, generators xl, . . ., xd of G can be found such that the relation deting G takes on the form 431,x,]
. . . [xdl,xd]
=
1,
square brackets denoting commutators, If P = 2, there are several group types depending on a more subtle invariant. For details, see Labute (1965) and Serre (1964). In the present context, all this machinery is necessary only if 1 = p. If Z # p, the cardinality q of the residue class field of k must be = 1 (mod Z) in order for 6 to be = 1. Let Z’(q 1, s maximal; C be a primitive I”th root of unity in k. Then G has two generators u and r obtained by arbitrary extension to K of the norm residue symbols for x (prime element) and C. Every finite subextension L/k is metabelian with the unramiCed part T/k left fixed by r. Therefore 7 = ({, LIT), where [ E T may be chosen to be a root of unity. Hence CRC~ = (a& L/T) = (Cq,L/T) = 7q, this relation holding on every finite L/k and hence on K/k. 3. Global Fields Let k be a finite extension of Q, and K the maximal Zextension of k with Galois group G. For each prime ZJof k consider the completion k, and its maximal Eextension K, (here we deviate from standard notation!) with Galois group G,. We flx an extension w of t, to K, obtaining an injection q,,:G,+G whose image is the decomposition group of w. Let 6,6, be the Ztorsion rank of k*, k: resp. THEOREM 2. The (pv induce a monomorphism p* : H’(G) + u H’(G,,) ” whose cokernel has rank 6.
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301
Proof. We proceed as in the proof of Theorem 1. (1) 6 = 1. An injection i: R + K* yields corresponding injections iv: R + K,* to make commutative diagrams H’(G) + H’(G,K*), 1 1 H2(Gv) ) H2(G,K:), bijectivity of the horizontal arrows being furnished by Hilbert (Theorem 90) via the exact sequence 1 1 l+SI+KK*+P,l and its local counterpart. We use the righthand sides of these diagrams and Hasse’s characterization of Brauer classes by their local invariants to obtain the exact sequence 1+ H’(G) 2 u H’(G,) : Cl + 1. ” where x denotes the sum of invariants written multiplicatively. (2) 6 = 0. Again we adjoin Zth roots of unity obtaining a field k’. We extend each place u’ of k’ to a place w’ of the maximal Zextension K’, so that w’ agrees with the already chosen w on K. We now have a diagram as in Lemma 2 for the global fields and, for each v’, a similar local one, showing Ku/k, and Kj/k:# with Galois groups G, and GLSresp. The commutation rule: cpV0 8, = 8 0 (py. (0’1~) yields a commutative diagram 9.W’
H2(G’) + H’(G:,) pv* 0.1 H’(G) ‘2 H2(Gv) for each u’. Taking all these diagrams together and noting the injectivity of 8* (since H’(K’/K) = 1, again by maximality of K), we deduce the injectivity of ‘p* = n cp: from that of n cp$, which was established in (1). To ;rove surjectivity, Ge note that, by Lemma 2, the image of 8* is Hz(G)“, hypothesis (b) being verified as before by identifying H2(K’/k’), H’(K’/L) with the Itorsion of Brauer groups over k’ and L resp. and watching the former die (locally everywhere) as we restrict them to L. Next we observe that H’(G,) # 1 e 6, = 1 o u splits completely in k’, and that we can limit our attention to the set Se of these places. We now have a diagram H;!;“E , g ~2WtJ H’(G)
2 ~~.Hf”GJ
302 where 8* is a bijection, that the map
K. HOECHSMANN
and propose to prove the surjectivity of ‘p* by showing H*(G) +L&~*(G.)
is no longer injective, if we leave out any b E So from the product on the right. This amounts to finding a nontrivial CIE H2(G’)E such that (p,‘(a) = 1 for all u’ not lying over b. Arguing as at the end of the proof of Theorem 1 and using the notation introduced there, we find inv,,, [ i(a”)] = m . inv,. [ icr] (N.B. this time E operates on the places, too.) Therefore a E H2(G)E e inv#,, [ia] = m. inv,e [ia]. We define the desired a by prescribing the invariants of ia: if v’J’b (0 nv,. [iu] = mi if')'= db (j=O,...,r1)
i
where b’ is a fixed place above b, and r = [k’ : k]. Since rl Cm' j=O
z 0 (modI),
this does define a Brauer class ia of order I. Remark. By Lemma 1, this result can be translated into a theorem of H. Koch (1965) about relations defining the group G. If we present G, G. as factor groups of free groups F, I;, and extend cpVto a morphism of the corresponding exact sequences as in Lemma 1, it follows from Theorem 2 that R is generated by the images p>,(R”). Now, R, is generated by a single relation r,, which is trivial if S, = 0 (cf. Theorem 1). Theorem 2 also asserts that the cp,(r,) form a minimal system of relations if 6 = 0 and that, if 6 = 1, a minimal system is obtained by leaving one of them out. APPENDIX
Restricted Ramification Let S be a set of places on the field k considered in the preceding paragraph, and let K(S) be the maximal Zextension of k unramified outside of S. So far, we have been dealing with the case S = all places; the other extreme, S = 0, is the topic of Roquette’s Chapter IX. We wish to conclude with some remarks about the general situation. In Koch (1965), G(S) = Gal (K(s)/k) is studied as a factor group of G. Generators of G are obtained from the norm residue symbols for generators of I/k*I’ (I= ideles of k). More precisely, one considers the exact sequence 1 + U/k*I’ n U ) I/k*l’ + Cl/Cl’ , 1
303
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(U = idele units; Cl = ideal classes), and chooses for generators: (1) preimages of a basis of the finite cokernel and (2) for each place u a basis B. of UJJL. The natural map G+%9 is then given by setting z = 1 whenever r E B,, v # S (the elements of B, lie in the inertia group of u). If we had a description of G in terms of these generators and some defining relations, we should get a corresponding description of G(S) by simply crossing out the unwanted r’s everywhere. But Theorem 2, gives relations for G only if we start with a minimal system of generators; and our system is not minimal! Indeed, a certain finite number of elements of u B, has to be eliminated to make it minimal, and the outlined procedure Gorks only if S is large enough to include the D’S corresponding to these elements. In gensral this approach is still strong enough, to yield a fundamental inequality of SafareviE (1963, Theorem 5), and in favourable special cases it leads to a satisfactory description of G(S). For details, see (Koch, 1965). In Brumer (196?), the methods used by us to prove Theorem 2 (case (1)) are directly applied to the more general problem. It is assumed that & = 1 and that S contains all primes lying above I. Then the extraction of Zth roots of Sunits leads to extensions unramified outside of S and thus to an exact sequence 142~E&E(s)+1. where E(S) denotes Sunits of K(S). Passing to cohomology,
we have
1+ ,H’WS), E(S)) ) ~2(W9~ f, ~2(G(s>, JW)z , 1. where, for abelian groups X, ,X means X/X’. It is not difficult, to identify the last term with a subgroup of Br (k)l, more precisely: with those Brauer classes of order I whose localizations are trivial outside of S. Furthermore, H’(G(S), E(S)) turns out to be isomorphic to ideal classes of k modulo classes containing elements of S; we shall call this group A(S). We obtain an exact sequence 1 + J(S) + H2(G(S)) +yL6sH2(GJ 1: Q + 1 analogous to the one in the proof of Theorem 1 (part (1)). The more difbcult case (8 = 0) has not yet been successfully dealt with by this method. REFERENCES
Brumer, A. (1961). Galois groups of extensions of algebraic number fields with given ramification. (Unpublished.) Hasse, H. (1962). “Zahlentheorie”, AkademieVerlag, Berlin.
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Koch, H. (1965). IErweiterungen mit vorgegebenen Verzweigungsstellen. J. reine angew. Math. 219, 3061. Labute, J. (1965). Classification des groupes de Demu&in. Comptes rendues, 260, 10431046. Serre, J.P. (1963). Structure de certains propgroupes. Skminaire Bourbaki, exp. 252. Serre, J.P. (1964). Cohomologie galoisienne. Lecture notes in Mathematics, 5. SpringerVerlag. SafareviE, I. R. (1963). Extensions with given ramification. Publs. math. Inst. ht. Etud. scient. 18, 7195. (Russian with French summary.)
/
xv Fourier
Analysis in Number Fields and Hecke’s ZetaFunctionst J. T. TATE
1. Introduction . . . . . . . . 1.1. Relevant History . . . . . . 1.2. This Thesis . . . . . . . 1.3. “Prerequisites” . . . . . . 2. The Local Theory . . . . . . . 2.1. Introduction . . . . . . . 2.2. Additive Characters and Measure . . . . 2.3. Multiplicative Characters and Measure . 2.4. The Local Cfunction; Functional Equation . 2.5. Computation of p(c) by Special Cfunctions . . 3. Abstract Restricted Direct Product . . . . 3.1. Introduction . . . . . . . 3.2. Characters . . . . . . . 3.3. Measure . . . . . . 4. The Theory in the Large . . . . . . 4.1. Additive Theory . . 4.2. RiemannRoth Theorem 1 1 1 1 . 4.3. Multiplicative Theory . . . . . 4.4. The Cfunctions; Functional Equation . . 4.5. Comparison with the Classical Theory A Few Comments on Recent Related Literature (Jan: 1967) References . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . .
. . . :
306 306 306 307 308
. 308 :
311 308
: . . . . . . . :
313 316 322 322 324 325 327 327 331 334 338
.
341
.* Z18
ABSTRACT We lay the foundations for abstract analysis in the groups of valuation vectors and ideles associated with a number field. This allows us to replace the classical notion of Cfunction, as the sum over integral ideals of a certain type of ideal character, by the corresponding notion for ideles, namely, the integral over the idele group of a rather general weight function times an idele character which is trivial on field elements. The role of Hecke’s complicated thetaformulas for theta functions formed over a lattice in the ndimensional space of classical number theory can be played by a simple Poisson formula t This is an unaltered reproduction of Tate’s doctoral thesis (Princeton, May 1950). The editors were urged by several of the participants of the Conference to include this thesis in the published proceedings because although it has been widely quoted it has never been published. The editors are very grateful to Tate for his permission to do this and for the comments on recent literature (p. 346). 305
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for general functions of valuation vectors, summed over the discrete subgroup of field elements. With this Poisson formula, which is of great importance in itself, inasmuch as it is the number theoretic analogue of the RiemannRoth theorem, an analytic continuation can be given at one stroke for all of the generalized Cfunctions, and an elegant functional equation can be established for them. Translating these results back into classical terms one obtains the Hecke functional equation, together with an interpretation of the complicated factor in it as a product of certain local factors coming from the archimedean primes and the primes of the conductor. The notion of local Cfunction has been introduced to give local definition of these factors, and a table of them has been computed.
1. Introduction 1.1. ReIevant History Hecke was the first to prove that the Dedekind cfunction of any algebraic number field has an analytic continuation over the whole plane and satisfies a simple functional equation. He soon realized that his method would work, not only for the Dedekind cfunction and Lseries, but also for a cfunction formed with a new type of ideal character which, for principal ideals depends not only on the residue class of the number modulo the “conductor”, but also on the position of the conjugates of the number in the complex field. Overcoming rather extraordinary technical complications, he showed (19 18 and 1920) that these “Hecke” cfunctions satisfied the same type of functional equation as the Dedekind cfunction, but with a much more complicated factor. In a work (Chevalley, 1940) the main purpose of which was to take analysis out of class field theory, Chevalley introduced the excellent notion of the idble group, as a refinement of the ideal group. In idbles Chevalley had not only found the best approach to class field theory, but to algebraic number theory generally. This is shown by Artin and Whaples (1945). They defined valuation vectors as the additive counterpart of ideles, and used these notions to derive from simple axioms all of the basic statements of algebraic number theory. Matchett, a student of Artin’s, made a first attempt (1946) to continue this program and do analytic number theory by means of ideles and vectors. She succeeded in redefining the classical cfunctions in terms of integrals over the idUe group, and in interpreting the characters of Hecke as exactly those characters of the ideal group which can be derived from idele characters. But in proving the functional equation she followed Hecke. 1.2. This Thesis Artin suggested to me the possibility of generalizing the notion of cfunction, and simplifying the proof of the analytic continuation and functional
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equation for it, by making fuller use of analysis in the spaces of valuation vectors and ideles themselves than Matchett had done. This thesis is the result of my work on his suggestion. I replace the classical notion of [function, as the sum over integral ideals of a certain type of ideal character, by the corresponding notion for ideles, namely, the integral over the idele group of, a rather general weight function times an idele character which is trivial on field elements. The role of Hecke’s complicated thetaformulas for theta functions formed over a lattice in the ndimensional space of classical number theory can be played by a simple Poisson Formula for general functions of valuation vectors, summed over the discrete subgroup of field elements. With this Poisson Formula, which is of great importance in itself, inasmuch as it is the number theoretic analogue of the RiemannRoth theorem, an analytic continuation can be given at one stroke for all of the generalized cfunctions, and an elegant functional equation can be established for them. Translating these results back into classical terms one obtains the Hecke functional equation, together with an interpretation of the complicated factor in it as a product of certain local factors coming from the archimedean primes and the primes of the conductor. The notion of local cfunction has been introduced to give a local definition of these factors, and a table of them has been computed. I wish to express to Artin my great appreciation for his suggestion of this topic and for the continued encouragement he has given me in my work. 1.3. “Prerequisites” In number theory we assume only the knowledge of the classical algebraic number theory, and its relation to the local theory. No knowledge of the idele and valuation vector point of view is required, because, in order to introduce abstract analysis on the idele and vector groups we redefine them and discuss their structure in detail. Concerning analysis, we assume only the most elementary facts and definitions in the theory of analytic functions of a complex variable. No knowledge whatsoever of classical analytic number theory is required. Instead, the reader must know the basic facts of abstract Fourier analysis in a locally compact abelian group G: (1) The existence and uniqueness of a Haar measure on such a group, and its equivalence with a positive invariant functional on the space L(G) of continuous functions on G which vanish outside a compact. (2) The duality between G and its character group, G, and between subgroups of G and factor groups of G. (3) The del?nition of the Fourier transform, j, of a function f E L,(G), together with the fact that, if we choose in G the measure which is dual to the measure in G, the Fourier Inversion Formula holds (in the ntive sense) for all functions for which it could be expected to hold; namely, for functions f EL,(G)
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such that f is continuous and f~ L,(e>. (This class of functions we denote by ‘Z&(G).) An elegant account of this theory can be found for example in Cartan and Godement (1947). 2. The Lmal Theory 2.1. Introduction Throughout this section, k denotes the completion of an algebraic number field at a prime divisor p. Accordingly, k is either the real or complex field if p is archimedean, while k is a “padic” field if p is discrete. In the latter case k contains a ring of integers o having a single prime ideal p with a finite residue class field o/p of Np elements. In both cases k is a complete topological field in the topology associated with the prime divisor p. From the infinity of equivalent valuations of k belonging to p we select the normed valuation defined by: a = ordinary absolute value if k is real. 1:a = s9 uare of ordinary absolute value if k is complex. Ial = (A+)‘, where Y is the ordinal number of a, if k is padic. We know that k is locally compact. The more exact statement which one can prove is: a subset B c k is relatively compact (has a compact closure) if and only if it is bounded in absolute value. Indeed, this is a well known fact for subsets of the line or plane if k is the real or complex field; and one can prove it in a similar manner in case k is padic by using a “Schubfachschluss” involving the finiteness of the residue class field. 2.2. Additive Characters and Measure Denote by k+ the additive group of k, as a locally compact commutative group, and by t its general element. We wish to determine the character group of k+, and are happy to see that this task is essentially accomplished by the following: LEMMA 2.2.1. If e + X(r) is one nontrivial character of k+, then for each is also a character. The correspondence q c) X(q<) is rl E k+, t + Wtl an isomorphism, both topological and algebraic, between k’ and its character group. Proqf. (1) X(qr) is a character for any fixed q because the map r + qr is a continuous homomorphism of k+ into itself. (2) X(h +rlN = Xh 5 +v2 0 = WI W(tl, 0 shows that the map v + X($) is an algebraic homomorphism of k+ into its character group. (3) X(qt) = 1, all l =z=qk+ # k+ sj tf = 0. Hence it is an algebraic isomorphism into. (4) X(qr) = 1, all q * k+r # k+ => t = 0. Therefore the characters of the form X(q[) are everywhere dense in the character group.
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(5) Denote by B the (compact) set of all r E k+ with ItI < Mfor a large M. Then: q close to 0 in k+ * qB close to 0 in k+ + X(qB) close to 1 in complex plane * X(q<) close to the identity character in the character group. On the other hand, if &, is a fixed element with X(&J # 1, then: X(@ close to identity character =z. X(vB) close to 1, closer, say, than X(&,) =E&., 4 qB =S q close to 0 in k+. Therefore the correspondence q c, X(qr) is bicontinuous. (6) Hence the characters of the form X(qa comprise a locally compact subgroup of the character group. Local compactness implies completeness and therefore closure, which together with (4) shows that the mapping is onto. To fix the identification of k’ with its character group promised by the preceding lemma, we must construct a special nontrivial character. Let p be the rational prime divisor which p divides, and R the completion of the rational field at p. Define a map x + A(x) of R into the reals mod 1 as follows : Case 1. p archimedean, and therefore R the real numbers. A(x) =  x (mod 1) (Note the minus sign!) Case 2. p discrete, R the field ofpadic numbers. by the properties:
A(x) shall be determined
(a) A(x) is a rational number with only a ppower in the denominator. (b) A(x)  x is a padic integer. (‘IO tid such a A(x), let p’x be integral, and choose an ordinary integer n such that n = p’x (modpy. Then put A(x) = n/p’; A(x) is obviously uniquely determined modulo 1.) LEMMA 2.2.2. x + A(x) is a nontrivial, the group of reals (mod 1).
continuous additive map of R into
Proof. In case 1 this is trivial. In case 2 we check that the number A(x)+@) satisfies properties (a) and (b) for x+y, so the map is additive. It is continuous at 0, yet nontrivial because of the obvious property: A(x) = 0 o x is padic integer. Define now for r E k+, A(r) = A(S,,,t). Recalling that ,!& is an additive continuous map of k onto R, we see that t: + e2nfh(C) is a nontrivial character of k’. We have proved: THEOREM 2.2.1. k’ is naturally its own character group if we identifr the character e + eZniAtqC)with the element q E k+.
LEMMA 2.2.3. In case p is discrete, the character eZniACqC) associated with qistrivialonoifandonlyifq~b l, b denoting the absolute d@Terent of k.
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Proof. h(tjo) = 0 * &,(~‘,)) = 0 e &,R(~o) c OR* q E b ‘. Let now ,u be a Haar measure for k+. LEMMA 2.2.4. If we define ,uu,(M) = p(arM) for a # 0 E k, and M a measurable set in k’, then p1 is a Haar measure, and consequently there exists a number q(a) > 0 such that p, = cp(a),u. Proof. 5 + a5 is an automorphism of k+, both topological and algebraic. Haar measure is determined, up to a positive constant, by the topological and algebraic structure of k+. LEMMA 2.2.5. The constant &a) of the preceding lemma is lal, i.e. we have Pww = I4AWPro@I If k is the real field, this is obvious. If k is complex, it is just as obvious since in that case we chose 1~1to be the square of the ordinary absolute value. If k is padic, we notice that since o is both compact and open, 0 < p(o) c co, and it therefore suffices to compare the size of o with that of ao. For u integral, there are N(oro) cosets of ao in o, hence ~(CIO) = (N(ao))‘p(o) = It+(o). For nonintegral CI, replace a by al. We have now another reason for calling the normed valuation the natural one. lcrl may be interpreted as the factor by which the additive group k+ is “stretched” under the transformation < + at. For the integral, the meaning of the preceding lemma is clearly:
444
= I4 445);
or more fully:
Jf<0 40
= I4 .If(d) 440.
So much for a general Haar measure ,u. Let us now select a fixed Haar measure for our additive group k’. Theorem 2.2.1 enables us to do this in an invariant way by selecting the measure which is its own Fourier transform under the interpretation of k+ as its own character group established in that theorem. We state the choice of measure which does this, writing d5 instead of dp(l), for simplicity: dr = ordinary Lebesgue measure on real line if k is real. d5 = twice ordinary Lebesgue measure in the plane if k is complex. d< = that measure for which o gets measure (A%)* if k is padic. THEOREM 2.2.2. If we define the Fourier transform f of a function f e L,(k+) by: f(q) = 1 f(Q e2niA(qC)de, then with our choice of measure, the inversion formula f(5) = If(q) e2”“‘(cq) dq = f(  0 holdr for f e ?&(k+).
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Proof. We need only establish the inversion formula for one nontrivial function, since from abstract Fourier analysis we know it is true, save possibly for a constant factor. For k real we can take f(c) = e‘lcl’, for k complex, f(5) = e2n151; and for k padic, f(t) = the characteristic function of o, for instance. For the details of the computations, the reader isreferred to Section 2.6 below. 2.3. Multiplicative Characters and Measure Our Grst insight into the structure of the multiplicative group k* of k is given by the continuous homomorphism a + Ial of k* into the multiplicative group of positive real numbers. The kernel of this homomorphism, the subgroup of all a with jcrl = 1 will obviously play an important role. Let us denote it by u. u is compact in all cases, and in case k is padic, u is also open. Concerning the characters of k*, the situation is different from that of k+. First of all, we are interested in all continuous multiplicative maps a P c(a) of k* into the complex numbers, not only in the bounded ones, and shall call such a map a quasicharacter, reserving the word “character” for the conventional character of absolute value 1. Secondly, we shall find no model for the group of quasicharacters, or even for the group of characters, though such a model would be of the utmost importance. We call a quasicharacter unramified if it is trivial on U, and first determine the unramified quasicharacters. LEMMA 2.3.1. The unramified quasicharacters are the maps of the form c(a) = Ial” E eslOglul, where s is any complex number, s is determined by c if p is archimedean, while for discrete p, s is determined only mod 2nijlog Np. Proof. For any s, Ial” is obviously an unramified quasicharacter. On the other hand, any unramified quasicharacter will depend only on Ial, and as function of Ial will be a quasicharacter of the value group of k. This value group is the multiplicative group of all positive real numbers, or of all powers of Np, according to whether p is archimedean or discrete; it is well known that the quasicharacters of these groups are those described. If p is archimedean, we may write the general element a E k* uniquely in the form a = @, with d E u, p > 0. For discrete p, we must select a fixed element II of ordinal number 1 in order to write, again uniquely, a = &, with a”E u and, this time, p a power of K. In either case the map a + d is a continuous homomorphism of k* onto u which is identity on u. THEOREM 2.3.1. The quasicharacters of k* are the maps of the form a + c(a) = Z’(E)lal”, where c” is any character of u. Z is uniqueZy determined by c. s is determined as in the preceding lemma. Proof. A map of the given type is obviously a quasicharacter. Conversely, if c is a given quasicharacter and we define c”to be the restriction
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of c to U, then E is a quasicharacter of u and is therefore a character of u since u is compact. a P c(a)/E(C) is an umamiGed quasicharacter, and therefore is of the form Ial” according to the preceding lemma. The problem of quasicharacters c of k* therefore boils down to that of the characters E of U. If k is the real field, u = { 1,  1) and the characters are t((a) = B, n = 0, 1. If k is complex, I( is the unitcircle, and the characters are F(C) = d”, n any integer. In case k is padic, the subgroups 1 +p”, v > 0, of u form a fundamental system of neighborhoods of 1 in U. We must have therefore c”(1+ p”) = 1 for sufficiently large Y. Selecting v minimal (v = 0 if c’ = l), we call the ideal i = p’ the conductor of Z. Then E is a character of the finite factor group u/(1 +f) and may be described by a finite table of data. From the expression c(a) = Z(Z)ja[s for the general quasicharacter given in Theorem 2.3.1, we see that [c(a)1 = lalb, where c = Re (s) is uniquely determined by c(a). It will be convenient to call cr the exponent of c. A quasicharacter is a character if and only if its exponent is 0. We will be able to select a Haar measure da on k* by relating it to the measure dt on k+. If g(a) EL(k*), then g(<)lSl’ EL(k+0). So we may define on L(k*) a functional Wd =
J sco[epe. k+0
If h(a) = g(pa) (B E k*, fixed) is a multiplicative Wd =
1 sW>ltl’
translation
of g(a), then
dt = Wd,
k+0
as we see by the substitution t + /3‘r; 2.2.5. Therefore our functional @ which is also invariant under translation. lt measure on k+. Denoting this measure ~doc)4 a =
dr + I/31’ d{ discussed in Lemma is obviously nontrivial and positive, must therefore come from a Haar by d 1a, we may write
1 dOltI’
&.
k+0
Obviously, the correspondence g(a) t, g(t)l{l’ is a ll correspondence between L(k*) and L(k’  0). Viewing the functions of L,(k*) and Ll(k+  0 as limits of these basic functions we obtain: LEMMA 2.3.2. functions
g(a) E L,(k*) s g(aNa
o g({)lrlml
E L,(k+0),
and for
these
= k+~odt~ltlld~~
For later use, we need a multiplicative measure which will in general give the subgroup u the measure 1. To this effect we choose as our standard
FOURIER ANALYSIS INNUMBWFIE~S
AND HFXXJ?S ZETAFUNCTIONS
313
Haar measure on k* : du=d,a=i,
if p is archimedean.
a
Np da da =  Np dla=Np1 Np  1 Ial’ LEMMA 2.3.3. In case p is discrete,
if p is discrete.
I” d a = (Nb)*. ProoJ
Therefore
s Y
Y
2.4. The Local Cfunction; Functional Equation In this section f(c) will denote a complex valued function defined on k+ ; f(a) its restriction to k*. We let 3 denote the class of all these functions which satisfy the two conditions: 3J f(C), a&CT(C) continuous, E &(k+); i.e.f(t) E 23,(k+) 3J j(a)lal” and~(a)jal” E L,(k*) for Q > 0. A cfunction of k will be what one might call a multiplicative quasiFourier transform of a function f o 3. Precisely what we mean is stated in DEFINITION 2.4.1. Corresponding to each f E 3, we introduce a function Ccf, c) of quasicharacters c, defined for all quasicharacters of exponent greater than 0 by U
4 = J’ fWc(aYa,
and call such a function a cfunction of k. Let us call two quasicharacters equivalent if their quotient is an unramified quasicharacter. According to Lemma 2.3.1, an equivalence class of quasicharacters consists of all quasicharacters of the form c(a) = q,(a)[aI’, where c,(a) is a tied representative of the class, s a complex variable. It is apparent that by introducing the complex parameter s we may view an equivalence class of quasicharacters as a Riemann surface. In case p is archimedean, s is uniquely determined by c, and the surface will be isomorphic to the complex plane. In case p is discrete, s is determined only mod 2xi/log JTp, so the surface is isomorphic to a complex plane in which
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points differing by an integral multiple of Zai/log Np are identifiedthe type of surface on which singly periodic functions are really defined. Looking at the set of all quasicharacters as a collection of Riemann surfaces, it becomes clear what we mean when we talk of the regularity of a function of quasicharacters at a point or in a region, or of singularities. We may also consider the question of analytic continuation of such a function, though this must of course be carried out on each surface (equivalence class of quasicharacters) separately. LEMMA 2.4.1. A ~function is regular in the “domain” of all quasicharacters of exponent greater than 0. Proof. We must show that for each c of exponent > 0 the integral a re Presents a regular function of s for s near 0. Using the jfb4c(alal”d fact that the integral is absolutely convergent for s near 0 to make estimates, it is a routine matter to show that the function has a derivative for s near 0. The derivative can in fact be computed by “differentiating under the integral sign”. It is our aim to show that the cfunctions have a singlevalued meromorphic analytic continuation to the domain of all quasicharacters by means of a simple functional equation. We start out from LEMMA 2.4.2. For c in the domain 0 < exponent c < 1 and 9(a) = Ialc‘(a) we have C(f, m?9 21 = r
absolutely convergent for c in the region we are considering. We may write this as an absolutely convergent “double integral” over the direct product, k* x k*, of k* with itself: ‘)[Pld(a, P). I I fW(P>c(aF Subjecting k* x k* to the “shearing” automorphism (a, /I) + (a, ap), under which the measure d(a, b) is invariant we obtain LO. If f(a)~(aP>c(P‘>[aPld(a, According to Fubini this is equal to the repeated integral 1 (sf(a)P(aB)lalda)c~l)[~~d~. To prove our contention it suffices to show that the inner integral da is symmetric in f and g. This we do by writing down the obviously symmetric additive double integral Jf(a)&afi)lal
I I f(tkh)
e2niA(CBs)d(t, ~9,
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changing it with the Fubini theorem into dv) d5 = ~.fW(5B) 8, f f(t) (I B(V) e znrA(CBs) and observing that according to Lemma 2.3.2 this last expression is equal to the multiplicative integral f(c#(c@)[aldl a = constant. /f(a)d(@) laida I We can now announce the Main Theorem of the local theory. THEOREM 2.4.1. A cfunction has an analytic continuation to the domain of all quasicharacters given by a functional equation of the type WY 4 = P(CN3Y 2). The factor p(c), which is independent of the function f, is a meromorphic function of quasicharacters defined in the domain 0 < exponent c < 1 by the functional equation itself, and for all quasicharacters by analytic continuation. Proof In the next section we will exhibit for each equivalence class C of quasicharacters an explicit function fc E 3 such that the function P(C) = C(fc9 4/lltfc, 2) is defined (i.e. has denominator not identically 0) for c in the strip 0 < exponent c < 1 on C. The function p(c) defined in this manner will turn out to be a familiar meromorphic function of the parameter s with which we describe the surface C, and therefore will have an analytic continuation over all of C. From these facts, which will be proved in $2.5, the theorem follows directly. For since C was in any equivalence class, p(c) is defined for all quasicharacters. And if f (r) is any function of 3 we have according to the preceding lemma WY WC‘,, 21 = KfY wcfc~ 49 therefore uf, 4 = PWUY 219 if c is any quasicharacter in the domain of 0 < exponent c < 1, where Cl.6 4 and Kf’, 2) are originally both defined, and C is the equivalence class of c. Before going on to the computations of the next section which will put this theory on a sound basis, we can prove some simple properties of the factor p(c) in the functional equation which follow directly from the functional equation itself. LEMMA 2.4.3. (1)
p(2) = c$ft
(2)
P(E)
= cc
l)P(C)*
J. T.TATE
316 Proof. (1) cu 4 = PM.t because](a) = f(a)
a = Pw.wdT, 6 = c( W(fa 4 and &a) = c(a). Therefore p(c)p(Q = c( 1).
(2) CUP 4 = ccf, c3 = PGW
8 = PM1N.l Q = P(Mw39 Q because f(a) = I(  a) and &a) = E(a). On the other hand, UC) =po am. Therefore p(E) = c( 1)s). COROLLARY 2.4.1. Ip( = 1 for c of exponent 3. Proof. (exponent c) = 3 * c(a)E(a) = Ic(a)12 = Ial = c(a)t(a) * E(a) = e(a). Equating the two expressions for p(E) and p(e) given in the preceding lemma yields p(c)p(c) = 1. 2.5. Computation of p(c) by Special Cfunctions This section contains the computations promised in the proof of theorem 2.4.1. For each equivalence class C of quasicharacters we give an especially simple function fc E 3 with which it is easy to compute p(c) on the surface C. Carrying out this computation we obtain a table which gives the analytic expression for p(c) in terms of the parameters on each surface C. It will be necessary to treat the cases k real; k complex; and k padic separately. k Real t is a real variable. II((f) =t. & means ordinary Lebesgue measure.
a is a nonzero real variable. Ial is the ordinary absolute value. ah da = m.
The Equivalence Classes of QuasiCharacters. The quasicharacters of the form Ial’, which we denote simply by IIS, comprise one equivalence class. Those of the form (sign a)laI’, which we denote by f I I”, comprise the other. The Corresponding Functions of 3. We put f(r) = e“C’ and f,(e) = r e‘@. Their Fourier Transforms. We contend
f(t) f(C) and f&l = if&). Indeed, these are simply the two identities co 03 4+2+d,, = e“C’ and l=+2~&ld,, e I J tte familiarfyom
classical Fourier analysis. iie
= ite“t=
first of these can be established
FOURIER
ANALYSIS
IN NUMBER
directly by completing substitution q P 0 +i& and replacing
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the square in the exponent, making the complex which is allowed by Cauchy’s integral theorem,
the definite integral
The second identity
AND
is obtained
1 e“C’ dt by its wellknown value 1 CO 1 d by applying the operation   to the 2ni d<
first.
The (;functions. We readily compute: C(f, 113 = j)(c+$
dcr = j Pa2]#; 00
m
Explicit
Expressions for p(c):
n:r s P(ll3= Tr1, 0122 n
= 21%*cos
l(s)
( 2 > J+l 3r
d*ll?
; 0
= i
,: n
I
s+l (
)+1
> ~c’~+~~ 2
=i2’3‘sin(~)~(s).
2
Here the quotient expressions for p come directly from the definition of p as quotient of suitable Cfunctions; the second form follows from elementary rfunction identities.
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k CompIex a = rele is a nonzero t: = x + & is a complex variable. variable. M)
= 2Re(o
complex
I4 = r2 is the square of the ordinary absolute value.
=2x.
2rldrdOl da = & ,a, = p
dr = 21dxdyj is twice the ordinary Lebesgue measure.
2 = ; Idr dOI.
Equivalence Classes of QuasiCharacters. The characters c,(a) defined by c,(re”) = ein’, n any integer, represent the different equivalence classes. The nth class consists of the characters c,(cr)[a[‘, which we denote by c.11’. The Corresponding Functions of 3. We put (xf,(O
=
jy)lnl
{(x+iy)l”l
Their Fourier Transforms.
e2rW+Y’) ez’w+Y9;
;.
We contend
f”(r) = i’“y,(r), Let us first establish contention is simply This can be shown plane into a product
:: :
for all n.
this formula for n 2 0 by induction. For n = 0, the that fO(r) = e 2n(x’+yf) is its own Fourier transform. by breaking up the Fourier integral over the complex of two reals and using again the classical formula +m e nu=+2nixu du = eVnx2. s
(The factor 2 in the expiient of our function fo(T) just compensates the factor 2 in d5 and in A(<).) Assume now we have proved the contention for some n 2 0. This means we have established the formula I f.GW which, written out, becomes
2nra(C")dq= j"f.(c),
w 7 ujit)).e2"(U1'~')+4~1(XYY~)2du II co m Applying the operator
dv =
j"(X+iy)ne2x(%'+Y').
D=$.(&+i$ to both sides (a simple task in view of the fact that since z” is analytic,
FOURIER
ANALYSIS
IN NUMBER
FIELDS
= 0), we obtain ce 01 2n(u’+~‘)+4nl(xuy~)2dudv (u+iu)“+‘e II 02 03
AND
HECKE’S
3 19
ZETAFUNCTIONS
qx+iyy
=
jn+l(X+iy)"+le2"(~'+y').
This is the contention for n+ 1. The induction step is carried out. To handle the case n < 0, put a roof on the formula j’,,(e) = il”x(l) which we have already proved, and remember that Lto =fA5) = ( wL.“(0. The CFunctions. For a = r eie we have fn(a) = ,I4 einee2nr2 IuI” = rzS 2r dr dO
c,(a) = eine
dcr=rl.
Therefore m 2% C(f.,c”Il”>
= J’fn(ct)c,,(a)[a~da
= Is r2(s1)‘I”le2nrz2rdrde 00
= zn ~(r2~1)+‘~1ez”zd(r2) = (2,90’“+$$s
,
0
and
Explicit
Expressions
for p(c).
k pa&c e = a padic variable. NO = wc3>dt is chosen so that o gets measure (JV~)~. a = CC’, nonzero padic variable, n a fixed element of ordinal number 1, v an integer. Ial = (Jp)“. ./yp
da = Np
da
  so that u gets multiplicative 1 Ial’
measure (~0)~*.
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The Equivalence Classes of QuasiCharacters. c,(a), for n 2 0 shall denote any character of k* with conductor exactly p”, such that c.(n) = 1. These characters represent the different equivalence classes of quasicharacters. The Corresponding Functions of 3. We put
j(t) = {;sfA(c)’ ;; , Their Fourier Transforms. We contend f”
; ; 1 . ;:;
; mm& .
Proof.
3”(c)=
jj(q) ,2=ih(h) drt =
j e2nih((t lh) drl. b$”
This is the integral, over the compact subgroup b‘p* c k+, of the additive character q + e2n*h((C1)“). If t: z 1 (mod p”), this character is trivial on the subgroup, and the integral is simply the measure of the subgroup: (Nb)*(Nn)“. In case { $ 1 (mod p”), this character is not trivial on the subgroup and the integral is 0. The CFunctions. First we treat the unramified case: n = 0. The only character of type ca is the identity character, and f. is the characteristic function of the set bl. We shall therefore compute
CCr,,l]“> = I IaI”da. b1 Denote by A, the “annulus” of elements of order v, and let b = pd. Then bl = uz ,, A,, a disjoint union, and
3e is JYb* times the characteristic
function of o, so we have, similarly,
C(30,t?= ~cT~,ll’~>= JQ*.l
0
Ial’“da
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In the ramified case, n > 0, cu&lll”)
=
1 eZ+iA(a)c,(a) IaI’ da b$”
= vafmNpmvJ~
e2xi”(a) c,,(a) da. A” We assert that all terms in this sum after the first are 0. In other words that e2*“‘@) c,(a) da = 0 for Y >  dn. I AV Proof. Case 1. v 2 d. Then A, c b‘, so e2*‘*@) = 1 on A,, and the integral is
since c,(a) Case 2. i.e. if n > type a,+b’ A(a,) and
1 c,(a) da = 1 c,(an”) da = j c,(a) da = 0 Y Y A” is ramified and therefore nontrivial on the subgroup U. d > Y > dn. (Occurs only if there is “higher ramification”; 1.) To handle this case we break up A, into disjoint sets of the = a,, +pSd = ao(l +pBdy). On such a set, A is constant =
I cltJ+b’ This is 0 because I
ao+tl’
e2”‘A(“)c,(a)
da =
e2~lAk7~)
c,(a) da.
I
cao+b1
c,(a) da =
c,(a) da =
I
= c,W
I
c&ad
da
l+pd”
ao(l+pd‘) 1 l+pd”
44
da,
and this last integral is the integral over a multiplicative subgroup 1 + pmdVv of a character c,(a) which is not trivial on the subgroup. Namely, d > v =S &J~’ =E1 +pmd* is a subgroup of k*, and v > dn * the conductor P”&I~* * c,(a) not trivial on it. We have now shown eZniA@)c,(a) da. I A4n To write this in a better form, let {E} be a set of representatives of the elements of the factor group u/(1 +p”), so that u = us(l +p”), a disjoint union. Then = U.&X d“(l+p”) = ~.(&nd“+bl). A dn = undn rV,, %I13 = m+d+n)r
On each of these sets into which we have dissected Awdmn, c,, is constant
322 =
C,(&K
J. T. TATE
+“)
= c,(s), and A is constant = A(sr~~“).
We therefore have
((f,, cnlls) = .Xp(d+nf(~
C,(E) e2’+‘@lnd+“)) j du, * 1 +p” a form which will be convenient enough. The payoff comes in computing r<.L $ = r<xv c,‘ll’“>. For 3” is A%*JVp” times the characteristic function on which c; ‘(a)laj’ VS= 1. Therefore
c(J~,~?
= JQ+J~”
of the set 1 +p”, a set
1 da, a constant! 1+&n
ExpIicit
Expressions for p(c)
AC113= ~@D”*PcM 1‘f
c is a ramified
character
with conductor
f,
such that c(x) = 1.
is a socalled root number and has absolute value 1. {E) is a set of representatives of the cosets of 1 +f in U. Taking the quotients of the Cfunctions we have worked out yields these expressions directly if we remember that b = pd and, in the ramified case, that the conductor of c, was f = p”. The fact that constant p,(c) has absolute value 1 follows from Corollary 2.4.1. Namely, since c is a character, cl/* has exponent 3, so we must have jp(~lj~)l = Ip,,(c)l = 1. 3. Abstract Restricted Direct Product 3.1. Introduction. Let {p] be a set of indices. Suppose we are given for each p a locally compact abelian group GP, and for almost all p (meaning for all but a finite number of p), a tied subgroup HP c GP which is open and compact. We may then form a new abstract group G whose elements a = ( . . . , up,. . . ) are “vectors” having one component aP E G,, for each p, with a, E HP for almost ail p. Multiplication is defined componentwise. Let S be a finite set of indices p, including at least all those p for which HP is not defined. The elements a E G such that aP E HP for p 4 S comprise a subgroup of G which we denote by Gs. Gs is naturally isomorphic to a direct product n G, x n HP of locally compact groups, almost all of PSS oes which are compact, and is therefore a locally compact group in the product
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topology. We define a topology in G by taking as a fundamental system of neighborhoods of 1 in G, the set of neighborhoods of 1 in G,. The resulting topology in G does not depend on the set of indices, S, which we selected. This can be seen from LEMMA 3.1.1. The totality of all “~arallelotopes” of the form N = n N,, where N+, is a neighborhood of 1 in GQfor all p, and NQ = HP for almost all premember the HP are open by hypothesisis a fundamental system of neighborhoodsof 1 in G. Proof. By the definition of product topology a neighborhood of 1 in Gs
contains a parallelotope of the type described. On the other hand, since N, = HP for almost all p, the intersection Np n Gs = n Np x Q$‘“p n 4) P ) Q=s is a neighborhood of’1 in Gs. It is obvious that Gs is open in G and that the topology induced in Gs as a subspace of G is the same as the product topology we imposed on Gs to begin with. Therefore a compact neighborhood of 1 in Gs is a compact neighborhood of 1 in G. It follows that G is locally compact. DEFINITION 3.1.1. We call G (as locally compact abelian group) the restricted direct product of the groups GQrelative to the subgroupsHp.
It will, of course, be convenient to identify the basic group G, with the subgroup of G consisting of the elements “0 = (1, 1,. . . , a,, . . .) having all components but the pth equal to 1. For that subgroup of G is naturally isomorphic, both topologically and algebraically to GQ. Since the components, aQ, of any element a of G lie in HP for almost all p, G is the union of the subgroups of the type G,. This fact will allow us to reduce our investigations of G to a study of the subgroups G,. These Gs in turn may be effectively analyzed by introducing the subgroup GS c Gs consisting of all elements a E G such that a, = 1 for p E S; a, E H,, p 4 S. GS is compact since it is naturally isomorphic to a direct product n H, of compact groups. Gs can be considered as the direct product PCS Gs = n G, x GS of a finite number of our basic groups G, and the ( QeS > compact group Gs. We close our introduction of the restricted direct product with LEMMA 3.1.2. A subset C c G is relatively compact (hasa compact closure) if, and only if, it is contained in a parallelotope of the type fl BP, where BP is a compact subset of GQfor all p, and BP = HP for almost a;1 p. Proof: Any compact subset of G is contained in some G,, because the
Gs are open sets covering G, and the union of a finite number of subgroups G, is again a Gs. Any compact subset of a G, is contained in a parallelotope
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of the type described, for it is contained in the Cartesian product of its “projections” onto the component groups G+,. These projections are compact since they are continuous images, and are contained in HP for p 4 S. On the other hand, any parallelotope n B,, is obviously a compact subset P of some Gs; therefore of G. 3.2. Characters Let c(a) be a quasicharacter of G, i.e. a continuous multiplicative mapping of G into the complex numbers. We denote by cP the restriction of c to cP is obviously a G,: (cJap) = ~(a,,) = ~$1, 1,. . ., a*,. . .) for a, E G,). quasicharacter of G,,. LEMMA 3.2.1. c,, is trivial on H,, for almost all p, and we have for any tlEG 44 = II c&J P almost all factors of the product being 1. Proof. Let U be a neighbourhood of 1 in the complex numbers containing no multiplicative subgroup except (1). Let N = n NP be a neighborhood of 1 in G such that c(N) c U. Select an S containinlall p for which NP # H,. Then GScN~~GS)cUe,c(GS)=l~C(Hp)=l for p#S. Ifa is a fixed element of G we impose on S the further condition that a E Gs and write a as with as ta G’. Then +!Asc(a) =Vvsc(aP). c(a”) =P~scP(u~) = II %(Ca)* P since for p # S, cP(ap) = 1. LEMMA 3.2.2. Let c,, be a given quasicharacter of Gp for each p, with cp trivial on HP for almost all p. Then if we define c(a) = n c,,(u,,) we obtain P a quasicharacter of G. Proof. c(a) is obviously multiplicative. To see that it is continuous select an S containing all p for which cJH+,) # 1. Let s be the number of p in S. Given a neighborhood, U, of 1 in the complex numbers, choose a neighborhood V such that V” c U. Let N,, be a neighbourhood of 1 in G, such that c,,(N,,) c V for p E S, and let NP = HP for p # S. Then c vs c u. c IIN, ( P ) Restricting our consideration to characters, we notice first of all that c(a) = n c,(a+,) is a character if, and only if, all cP are characters. Denote by G, tie character group of GP, for all p ; for the p where HP is defined let HP+ c GP be the subgroup of all c+,E GP which are trivial on Hp. Then
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discrete  H,* open, and HP open * GJH,
z H,* compact.
THEOREM 3.2.1. The restricted direct product of the groups eP relative to the subgroups H,* is naturally isomorphic, both topologically and algebraically, to the character group 8 of G.
Proof. Of course we mean to identify c = (. . ., cP,. . .) with the character c(a) = n cP(aP). The two preceding lemmas, applied to characters, show
that this’ is an algebraic isomorphism between the two groups. We have only to check that the topology is the same. To this effect we reason as follows: c = (. .., CP,. ..> is close to 1 as a character o c(B) close to 1 for a large compact B c G o c n BP close to 1 for B,, c GP, compact, (
)
4 = HP for almost all p o c’,(&,) close to 1 wherever BP # HP and q,(BJ = c&Q = 1 at the remaining p (since HP is a subgroup, c,,(H& can be close to 1 only if cP(HP) = 1) o cP close to 1 in GP for a finite number of p and cP E H,* at the other p o c close to 1 in the restricted direct product of the G,. 3.3. Measure Assume now that we have chosen a Haar measure da, on each GP such thatsds = 1 for almost all p. We wish to deCne a Haar measure da on G for &ch,
in some sense, da = n da,.
consider Ga as the finite direct iroduct
To do this, we select an S; then Gr =
G, x G”, in order to PPES ) define on Gr a measure das = (dh,“(a) .da’, where da’ is that measure on the compact group Gs for which j da’ =
Since Gs is an open pIsls(il&?v)* subgroup of G, a Haar measure da on G is now determined by the requirement that da = da, on Gr. To see that the da we have just chosen is really independent of the set S, let T 3 S be a larger set of indices. Then Gs c Gr, and we have only to check that the daT constructed with T coincides on Gs with the da, constructed with S. Now one sees from the decomposition Gs = (, Eg&) x GT that da’ = (P ,l$$~~). daT; for the measure on the G‘
righthand Therefore
side gives to the compact da, =prlIsda,.das
group
Gs the required
measure.
=PJJsda+, .P ejJsdad. da’ = da,.
We have therefore determined a unique Haar measure da on G which we may denote symbolically by da = n da,. P
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If (p(S) is any function of the finite sets of indices S, with values in a topological space, we shall mean by the expression lim (p(S) = cpO the statement: “given any neighborhood V of qo, there exisi a set S(v) such that S 3 S(V) * q(S) E V”. Intuitively, lim (p(S) means the limit of q(S) S
as S becomes larger and larger. LEMMA 3.3.1. Iff(a)
is a function on G, I
f(a) da = lim S
I
f(a) da,
GS
if either (1) f(a) measurable, f(a) 2 0, in which case + co is allowed as value of the integrals; or (2) f(a) E&(G), in which case the values of the integrals are complex numbers. Proof. In either case (1) or (2) j f ( a) d a is the limit of j f (a) da for larger B
and larger compacts B c G. Since any compact, B, is contained in some G,, the statement follows. LEMMA 3.3.2. Assume we are given for each p a continuous function f, E&(G& such thatf,(aJ = 1 on HP for almost all p. We define on G the (th is is really a finite product), and contend: function f(a) = nf,(a,), P (1) f(a) is continuous on G. (2) For any set S containing at Zeast those p for which either f,(H,) # 1, or Ida, # 1, wehave =v /fbMa =~s[~(avW,]. GS Proof. (1) f(a) is obviously continuous on any G,; therefore on G. (2) For a E Gs, f(a)
=p~fp(d,).
f(a) da = f(a) da, = j (Jp&J) J‘ I G.9 GS GS =vlJ
Hence
(p% [pwds]
. daS) * J &Is =,rjs [f.fv’~v)d~v]. GS
THEOREM 3.3.1. If fp(a,) and f(a) are the functions of the preceding lemma and iffurthermore
then f(a) E L,(G) and, I
f(a) da = v [j f,(a,) da,].
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Proof. Combine the two preceding lemmas; tist for the function IfW = lI I&
Apply Theorem
3.3.1 to the function f(a)c(a> = n &(a,)c+,(a,)
to see that the Fourier transform of the product is the product Jf the Fourier transforms. Since f,(a,) E B,(G,), we have J)‘(cJ E&(G& for all p. For almost all p, &(c,) is the characteristic function of H,* according to the remark above. From this we see thatf(c) E &(G), hence f(a) E Bl(G). COROLLARY 3.3.1. The measure dc = n dc, is dual to da = n da,. Proof. Applying the preceding lemma :o the group G with the’measure dc, we obtain for our “product” functions the inversion formula f(a) = ~f~c)c(a) dc from the componentwise
inversion formulas.
4. The Theory in the Large 4.1. Additive Theory In this chapter, k denotes a finite algebraic number field, p is the generic prime divisor of k. The completion of k at the prime divisor p shall from now on be denoted by k,, and all the symbols o, A, b, 11,c, etc., defined in Chapter 2 for this local field kp shall also receive the subscript p: oP, A,, b,, I&, cp, . . . , etc.
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DFSINITION 4.1.1. The additive group V of valuation vectors of k is the restricted direct sum, over all prime divisors p, of the groups k,’ relative to the subgroups op. We shall denote the generic element of V (= valuation vector) by X= ( . . . . xp,... ). F rom Theorems 3.2.1. and 2.2.1 and Lemma 2.2.3 we see that the character group of V is naturally the restricted direct sum of the groups k,’ relative to the subgroups (b;l. Since bP = o, for almost all p this sum is simply V again! Looking more closely at the identifications set up in these theorems we see that the element t) = (. . . , up, . . .) E V is to be identified with the character x = (. . . , xp, . . .) + v exp (2niA,(9& = exp $i 1 AJs,s)] P of V. This suggests that we define the additive function A(x) = CA, (%I) P on V, and introduce componentwise multiplication qx = (. . ., q),. . .)(. . .,x,,. . .) = (. . ., t)&, . . .) of elements of V in order to be able to assert neatly: THEOREM 4.1.1. V is naturally its own character group if we identifv the element t) E V with the character x + e2n’A(9*) of V. On V we shall, of course, take the measure dx = n dx, described in Section 3.3, dx, being the local additive measure defined i:Section 2.2. Since these local measures dx, were chosen to be selfdual, the same is true of dx, according to Corollary 3.3.1. We state this fact formally in THEOREM 4.1.2. If for a function f (x) E L,(V) we define the Fourier transform f(q) = s j(x) e2n’A(9x)dx,
then for f (x) E B,(V)
the inversion formzda
f(x) = I f(q) e2n”‘(x9)dt) holds. What is the analogue in the large of the local Lemma 2.2.4 and 2.2.5, that is, of the statement d(a<) = ial dC: for a E k* ? In that local consideration, a played the role of an automorphism of kz, namely the automorphism c + a& This leads us to investigate the question: for what a E V is x + ax an automorphism of V? We first observe that for any a E V, x + ax is a continuous homomorphism of V into V. A necessary condition for it to be an automorphism is the existence of a b E V such that ab = 1 = (1,1, . . .). But this is also suflticient, for with this b we obtain an inverse map x + bx of the same form. Now for such a b to exist at all as an “unrestricted” vector, we need op # 0 for all p, and then b, = a; l. The further condition
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bo Vmeans apl E eP for almost all p, therefore jaJ, = 1 for almost all x. These two conditions mean simply that a is an id&le in the sense of Chevalley. We have proved LEMMA 4.1.1. The map x + ax is an automorphism of V if and on& if a is an idtile. At present we shall consider ideles only in this role. Later we shall study the multiplicative group of idMes as a group in its own right, with its own topology, as the restricted direct product of the groups k,* relative to the subgroups up. To answer the original question concerning the transformation of the measure under these automorphisms we state LEMMA 4.1.2. For an idPIe, a, d(ax) = Ia/ dx, where
I4 = II QI4
(really a finite product). Proof. If N = n NP is a compact neighborhood Theorem
of 0 in V, then by
3.3.1 and ‘Lemma 2.2.5
The last, and most important thing we must do in our preliminary discussion of V is to see how the field k is imbedded in V. We identify the element t E k with the valuation vector e = (& c, . . . , e, . . .) having all components equal to I$, and view k as subgroup of V. What hind of subgroup is it? LEMMA 4.1.3. If S, denotes the set of archimedean primes of k, then (1) k n k = o, the ring of algebraic integers in k, and (2) k+ V& = V. Proof. (1) This is simply the statement that an element C E k is an algebraic integer if and only if it is an integer at all finite primes. (2) k+Jcs, = V means: given any x E V, there exists a { E k approximating it in the sense that ex, E D, for all finite p. Such a t can be found by solving simultaneous congruences in o. The existence of a solution is guaranteed by the Chinese Remainder theorem. Let now ? denote the “infinite part” of V, i.e. the Cartesian product n k+, of the archimedean completions of k. If a generating equation for k QSS,
over the rational
field has rl real roots and r, pairs of conjugate complex
roots, then t is the product of ri real lines and r, complex planes. As such
330 it is naturally = absolute
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a vector space over the real numbers of dimension n = rl + 2rz degree of k.
For any x E V denote by f the projection
x”= (..., rp ,... 1+);s, ofx on ?. LEMMA 4.1.4. If@,, C02,. . . , o,,) is a minimal basis for the ring of integers o OD of k over the rational integers, then 1“0 l, 02, . , ., & 1 is a basis for the vector space “v over the real numbers. The parallelotope 5 spanned by this basis 6 = set of all x” =il xv& with 0 I; x, < 1) has the volume ,/Id] (where d = (det (OF)))” = absolute discriminant of k) of measured in the measure di =p g dx, which is natural in our setup. 00 Proof. The projection r + T of k into “v is just the classical imbedding of a number field into nspace. The reader will remember the classical argument which runs: k separable => d = (det (u#)))~ # 0 3 {&, G,, . . . , &} linearly independent, and (with a simple determinant computation) E has volume 2“,//ldl. For us the volume is 2ra times as much because we have chosen for complex p a measure which is twice the ordinary measure in the complex plane. DEFINITION 4.1.2. The additive fundamental domain D c V is the set of allxsuchthatxEVs_and~E& THEOREM 4.1.3. (1) D deserves its name because any vector x E V is congruent to one and only one vector of D module the field elements <. In other words, V .s,U~({+ D), a disjoint union. (2) D has (measure 1. Proof (1) Starting with an arbitrary x E V we can bring it into Vs by the addition of a field element which is unique mod o (Lemma 4.1.3). Once in Vs we can find a unique element of o, by the addition of which we can stay in V,, and adjust the infinite (Lemma 4.1.4).
components
so that they lie in 5
(2) To compute the measure of D, notice that D c Vs and D = 5 x Vsm. Therefore
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Now since the discriminant d (as ideal) is the norm of the absolute different b of k, and since b is the product of the local differents b,, we have IdI = n (XPbp). Therefore the measure which we have computed is 1. P6SoD COROLLARY 4.1.1. k is a discrete subgroup of V. Thefactor group V mod k is compact. ProoJ k is discrete, since D has an interior. D is relatively compact. LEMMA 4.1.5. A(c) = 0 for all c E k. ProoJ NO = c A,(t) P
V mod k is compact,
since
= c ~p,
because “the trace is the sum of the local traces”. SincePS(D is a rational number, the problem is reduced to proving that c I,(x) E 0 (mod 1) for rational
X. This we do by observing that the ratikral
number 1 n,(x) is P
integral with respect to each fixed rational C Apt4 = (, +Ip,lzpw)
+ 464 + n,bj
prime q. Namely = (, +Ip,sb))
+ (4(x)  4
P
expresses 2(x) as sum of qadic integers. THEOREMS 4.1.4. k* = k, that is A(xr) = 0 for all < e x E k. Proof. Since k* is the character group of the compact factor group V mod k, k* is discrete. k* contains k according to the preceding lemma, and therefore we may consider the factor group k* mod k. As discrete subgroup of the compact group V mod k, k* mod k is a finite group. But since it is a priori clear that k* is a vector space over k, and since k is not a finite field, the index (k* : k) cannot be finite unless it is 1. 4.2. RiemannRoth Theorem
We shall call a function C&W)periodic if cp(x+ <) = (p(x) for all 5: E k. The periodic functions represent in a natural way all functions on the compact factor group V mod k. cp(x) represents a continuous function on V mod k if and only if it is itself continuous on V. LEMMA 4.2.1. If q(x) is continuous and periodic, then j cp(x)dx is equal to: D
the integral over the factor group V mod k of the function on that group which C&S) represents with respect to that Haar measure on V mod k which gives the whole group V mod k the measure 1. Proof. Define I(q) = 1 cp(x) dx and consider it as functional on L( V mod k).
Observe that it has the properties
characterizing
the Haar integral.
t Here k* is the dual of k, not the multiplicativegroup of k (!).
(To
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check invariance under translation merely requires breaking D up into a disjoint sum of its intersections with a translation of itself.) The functional is normed to 1 because dx = 1. d k is naturally the character group of V mod k in view of Theorem 4.1.4. The Fourier transform, e(t), of the continuous function on V mod k which is represented by cp(x) is e(e) = J q(x) ezn*A(Cr)dx. D
LEMMA 4.2.2.
Proof.
If q(x) is continuous and periodic and c I@(01 < 00, then
e(C) is summable on k, $:&teeing that the inversion formula holds. The asserted equality is simply the inversion formula explicitly written out. LEMMA 4.2.3. If f(x) is continuous, c L,( V), and c f(x + q) is uniformly vok convergent for x 5 D (convergence means absolute convergence because k is nor ordered in any way), then for the resulting continuous periodic function cp(4 = C f(x+ rt) we have 4X0 = 3X0. risk Proof. cj3(&)= 1 q(x) eZniA(Cx)dx D
~Ekj(zt+q) e2*mclc))dx
= IP
=~kll(l+n)e2x~A(x”d. (The interchange is justified because we assumed the convergence to be uniform on D, and D has finite measure.) = zk
jDj(x) e2”fA(sCqC)dx
=~k:i~(~)e2xiA(~~)d~ (since A(q<) = 0) = I f(x)e  2nf4=0 dx
= 3m. Combining the last two Lemmas 4.2.2. and 4.2.3, and putting the assertion of Lemma 4.2.2 we obtain
x = 0 in
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LEMMA 4.2.4. (Poisson Formula.) Iff(r) satisfies the conditions: (1) f(x) continuous, E L,(V); (2) 1 f(x + r) uniformly convergent for x E D; C=k (3) C If<01 convergent; C=k then g(t)
=~&m.
If we replacef(x) by f(ax) (a an idele) we obtain a theorem which may be looked upon as the number theoretic analogue of the RiemannRoth theorem. THEOREM 4.2.1. (RiemannRoth Theorem.) Iff(x) satisfis the conditions: (1) f(x) continuous, B L,( v); (2) c~kfb@ + 0) convergent for all ids/es a and valuation vectors z,
then
uniformly for x E D; (3) C 1 3(4 convergent for aN hfdes a
Proof: The function g(z) = f(ax) satisfies the conditions lemma because
of the preceding
gj(x) = 1 f(at)) e2ani”(rp) dq
(Under the transformation
r) + n/a, dq + (l/[al) dtp)
We may therefore conclude
that is,
It is amusing to remark that, had we never bothered to compute the exact measure of D, we would now know it is 1. For we could have carried out all the arguments of this section with an unknown measure, say p(D), of D.
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The only change would be that in order to have the inversion formula of Lemma 4.2.2 we would have to have given each element of k the weight l/p(D). The Poisson Formula would then have read, $jgo~=,Ip. Iteration
of this would yield @(D))’ = 1, ;herefore p(D) = 1 I
4.3. Multiplicative Theory In this section we shall discuss the basic features of the multiplicative group of ideles. DEFINITION 4.3.1. The multiplicative group, I, of id2les is the restricted direct product of the groups k,* relative to the subgroups u,,. We shall denote the generic idble by a = (. . . , a,, . . .). The name idele is explained (at least partly explained!) by the fact that the idele group may be considered as a refinement of the ideal group of k. For if we associate with an idtle a the ideal p(a) = n pord@*, then the map a + (p(a) is DC.% obviously a continuous homomorphism of the idele group onto the discrete group of ideals of k. Since the kernel of this homomorphism is Is*, we may say that an idele is a refinement of an ideal in two ways. First, the archimedean primes figure in its makeup, and second, it takes into account the units at the discrete primes. Concerning quasicharacters of I, we can only state, according to Section 3.2, that the general quasicharacter c(a) is of the form c(a) = n c,(a,), where c,(a,) is a local quasicharacter (described in Section 2.3) and c,(i+,)isunramified at almost all p. For a measure, da, on I we shall of course choose da = n da,, the da, P being the local multiplicative measure defined in Section 2.3. We can do nothing really significant with the idele group until we imbed the multiplicative group k* of k in it, by identifying the element CIE k* with the idele a = (a, a,. . ., a, . . . ). Throughout the remainder of this section our discussion will center about the structure of I relative to the subgroup k*. The first fact to notice is that the ideal q(a) associated with an idtle a E k* is the principal ideal ao generated by a, as it should be. Next we have the “product formula” for elements a E k*. Though this is well known, we state it formally in a theorem in order to present an amusing proof. THEOREM 4.3.1. lal(= lj Ia&) = 1 fir aE k*. Proof. According to Lemma 4.1.2 the (additive) measure of aD is Ial times the measure of D. Since ak’ = k+, aD would serve as additive fundamental domain just as well as D. From this it is intuitively clear that aD has the
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same measure as D and therefore Ial = 1. To make a formal proof one has simply to chop up D and aD into congruent pieces of the form D n (t +aD) and ( {+ D) n aD respectively, < running through k. This theorem reminds us to mention explicitly the continuous homomorphism a + Ial = n Ialp of Z onto th e multiplicative group of positive real numbers. The k&nel is a closed subgroup of Z which will play an important role. We denote this subgroup by J, and its generic element (idele of absolute value 1) by b. It will be convenient (although it is aesthetically disturbing and not really necessary) to select arbitrarily a subgroup T of Z with which we can write Z = T x J (direct product). To this effect we choose at random one of the archimedean primes of kcall it p,and let T be the subgroup of all ideles a such that aPO> 0 and aP = 1 for p # p,,. Such an idele is obviously uniquely determined by its absolute value; indeed the map a + lal, restricted to T, is an isomorphism between T and the multiplicative group of positive real numbers, and it will cause no confusion if we denote an idele of T simply by the real number which is its absolute value. Thus a real number t > 0 also stands either for the idble (t, 1, 1, . . .) or for the idele (Jr, 1, 1, . . .>, according to whether p,, is real or complex, if we write the nocomponent first. Since we can write any idble a uniquely in the form a = Ial .b with la/ E T and b = alai’ E J, it is clear that Z = TX J (direct product). In order to select a fixed measure db on J we take on T the measure dt = dtlt and require da = d t. db. Then for computational purposes we have (in the sense of Fubini) the formulas BOWL
=I
[/kWb]
t =!
[b$]
db
for a summable idele functionf(a). The product formula means that k* c J, and we wish now to describe a “fundamental domain” for J mod k*. The mapping of ideles onto ideals allows us to descend to the subgroup Js, = J n Ia. To study Js, we map the ideles b E Js, onto vectors Z(b) = (. . ., log lb!,,, . . .)peSb, having one component, log lbl, for each archimedean prime except no. (This set of r = rl +r, 1 primes is denoted by S’,.) It is obvious that the map b P Z(b) is a continuous homomorphism of Jsm onto the additive group of the Euclidean rspace. The ontoness results from the fact that although the infinite components of an idMe b E Js, are constrained by the condition n 1% =p rJ)lP = 1, they are completely free in the set Sb, since we can ldjust the p. component. k* n Js, is the group of all elements E E k* which are units at all finite
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primes; that is, which are units of the ring o. The units C for which Z(g) = 0 are the roots of unity in k and form a ftnite cyclic group. It is proved classically that the group of units e, modulo the group of roots of unity C, is a free abelian group on r generators. This proof is effected by showing that the images Z(s) of units form a lattice of highest dimension in the rspace. If, tl=refore, (&sSls;r is a basis for the group of units modulo roots of unity, the vectors Z(sJ are a basis for the rspace over the real numbers and we may write for any 6 E Js,, l(b) = &v
KG
with unique real numbers x,. Call P the parallelotope
in the Rspace spanned
by the vectors Z(eJ; that is, the set of all vectors i x,Z(eJ with 0 5 x, < 1. VP1 Call Q the “unit cube” in the rspace; that is the set of all vectors ( . . ..x.,... 1peS$ with0 5 xP < 1. LEMMA 4.3.1.
db _ ww* 44 I Iu9 where Z”(P)
R
’
is the set of all b E Js, such that Z(b) E P, and R = f det (log I~ilp)plw+r e
is the regulator of k. Proof. Because Z is a homomorphism, measure of Z‘(P) volume of P = kdet (log Is&) = R, measure of Z”(Q) = volume of Q and we have only to show & _ 2”Cw* 44 I I (0) Z‘(Q) is the set of all bcsJs, with 1 S 161, < e for no&. Let Q* be the set of all a E IS, with 1 5 lal, c e for p ES,. Then l
because tb E Q*  b E Z‘(Q) to show that
and 1 5 Itbl,, < e. We have therefore only da I P
2”(27rp =Jldl*
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Q* as the cartesian product
setofalla,ok~suchthatI
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337
Q* = ,g_e= x Iss where Q: is the Then
because for p real,
for p complex,
and 1 da, =vg (JY,b,)* = &. s daSm =,g w Is U? DEFINITION 4.3.2. Let h be the class number of k, and select idsles b(l) , . . . , bCh’E J such that the corresponding ideals (p(b(‘)), . . . , cp(@)) represent the diflerent ideal classes. Let w be the number of roots of unity in k. Let E. be the subset of all b E I l(P) (see preceding lemma) such that 2n We define the multiplicative fundamental domain, E, for 0 s argbpo < ;. Jmod
k* to be E=Eob”‘~Eob’2’~...~Eob’h’. 4.3.2. (1) J = U, aE, a disjoint union. 2”(21c)“hR
THEOREM
(2)
jdb
=
&lw
l
B Proof. (1) Starting with any idele b E J we can change it into an idele which represents a principal ideal by dividing it by a uniquely determined b(‘). If this principal ideal is ao (a uniquely determined modulo units), multiplication by etl brings us to an idble of J representing the ideal otherefore into Js,. Once in Js, we can find a unique power product of the fundamental units e, which lands us in J‘(P), with only a root of unity, i, at our disposal. This C is exactly what we need to adjust the argument of 2n the p. component to be in the interval 0, w . Lo and behold we are in E. ! ( > For our original id&le b we have found a unique /I E k* and a unique b(‘) such that b E fib(‘)E,.
338
J. T. TATE
(2) (measure E) = h . (measure E,) = t . (measure I‘(P))
=
2”(2x)“hR JW
according to the two disjoint decompositions E = lJ;a 1 b”‘E,, P(P)
= u CE,
and the preceding lemma. COROLLARY 4.3.1. k* is a discrete subgroup of J (therefore of I). J mod k* is compact. Proof. One sees easily that E has an interior in J. On the other hand, E is contained in a compact. We shall really be interested not in all quasicharacters of 1, but only in those which are trivial on k*. From now on when we use the word quasicharacter we mean one of this type. Let us close our introduction to the id&e group with a few remarks about these quasicharacters. The first thing to notice is that on the subgroup J, a quasicharacter is a character; i.e. lc(b)l = 1 for all b E J, because Jmod k* is compact. Next we mention that the quasicharacters which are trivial on J are exactly those of the form c(a) = lal”, where s is a complex number uniquely determined by c(a). For if c(a) is trivial on J, then c(a) depends only on 1al, and in this dependence is a continuous multiplicative map of the positive real numbers into the complex numbers. Such a map is of the form t + tS as is well known. To each quasicharacter c(a) there exists a unique real number CJsuch that Ic(a)l = lal”. Namely, lc(a)[ is a quasicharacter which is trivial on J. Therefore [c(a)1 = Ial’, for some complex s. Since [c(a)] > 0, s is real. We call c the exponent of c. A quasicharacter is a character if and only if its exponent is 0. 4.4. The [Functions; Functional Equation In this section f(x) will denote a complexvalued function of valuation vectors; f(a) its restriction to ideles. We let 3 denote the class of all functions f(x) satisfying the three conditions: (31) f(x>, andf( x) are continuous, E L,(V); i.e. f(x) E 5Ba,(V). (32)
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ZETAFUNCTIONS
In view of (3r) and (3J, the RiemannRoth theorem is valid for functions of 3. The purpose of (3J is to enable us to define cfunctions with them: DEFINITION 4.4.1. We associate with each f E 3 a function C(f, c) of quasicharacters, defined for all quasicharacters c of exponent greater than 1 by
KC 4 = / f(aMa> da. We call such a function a cfunction of k. Remember that we are now considering only those quasicharacters which are trivial on k*. These were discussed at the end of the preceding section, where the notion “exponent” is explained. If we call two quasicharacters which coincide on J equivalent, then an equivalence class of quasicharacters consists of all quasicharacters of the form c(a) = c,(a)lals, where c,,(a) is a fixed representative of the class and s is a complex number uniquely determined by c. Such a parametrization by the complex variable s allows us to view an equivalence class of quasicharacters as a Riemann surface, just as we did in the local theory (cf. Section 2.4). It is obvious from their definition as an integral that the Cfunctions are regular in the domain of all quasicharacters of exponent greater than 1 (see the corresponding local lemma). What about analytic continuation??? MAIN THEOREM 4.4.1. (Analytic Continuation and Functional Equation of the [Functions.) By analytic continuation we may extend the dejinition of any cfunction ccf, c) to the domain of all quasicharacters. The extended function is single valued and regular, except at c(a) = 1 and c(a) = Ial where it has simple poles with residues of and + r&O), respectively (K = 2”(2z)“hR/(,/ldlw) = voI ume of the multiplicative fundamental domain). [cf, c) satisfies the functional equation w5 4 = r<.L a, where e(a) = lalc‘(a) as in the local theory. Proof. For c of exponent greater than 1 we have [(f, c) =/f(a)c(a)
da = i .I
Ij(tb)c(tb)
db d_f= 7 &(f, c):, It.
say.
Here C,(J 4 = j WMb)
db
J
is absolutely convergent for c of any exponent, at least for almost all t, because it is convergent for some c, and Ic(tb)l = t exponent c is constant for b E J. The essential step in our proof consists in using the RiemanoRoth theorem to establish a functional equation for [,cf c):
340
J. T. TATB
LEMMA A. For all quasicharacters c we have (,CT.c)+~(O)~c(tb)db=Clil~4+~(O)S~(~b)db. E
E
Proof C,(f, c) +f(o) j c(tb) db = C 1 f(tb)c(tb) db +f(O) I c(rb) db E
crek*aE
6
(Because J = u, aE, a disjoint union) = C I f(atb)c(tb) db +f(O) J’ c(tb) db aek*E
E
(d(ab) = db; c(atb) = c(tb)) = I‘ [ C f(otrb)]c(tb)db E
ook*
+ /.f(O)c(fb)db. E
(By hypothesis (3J forf, the sum is uniformly tively compact subset E)
convergent for b in the rela
=/ [&M9] cW)db =J [&$)I @i,c(tb)db E
(RiemannRoth
theorem 4.2.1)
=/ [&3(+q $b)db E
(b + l/b; db + db). Reversing the steps completes the proof. LEMMA B.
1c(tb)db k
=
Id,
o
,
if c(a) = Ial if c(a) is non trivial on J.
Proof. j c(tb) db = c(t) 1 c(b) db. E
E
Here f c(b) db is the integral over the factor group J mod k* of the character E of this group which c(b) represents. Therefore it is either K (= measure of E), or 0, according to whether c(a) is trivial on J or not. In the former case we must notice that c(t) = jtr = t’.
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341
To prove the theorem write, for c of exponent greater than 1, UC)
=i,,,,s 0
=jctu4~ 0
+Jim$ 1
The .f is no problem. For it is equal to the integral of Aa) c(a) da over that half of I where Ial 2 1. Therefore it converges the better, the less the exponent of c is; and since it converges for c of exponent greater than 1, it must converge for all c. Now, the point is that we can use Lemma A (and the auxiliary Lemma B) to transform the d into an $ thereby obtaining an analytic expression for CV; c) which will be good for all c. Namely:
where the expression { { . . . >} is to be included only if c is trivial on J, in which case we assume c(a) = Ial”. W e are still looking only at c of exponent greater than 1. If c(a) = 1al” this means Re (s) > 1, which is just what is needed for the auxiliary integrals under the double bracket to make sense. Evaluating them and making the substitution t + l/t in the main part of the expression we obtain
and therefore
The two integrals are analytic for all c. This expression gives therefore the analytic continuation of [V; c) to the domain of all quasicharacters. From it we can read off the poles and residues directly. Noticing that for c(a) = Ial: e(a) = la]‘“, we see that even the form of the expression is unchanged by the substitution Cr; c) + (3, Q. Therefore the functional equation holds. The Main Theorem is proved! 4.5. Comparison with the Classical Theory We will now show that our theory is not without content, inasmuch as there do exist nontrivial cfunctions. In fact we shall exhibit for each
342
J. T. TATE
equivalence class C of quasicharacters an explicit function f E 3 such that the corresponding [function c(J c) is nontrivial on C. These special Cfunctions will turn out to be, essentially, the classical cfunctions and Lseries. The analytic continuation and the functional equation for our Ifunctions will yield the same for the classical functions. We can pattern our discussion after the computation of the special local cfunctions in Section 2.5. There we treated the cases k real, k complex, and k padic. Now we treat the case k in the large ! The Equivalence Classes of QuasiCharacters. According to a remark at the end of Section 4.3, each class of quasicharacters can be represented by a character. To describe the characters in detail, we will take an arbitrary. but fixed, finite set of primes, S (containing at least all archimedean primes) and discuss the characters which are unramified outside S. A character of this type is nothing more nor less than a product
c(a)= nPc&h,) of local characters, c+,,satisfying the two conditions (1) cP unramified outside S. (2) n c&x) = 1, for a E k*. P To construct such characters and express them in more concrete terms, we write for p fz S: CJQp) = qQlapI$ E, being a character of up, fP a real number (cf. Theorem 2.3.1). For p $ S, we throw all the local characters together into a single character, say c*(a) =v(jsqxap). and interpret c* as coming from an ideal character.
Namely:
The map
Q + (P,(a) =vJJsPorde= is a homomorphism of the idele group onto the multiplicative group of ideals prime to S. Its kernel is I,. c*(a) is identity on 1,. We have therefore c*(a) = ideW), where x is some character of the group of ideals prime to S. Our character c(a) is now written in the form 44 =v~s~v(~v) .vIjJal~~. x(cP&)). To construct such characters we must select our 2,, tP and x such that c(a) = 1, for a E k*. For this purpose we first look at the Sunits, E, of k,
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i.e. the primes; a basis c(a) to c(e,) = the ZP:
ANALYSIS
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ZETAFUNCTIONS
343
elements of k* n I,, for which (ps(s) = o. Assume S contains m+ 1 let e, be a primitive root of unity in k, and let (sr, .Q, . . . , E,} be for the free abelian group of Sunits modulo roots of unity. For be trivial on the Sunits it is then necessary and sufficient that 1, 0 % v 5 m. The requirement C(Q) = 1 is simply a condition on
We therefore fust select a set of i?,, for p E S, which satisfies A. The requirements c(s,) = 1, 1 5 v 5 m, give conditions on the tP: JJe,I~ =v~sE[l&J, 15 v< m which will be satisfied if and only if the numbers rP solve the real linear equations =ilog (pes flE(E”P vv)) ) (B> v~SiPw4v 1s v5 m for some value of the logarithms on the righthand side. We now select a set of values for those logarithms and a set of numbers rP solving the resulting equations B. Since, as is well known, the rank of the matrix (log le&) is m, there always exist solutions rp. And since c log l.Q, = 0 for all Y, the most V6S
general solution is then rp = rP+ r, for any r. While we are on the subject of existence and uniqueness of the rp we may remind the reader that if p is archimedean, different rP give different local characters c+, = ZJF; but if p is discrete, those rP which are congruent mod 2n/log Jyp give the same local cP. Having selected the f, and r+,,how much freedom is left for the ideal character x ? Not much. The requirement c(a) = 1 for all a E k* means that 1 must satisfy the condition
(C) for all ideals of the form rps(a), the ideals obtained from principal ideals by cancelling the powers of primes in S from their factorization. These ideals form a subgroup of finite index hs (less than or equal to the class number h; hs = 1 if S is large enough) in the group of all ideals prime to S. Since the multiplicative function of a on the righthand side of condition (C) has been fixed up to be trivial on the Sunits, it amounts to a character of this subgroup of ideals of the form &a). We must select x to be one of the finite number h, of extensions of this character to the group of all ideals prime to S. The Corresponding Functions of 3. Having selected a character
344
J. T. TATB
unramifled outside S, we wish to find a simple function f(x) E a whose Cfunction is nontrivial on the surface on which c(a) lies. To this effect we choose for each p E S some function f&J E 3P whose (local) Cfunction is nontrivial on the surface on which cP lies (for instance select f to be the function used to compute pP(cP 11:) in Section 2.5). For p 4 S, we let f,(x,) be the characteristic function of the set 0,. We then put f(x) = n fp@p>. P (We will show in the course of our computations that the f(r) is in the class 3.) Their Fourier Transforms. According to Lemma 3.3.3, Ad = ly&J, and moreover, f E ‘z$(V), i.e. f satisfies axiom (3J. Notice that f(x) is the same type of function as f(x), except for the fact that at those x # S where b, # 1, f&J equals Nb,* times the characteristic function of bpr, rather than the characteristic function of 0,. The CFunctions. Since 1f(a)] la]” = n &(a,)1 lal% is a product of local functions, almost all of which are 1 onPUp, we may use Theorem 3.3.1 to check the summability of If(a)1 la]” for Q > 1. A simple computation shows, for p 4 S,
J’ Ifp(a,>lla,l; da, = l fz+lo. kS The summability
follows therefore from the wellknown fact that the product
lI l4” l P$S, is convergent for Q > 1. Well known as this fact is, it should be stressed that it is a keystone of the whole theory. The existence of our Cfunctions just as that of the classical functions, depends on it. It is proved by descending directly to the basic field of rational numbers (see, for example, E. Landau, “Algebraische Zahlen”, 2nd edition, pages 55 and 56). Because!(x) is the same type of function as f(x), we see that If(a)1 lal” is also summable for Q > 1. Therefore f(x) satisfies axiom (33). Having established the .summability, we can also use Theorem 3.3.1 to express the Cfunction as a product of local Cfunctions. Namely C(fa 4 = !J cpv,, c&l) for any quasicharacter c = n cP of exponent greater than 1. If c now denotes P
our special character, 44 = ll cp(ap) =pl&e,(ap). P
xh(a)h
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IN NUMBER
we can compute explicitly
FIELDS
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ZETAFUNCTIONS
345
the local factors of [(f, c 113 for p $ S. Indeed
CJfp ~~11:) = J cv(ap)laplY~, *v
dn,+ = 1x(p)./y^;s’ because, for p 4: S, cJa,> = x(p ord*a+$. If therefore we introduce the classical Cfunction C(s, x), defined for Re (s) > 1 by the Euler product
we can write KfY cl13 =vlJs(&,
c,ll:> .,l&J%*.
C(s,xl
We see that our ccf ~11’) is, essentially, the classical function C(s, x). It may be remarked here that we could have obtained directly the additive expression for C(s, x), C(s,x) =
c
Q htz$Ldp
x(4 Jva”’
had we computed ccf, c 11’)by breaking up I into the cosets of I,, integrating over each coset, and summing the results, rather than by using Theorem 3.3.1 to express the cfunction integral as a product of local integrals. Treating the Cfunction off in the same way we Cnd u.f,~
=pI;l$fp9
c~hrJlsx@,w~;s
41 s, x9,
for Re (8) < 0. Before discussing the resulting analytic continuation and functional equation for [(s, x), we should set our minds completely at rest by checking that ourf(x) satisfies axiom (3J; that is, that the sum
is uniformly convergent for a in a compact subset of I and Y E D. We can do this easily under the assumption that, for p E S, the local functions & we chose in constructing f do not differ too much from the standard local functions which we wrote down in Section 2.5. Namely, we assume for discrete p E S, that f, vanishes outside a compact; and for archimedean p, that f,(x,) goes exponentially to zero as xP tends to inGnity. Under these assumptions one sees first that there is an ideal, a, of k such that f(a(x + t)) = 0 if t $ a,
346
J. T. TATE
for all a in the compact and x in D. The sum may then be viewed as a sum over a lattice in the ndimensional space which is the i&rite part of V, of the values of a function which goes exponentially to zero with the distance from the origin. The lattice depends on a and X, to be sure, but the restriction of a to a compact means that a certain tied small cube will always fit into the fundamental parallelotope of the lattice. The uniform convergence of the sum is then obvious. Analytic Continuation and Functional Equation for c(s, x). The analytic continuation which we have established for our Cfunctions, both in the large and locally, now gives directly the analytic continuation of &, x) into the whole plane. Our functional equations C
and SJ.& cp> = P,(c,)~,(& equation
=Jp,c%ll;+i”>
; Q “Jq*X‘@,)
tip) *&,x).
The explicit expressions for the local functions p,, are tabulated in Section 2.5. The meaning of the fi,, and tp, and their relationship to the ideal character x, is discussed in the first paragraph of this section. These ideal characters, x, which we have constructed out of idele characters, are exactly the characters which Hecke introduced in order to define his {(s, x) is that Cfunction; and the functional “new type of cfunction”. equation we have just written down is the functional equation Hecke proved for it. A Few Comments on Recent Related Literature [Added January 19671 In Lang’s book Algebraic Numbers (AddisonWesley, 1964) there is a review of the theory above, in which the global results are renormalized in a way which corresponds more closely to the classical theory and which is more suitable for applications. The applications to prime densities, the BrauerSiegel theorem, and the “explicit formulas” are also treated there. Lang has urged me to point out that in the main theorem on equidistribution of primes (Theorem 6, p. 130 lot. cit.) one must assume that a(Jp) = G, not only that a(J,) = G; thus the first application following that theorem, concerning the equidistribution of log p, is not valid. For a reinterpretation of the proof of the functional equation given above, see Weil, Fonction z&a et distributions, S&ninaire Bourbaki, No. 312, June 1966. Siegel’s work on quadratic forms has inspired much ad&c (addle is the modern term for valuationvector) analysis in recent years. For some
FOURIER
crucial
results
ANALYSIS
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and bibliography in that direction, 1964 and 1965.
347
ZETAFUNCTIONS
see Weil’s
papers
in
Acta Mathematics,
REFERENCES
Artin, E. and Whaples, G. (1945). Axiomatic characterization of fields by the product formula for valuations. Bull. Am. math. Sot. 51,469492. Cartan, H. and Godement, R. (1947). Thkorie de la dualitCet analyseharmonique dans les groupes abClienslocalement compacts. Ads. scient. EC. norm. sup., Paris (3), 64, 7999.
Chevalley, C. (1940). La thkorie du corps de classes.Ann. Math. 41, 394418. Hecke, E. Uber eine neue Art von Zetafunktionen und ihre Beziehungen zur Verteilung der Primzahlen. Math. Z. 1(1918), 357376; 4 (1920),1121. Reprinted in “Mathematische Werke”, Vandenhoeck und Ruprecht, GGttingen, 1959. Matchett, M. Thesis, Indiana University (1946), unpublished.
Exercises? Exercise Exercise Exercise Exercise Exercise Exercise Exercise Exercise
1: 2: 3: 4: 5: 6: 7: 8:
The power residue symbol . . The norm residue symbol . . The Hilbert class field . . . Numbers represented by quadratic forms Local norms not global norms. . On decomposition of primes . . A lemma on admissible maps . . Norms from nonabelian extensions .
. . . . . . . .
. . . . . . . .
. . . . . . . .
. 348 . 351 . 355 357 : 360 . 361 . 363 . 364
Exercise 1: The Power Residue Symbol (Legendre, Gauss, et al.) This exercise is based on Chapter VII, § 3, plus Kummer theory (Chapter III, 3 2). Let m be a tied natural number and K a lixed global field containing the group h of mth roots of unity. Let S denote the set of primes of Kconsisting of the archimedean ones and those dividing m. If al, . . . , a, are elements of K*, we let S(a,, . . ., a,) denote the set of primes in S, together with the primes v such that [Q& # 1 for some i. For a E K* and b E Is(‘) the symbol
0
f
is defined by the equation
where L is the field K(l;fla). EXERCISE 1.1. Show
0
z
is an mth root of 1, independent
of the choice
of “Ja. EXERCISE 1.2. Working in the field L’ = K(?/a, “Ja’) and using Chapter VII, $ 3.2 with K’ = K and L = K(mJa>, show ($)
=(;)
if b E I%“* 0’).
(fz)
EXERCISE 1.3. Show
(ii)=@(i)
if b E Ia(‘).
t These “exercises” refer primarily to Chapter VII, “Global class field theory*‘, and were prepared after the Conference by Tate with the connivance of Serre. They adumbrate some of the important results and interesting applications for which unfortunately there was not enough time in the Conference itself. 348
349
EXERCTISES Hence,
EXERCISE 1.4. (Generalized Euler criterion.) where NV = [k(v)], and
0
If’ u 4 S(u) then mj(Nv
l),
% is the unique mth root of 1 such that
a 0V
Nol EU
ii (mod
P,).
EXERCISE 1.5. (Explanation of the name “power residue symbol”.) v $ S(u) the following statements are equivalent:
For
a i =l. V () c1 (ii) The congruence a?’ = a (mod p,,) is solvable with x E 0,. (iii) The equation x” = a is solvable with x E &. (Use the fact that k(o)* is cyclic of order (Nol), Chapter II, App. C.)
and Hensel’s lemma,
EXERCISE 1.6. If b is an integral ideal prime to m, then m1
force&. (3i =[" (Do this first, using Exercise 1.4, in case b = 1) is prime. b = C n,v, note that, putting Nu = 1 +mr,,, we have Nb = n (1 +mr,,p EXERCISE 1.7. then6)
Then for general
s 1 + m C n, r,, (mod m’).)
If Q and b E Iz(“) are integral,
and if a’ E u (mod b),
=(i).
EXERCXE 1.8. Show that Artin’s reciprocity law (Chapter VII, § 3.3) for a simple Kummer extension L = K(“Ja) implies the following statement: rf b and 6’ E Is(“), and b’ 6l = (c) is the principal ideal of an element c E K* such that c E (K,*)” for aN u E S(u), then G)=(t). the condition
c E (II;*)”
Note that for v # S,
will certainly be satisfied if c = 1 (mod p,).
EXERCLSE 1.9. Specialize now to the case K = Q, m = 2. Let u, b, . . . denote arbitrary nonzero rational integers, and let P, Q, . . . denote positive, odd rational
integers.
For (a, P) = 1, the symbol (;)
= (&)
=*1
is
EXERCISES
350
defined, is multiplicative
in each argument
separately, and satisfies
(g=(i) 6)=(;)
if a = b (mod I’).
Artin’s reciprocity law for Q(Ja)/Q
implies
if P 3 Q (mod 84,
where a, denotes the “odd part of a”, i.e. a = 2’ao, with a, odd. (Use the fact that numbers = 1 (mod 8) are 2adic squares.) EXERCISE 1.10. From Exercise 1.9 it is easy to derive the classical law of quadratic reciprocity, namely Pl qp,
f
= (1)
p*1 8
P Q
, and
0
Indeed the formula
pQ =(1)
Pl
2
Ql
‘T
00
(*) above allows one to calculate
0
g
as function of P
for any tied a in a finite number of steps, and taking a =  1 and 2 one proves the first two assertions easily. For the last, define
1.
Then check first that if P E Q (mod 8) we have
(Writing
Q = P+8a
one finds using
Now, given arbitrary relatively prime P and Q, one can find R such that RP = Q (mod 8) and (R, Q) = 1 (even R E 1 (mod Q)), and then, by what we have seen,
.
Fixing R and varying P, keeping (P, Q) = 1, we see that
EXERCISES
351
Exercise 2: The Norm Residue Symbol (Hilbert, Hasse) We assume the reciprocity law for Kummer extensions, and use Chapter VII, $ 6. The symbols m, K, S, and S(ur, . . . , a,) have the same significance as in Exercise 1. For a and b E K* and an arbitrary prime u of K we define (a, b), by the equation (~u)~*(*) = (a, b),,ju, where $, : K: + G” is the local Artin map associated with the Kummer extension K(‘“Ja)/K. EXERCISE 2.1. Show that (a, b), is an mth root of 1 which is independent of the choice of vu. EXERCISE 2.2. Show (a, b),(u, b’), =(a, bb’), and (a, b),(u’, b), = (au’, b),.
Thus, for each prime v of K, we have a bilinear map of K* x K* into the group p,,, of mth roots of unity. EXERCISE 2.3. Show that (a, b), = 1 if either a or b E (K,*)m, and hence that there is a unique bilinear extension of (a, b), to K,* x K:. This extension is continuous in the vadic topology, and can be described by a finite table of values, because Kz/(K:)‘” is a finite group (of order m2/lmjo, where lrnlu is the normed absolute value of m at v). Moreover, the extended function on K: x K,* can be described purely locally, i.e. is independent of the field K of which K, is the completion (because the same is true of $3, and induces a nondegenerate pairing of K,*/(K:)” with itself into p,,,; however we will not use these local class field theoretic facts in most of this exercise. For a general discussion of (a, b),, and also for some explicit formulas for it in special cases, see Hasse’s “Bericht”, Part II, pp. 53123, Serre’s “Corps Locaux”, pp. 212221, and the ArtinTate notes, Ch. 12. The symbol (a, b), defined here coincides with that of Hasse and Serre, but is the opposite of that defined in ArtinTate. While we are on the subject, our local Artin maps $, coincide with those in Serre and in ArtinTate, but are the opposite of Hasse’s. EXERCISE 2.4. Show that (a, b), = 1 if b is a norm for the extension K,(vu)/K,. (See Chapter VII, 5 6.2; the converse is true also, by local class field theory, but this does not follow directly from the global reciprocity law.)
EXERCXE 2.5. We have (a, b), = 1 if a+ b E (Kz)“‘; in particular, a), = 1 = (a, 1  a),. (This follows from the purely algebraic lemma: (a, Let F be afield containing the group p,,, of mth roots of unity, and let a E FL. Then for every x E F the element ~“‘a is a norm from F(vu). Indeed, let am = a. The map d H au/a is an isomorphism of the Galois group onto a subgroup ,u~ of p,,, and is independent of the choice of a. Hence if (CJ is a
352
EIERcIsm
system of representatives of the COSets of & in &,,, we have for each x E F
Q.E.D.) EXERCISE 2.6. Show that (a, b),(b, a), = 1. (Just use bilinear@ on 1 = (ab, ab),.) EXERCISE 2.7. If u is archimedean, we have (a, b), = 1 unless K, is real, both a < 0 and b < 0 in &,, and m = 2. (In the latter case we do in fact have (a, b), =  1; see the remark in Exercise 2.4. Note that m > 2 implies that K,, is complex for every archimedean a.) EXERCISE 2.8. (Relation between normresidue and powerresidue symbols.) a o(b) If t, # S(u), then (a, b), = ; ; in particular, (a, b), = 1 for a 4 S(a, b). 0 (See ,the fist lines of Exercise 1 for the definition of S and S(u), etc. The result follows from the description of the local Artin map in terms of the Frobenius automorphism in the tutramified case. More generally, 0 # S =z(a, b), x
g , where c = ( l)o(a)u(b)&b)b~a~ 0 is a unit in & which depends bilinearly on a and b. To prove this, just write a = &%z,, and b  nuCb)bowhere v(x) = 1, and work out (a, b), by the previous rules; for the geometric analog discussed in remark 3.6 of Chapter VII, see Serre, lot. cit., Ch. III, Section 4.) EXERCISE 2.9. (Product Formuh.) For a, b E K* we have n (a, b), = 1, the product being taken over all primes u of K Exnncrsn 2.10. (i%e general powerreciprocity in K* we define
where (b)” is defined in Chapter VII,
law.) For arbitrary a and b
3 3.2.
Warning: With
% defined in this generality the rule 0 does not always hold, but it does hold if S(b) A S(a, a’) = S, and especially if b is relatively prime to a and a’. The other rule, ($=(;)($holds in general. Using Exercises 2.6, 2.8 and 2.9, prove that
353
EXERCISES
In particular
(*I
a iJ
00
and
bl a
=c~s(h a),,
A
0
c**>
b =v~s(l, b),,
if S(a) n S(b) = S,
if S(1) = S.
EXERCISE 2.11. If K= Q and m = 2, then S = (2, co}, and for P > 0 as in Exercise 1.10, we have (x, P), = 1. Hence the results of Exercise 1.10 are equivalent with PZ1
Pl
Ql
and (P, Q))z= ( l)y ‘2, (UT, for odd P and Q. On the other hand, these formulas are easily established working locally in Qz. In particular, the fact that (1+4c, b), = ( l)ol(*)‘, from which the value of (a, b), is easily derived for all a, b using Exercises 2.2,2.5 and 2.6, is a special case of the next exercise. (l,P)z=
(l)y
w%=
EXERCISE 2.12. An element a E K is called uprimary (for m) if K(“Ju)/K is unramified at u. For u 4 S, there is no problem: an element a is *primary if and only if u(a) I 0 (mod m). Suppose now u divides m and m = p is a prime number. Let C be a generator of q, and put 1 = 1 C. Check that Apzp‘/pis a unit at u, and more precisely, that rZp’ E p (mod PA), so that Apl/p z  1 (mod p,). Let u be such that a = 1 (modpllo,), so that we have a = 1 +Kc, with c E 0,. Prove that a is uprimary, and that for all b,
(a, b), = CS@)“(b),
where S denotes the trace from k(u) to the prime field and E is the uresidue oft. Also,ifu = 1 (mod p&), then u is uhyperprimary, i.e. u E (Kz)“. (Let ap = a, and write a = 1 +ti. Check that x is a root of a polynomial f(X) E o,[ X] such that j’(X) = Xp X c (mod p,). Thus f’(x) E  1 $0 And if c 3 0 (mod pJ (mod P,), so Ud = &c”J u > is indeed unratied. then f(X) splits by Hensel’s lemma, so &(“Ja) = &. Now xp = x+ c (mod p,), so if% = p*, then On the other hand, if a’ = [a = 1 +W, then x’ E x 1 (mod p,). bining these facts gives the formula for (a, b),.)
Com
2.13. Let p be an odd prime, [ a primitive pth root of unity, and m = p. Then p is totally ramified in K, and 1 = 1 C generates the prime ideal corresponding to the unique prime u of K lying over p. Let U, denote the group of units = 1 (mod 2’) in Kf, for i = 1,2, . . . . Then the image of vi = 1  rZ’ generates tJ,IU,, r, which is cyclic of order p, and the image of L generates K:/(K:)PIJ,. By the preceding exercise, EXERCSE
K = Q(&
354
EXERCISES
u p+l = uap. Hence the elements L, 5 = qr, ll2 = tf2,. . ., lL’= qp generate (Kf)/(K:)P. But that group is of order p2/1pI, = ~l+~, so these generators are independent mod pth powers. Show that (a>
(Vi3
rljh
=
(Vi,
Vi+j)u(Vi+j,
?j>v(Vi+
jt
J)i’,
for
all
kJ
2
1.
(b) Ifi+jrp+l,then(a,b),= 1foralla~U~andb~U~. 1, forl
(*) (;) = ($
for relatively prime a and b, each = + 1 (mod 3R),
and also (**I
, for a = f(l+
3(m+ rtc)).
As an application, prove: If q is a rational prime E 1 (mod 3), then 2 is a cubic residue (mod q) if and only if q is of the form x2 +27y2 with X, y E Z. (Write q = nR with rr E + 1 (mod 3R). Then Z/qZ z RjlrR, so 2 is a cubic residue (mod q) if and only if 2 = 1. Now use (*), and translate f 0 7T 0 into a statement about q.)
= 1
EXERIXE 2.15. Let L be the splitting field over Q of the polynomial X3  2. The Galois group of L/Q is the symmetric group on three letters. Using the preceding exercise, show that for p # 2,3 the Frobenius automorphism is given by the rules: FL&p) = (l), ifp = 1 (mod 3) and p of the form x2 +27y2, = 3cycle, if p E 1 (mod 3) and p not of the form x2 + 27u2, %&4 h/p(P) = 2cycle, if p E  1 (mod 3).
355
ExERclsEs
Hence, by Tchebotarov’s theorem, the densities of these sets of primes are l/6, l/3 and l/2, respectively. EXERCISE 2.16. Consider again an arbitrary K and m. Let a,, . . ., Us be a finite family of elements of K*, and let L be the Kummer extension generated by, the mth roots of those elements. Let T be a finite set of primes of K containing S(a,, . . . , a,), and big enough so that both JK = K*JR,=, and Jt = L*J=,=s, where T’ is the set of primes of L lying over T. Suppose we are given elements 5v,i E c(m, for u E T and 1 5 i 5 r, such that
(i) For each i, we have fl &
= 1, and
“ST
(ii) For each u E T, there exists an x, E K: for all i.
such that
(x,, a,)” = [y,l
Show then that there exists a Tunit x E Kr such that (x, a,)” = cu,i for all ooTandall1 ~ilr. The additional condition on T, involving T’, is necessary, as is shown by the example K = Q, m = 2, T = (00,2,7}, r = 1, a, =  14, co3,1=  1, To prove the statement, consider the group 52,l =I, L,l = 1. X = l’&~V*)/WV*)“‘~ th e subgroup A generated by the image of KT, and the smaller subgroup A, generated by the images of the elements a,, 1 5 i I r. The form (x, JJ) = nvsT (x,, y,), gives a nondegenerate pairing of .X with itself to p,,,, under which A is self orthogonal, and indeed exactly so, because [X] = mzt and [A] = m’, where t = [T]. (See step 4 in the proof of the second inequality in Chapter VII, 5 9, the notations S, n, and s there being replaced by T, m, and t here.) Thus X/A = Horn (A, pm) (note by the way that both groups are isomorphic to Gal (K(vKr)/K), by class field theory and Kummer theory, respectively), and, vice versa, A RJ Horn (X/A, /A,,,). So far, we have not used the condition that JL = L*JLsT.. Use it to show that if a E A and n,(a) E q(A,) for all v, where nv is the projection of X onto K,*/(K,*)“, then a E A,, i.e. vu EL. Now show that, in view of the dualities and orthogonalities discussed above, this last fact is equivalent to the statement to be proved. Exercise 3: The Hilbert Class Field Let L/K be a global abelian extension, u a prime of K, and iv : K: + JK the canonical injection. Show that v splits completely in L if and only if i,(K:) c K*NLIKJL, and, for nonarchimedean v, that v is tutramified in L if and only if i,(U,) c K*N,IKJL, where U,, is the group of units in K,,. (See Chapter VII, $ 5.1, 5 6.3.) Hence, the maximal abelian extension of K which is unramified at all nonarchimedean primes and is split completely at all archimedean ones is the class field to the group K*J,,,, where S now denotes the set of archimedean primes. (Use the Main Theorem
356
EXJZRCISES
(Chapter VII, 5 5.1) and the fact that K*NLIKJL is closed.) This extension is called the Hilbert class field of K; we will denote it by K’. Show that the Frobenius homomorphism FEIK induces an isomorphism of the ideal class group Hx = Ig[Px of K onto the Galois group G(K’/K). (Use the Main Theorem and the isomorphism JRIJR,s L$ I&) Thus the degree [K’ : K] is equal to the class number h, = [Hg] of K. The prime ideals in K decompose in K’ according to their ideal class, and, in particular, the ones which split completely are exactly the principal prime ideals. An arbitrary ideal a of K is principal if and only if F,,,,(a) = 1. The “class field tower”, K c K’ c K” = (K’)‘c . . . can be infinite (see Chapter IX). Using the first two steps of it, and the commutative diagram (see (11.3), diagram (13)) I K m?!5, G(K’I’K) con V I 1 I Ke FK”x’ + G(K”/K’), Artin realized that Hilbert’s conjecture, to the effect that every ideal in K becomes principal in K’, was equivalent to the statement that the Verlagerungt V was the zero map in this situation. Now G(K”/K’) is the commutator subgroup of G(K”/K) (Why?), and so Artin conjectured the “Principal ideal theorem” of group theory: If G is a finite group and G’ its commutator subgroup, then the map V: (G/G’) + G’/(Gy is the zero map. This theorem, and therewith Hilbert’s conjecture, was then proved by Furtwiingler. For a simple proof, see Witt, Proc. Intern. Conf. Math., Amsterdam, 1954, Vol. 2, pp. 7173. The tist five imaginary quadratic fields with class number # 1 are those with discriminants  15, 20,  23,  24, and  31, which have class numbers 2, 2, 3, 2, 3, respectively. Show that their Hilbert class fields are obtained by adjoining the roots of the equations X2 + 3, X2 + 1, X3  X 1, X2 + 3, and X3 + X 1, respectively. In general, if K is an imaginary quadratic field, its Hilbert class field K’ is generated over K by thejinvariants of the elliptic curves which have the ring of integers of K as ring of endomorphisms; see Chapter XIII. Let Jz denote the group of ideles which are positive at the real primes of K and are units at the nonarchimedean primes. The class field over K with norm group K*J& is the maximal abelian extension which is unramified at all nonarchimedean primes, but with no condition at the archimedean primes; let us denote it by Ki. Let Pg denote the group of principal ideals of the form (a), where a is a totally positive element of K. Show that F,,,, gives an isomorphism: &/Pi % G(K,/K). Thus, G(K,/K’) is an elementary t Called the
transfer
in Chapter IV, 5 6, Note after Prop. 7.
EXERCISES
357
abelian 2group, isomorphic to P=/Pz. Show that (PK : Pi)(Ks : Kg) = 2”, where Ki = K* n J& is the group of totally positive units in K, and rl is the number of real primes of K. We have Q1 = Q, clearly, but this is a poor result in view of Minkowski’s theorem, to the effect that Q has no nontrivial extension, abelian or not, which is unramified at all nonarchimedean primes (Minkowski, “Geometrie der Zahlen”, p. 130, or “Diophantische Approximationen” p. 127). Consider now the case in which Kis real quadratic, [K: Q] = 2, and rl = 2. Show that [Kl : K’] = 1 or 2, according to whether Ne =  1 or Ne = 1, where E is a fundamental unit in K, and N = Nx,o. For example, in case K = Q(,/2) or Q(J5) we have K’ = K, because the class number is 1, and consequently also Kl = K, because the units E = 1 + ,/2 and E = +(l + ,/5) have norm  1. On the other hand, if K = Q&/3), then again K’ = K, but Kl # K, because E = 2+,/3 has norm 1; show that K, = K(J 1). In general, when  1 is not a local norm everywhere (as in the case K = Q(J3) just considered), then Ns = 1, and Kl # K’. However, when  1 is a local norm everywhere, and is therefore the norm of a number in K, there is still no general rule for predicting whether or not it is the norm of a unit. Exercise 4. Numbers Represented by Quadratic Forms Let K be a field of characteristic different from 2, and a nondegenerate quadratic form in n variables with coefficients in K. We say that f represents an element c in K if the equation f(X) = c has a solution X = x EK” such that not all xi are zero. Zff represents 0 in K, then f represents all elements in K. Indeed, we have (tX+ Y) = t’f(x)+m(x, Y)+,f(Y). Iff(x)=Obutx#(O,O,..., 0), then by the nondegeneracy there is a y E K” such that B(x, JJ) # 0, so that f(tx +JJ) is a nonconstant linear function of I and takes all values in K as I runs through K. A linear change of coordinates does not affect questions of representability, and by such a change we can always bring f to diagonal form: f = c a, A’: with all a, # 0. If f = cX:  g(X,, . . ., X,) then f represents 0 if and only if g represents c, because if g represents 0 then it represents c. Hence, the question of representability of nonzero c’s by forms g in n  1 variables is equivalent to that of the representability of 0 by forms f in n variables. The latter question is not affected by multiplication off by a nonzero constant; hence we can suppose f in diagonal form with a1 = 1 in treating it: EXERCISE 4.1. The form f = X2 does not represent 0. EXERCISE 4.2. The form f = X*b
Y* represents 0 if and only if b E (K*)*.
358
EXERCISES
EXERCISE 4.3. The form f = X2b Y2cZ2 represents 0 if and only if c is a norm from the extension field K(Jb). EXERCISE 4.4. The following statements are equivalent: (i) Theformf= X2bY2  cZ2 + acT2 represents 0 in K. (ii) c is a product of a norm from K&/a) and a norm from K(Jb), (iii) c, as element of K(,/ab), is a norm from the field L = K(Ja, ,/b). (iv) The form 9 = X2bY2 cZ2 represents 0 in the field K(,/ab). (We may obviously assume neither a nor b is a square in K. Then the equivalence of(i) and (ii) is clear because the reciprocal of a norm is a norm, and the equivalence of (iii) and (iv) follows from Exercise 4.3 with K replaced therein by K(dab). It remains to prove (ii) o (iii), and we can assume ab 4 (K*)2, for otherwise the equivalence is obvious. Then Gal (L/K) is a fourgroup, consisting of elements 1, p, rr, r such that p, Q, and r leave fixed, respectively, Jab, ,/a, and ,/b, say. Now (ii) * (ii’): 3 x, y E L such that x” = x, y’ = y, and a?+“yl +p = c; and (iii) e (iii’) 3 z EL such that zI+~ = c. Hence (ii) * (iii) trivially. Therefore assume (iii’), put u  clzb+l and check that ub = u, i.e. u E K(Ja), and up+l = 1. Hence by Hilbert’s iheorem 90 (Chapter V, 5 2.7) for the extension K(Ja)/K, there exists x # 0 such that x0 = x and xp’ = u. Now put y = zp/x, and check that (ii’) is satisfied.) So far, we have done algebra, not arithmetic. From now on, we suppose K is a globalfield of characteristic # 2. EXERCISE 4.5. The form f of Exercise 4.3 represents 0 in a local field K,, if and only if the quadratic norm residue symbol (b, c), = 1. Hencefrepresents 0 in K, for all but a finite number of o, and the number of n’s for which it does not is even. Moreover, these last two statements are invariant under multiplication off by a scalar and consequently hold for an arbitrary nondegenerate form in three variables over K. EXERCISE 4.6. Let f be as in Exercise 4.4. Show that iff does not represent 0 in a local field K,, then a 4 (K:)2, and b 4 (Ki)‘, but ab E (Kz)2, and c is noi a norm from the quadratic extension K&/a) = K&/b). (Just use the fact that the norm groups from the different quadratic extensions of K,, are subgroups of index 2 in K:, no two of which coincide.) Now suppose conversely that those conditions are satisfied. Show that the set of elements in K,, which are represented by f is N cN, where N is the group of nonzero norms from K,(,/ a) , and in particular, that f does not represent 0 in K,. Show, furthermore, that if NcN # K:, then  1 $ N, and N+ N c N. Hence f represents every nonzero element of K, unless K, NNR and f is positive definite. EXERCISE 4.7. A form f in n 2 5 variables over a local field K, represents 0 unless K, is real and f definite.
EUIRCISES
359
EXERCWI 4.8. Theorem: Let K be a global jield and f a nondegenerate quadratic form in n variables over K which represents 0 in &for each prime v of K. Then f represents 0 in K. (For n = 1, trivial; n = 2, cf. Chapter VII, 5 8.8; n = 3, cf. Chapter VII, $ 9.6 and Exercise 4.3; n = 4, use Exercise 4.4 to reduce to the case n = 3; Anally, for n 2 5, proceed by induction: Let f(X) = ax: + bX2 9(X3,. . ., XJ, where g has n2 2 3 variables. From Exercise 4.5 we know that g represents 0 and hence every number in K, for all v outside a kite set S. Now (K,*)’ is open in K:. Hence, by the approximation theorem there exist elements x1 and x2 in K, such that the element c = ax: + bxj: # 0 is represented by g in K, for all v in S, and hence for all v. By induction, the form cY2g(&. . . ., XJ in n 1 variables represents 0 in K. Hence f does.) EXERCISE 4.9. Corollary: If n 2 5, then f represents 0 in K unless there is a real prime v at which f is definite. EXERCISE 4.10. A rational number c is the sum of three rational squares if and only if c = 4”r where r is a rational number > 0 and $ 7 (mod 8); every rational number is the sum of four rational squares. EXERCISE 4.11. The statements in the preceding exercise are true if we replace “rational” by “rational integral” throughout. (The 4 squares one is an immediate consequence of the 3 squares one, so we will discuss only the latter, although there are more elementary proofs of the four square statement not involving the “deeper” three square one. Let c be a positive integer as in 4.10, so that the sphere IX]’ = X:+Xf+Xz = c has a point x = (x,, x2, xs) with rational coordinates. We must show it has a point with integral coordinates. Assuming x itself not integral, let z be an integral point in 3space which is as close as possible to x, so that x = z+a, with 0 < lal2 I 3/4 < 1. The line I joining x to z is not tangent to the sphere; if it were then we would have lal2 = lzi2 1x1’ = jzl’c, an integer, contradiction. Hence the line I meets the sphere in a rational point x’ # x. Now show that if the coordinate of x can be written with the common denominator d > 0, then those of x’ can be written with the common denominator d = lal’d < d, so that the sequence x, x’, (x’)‘, . . . must lead eventually to an integral point. Note that d’ is in fact an integer, because d’= jal’d = Ixzj2d = (1x1’2(x,z)+Iz12)d = cd2(dx,z)+lzl’d.) EXERCISE 4.12. Let f be a form in three variables over K. Show that if f does not represent 0 locally in K,, then the other numbers in K, not represented by f constitute one coset of (K,*)2 in K:. (Clearly one can assume f = X2b Y2 ~2’; now use Exercise 4.6.) Using this, show that if K = Q and f is positive definite, then f does not represent all positive integers.
(Note the last sentence in Exercise 4.5.)
EXERCISES
360
For further developments and related work see 0. T. O’Meara: “Introduction to Quadratic Forms” (Springer, 1963) or Z. I. Borevii? and I. R. SafareviE, “Teorija Cisel” (“Nauka”, Moskva, 1964). [English translation, Z. I. Borevich and I. R. Shafarevich, “Number Theory”, Academic Press, New York: German translation, S. I. Borevicz and I. R. SafareviE, “Zahlentheorie”, Birkhluser Verlag, Basel.] Exercise 5: Local Norms Not Global Norms, etc. Let L/K be Galois with group G = (1, p, cr, z) z (Z/ZZ)*, and let K,, K,, and K3 be the three quadratic intermediate fields left fixed by p, 6, and z, respectively. Let N, = Nz,,,&‘) for i = 1, 2, 3, and let N = N,,,(L*). EXERCISE 5.1. Show that N1 N2N3 = (X E K*]x2 E N}. (This is pure algebra, not arithmetic; one inclusion is trivial, and the other can be proved by the methods used in Exercise 4.3.) EXERCLSE 5.2. Now assume K is a global field. Show that if the local degree of L over K is 4 for some prime, then N1 N, N3 = K* (cf. Chapter VII, 0 11.4). Suppose now that all local degrees are 1 or 2. For simplicity, suppose K of characteristic # 2, and let Kl = K(,/uJ for i = 1,2,3. For each i, let S, be the (infinite) set of primes of K which split in 4, and for x e K* put
=~g~1A
=orIy2JLa
=
fL
where (x, y),, is the quadratic norm residue symbol. Show that Ni. N2 N3 = Ker 9~ and is a subgroup of index 2 in K*. (The inclusion Nl N2 N3 c Ker Q is trivial. From Exercise 5.1 above and Chapter VII, 5 11.4 one sees that the index of N1 N2N3 in K* is at most 2. But there exists an x with (p(x) =  1 by Exercise 2.16.) EXERCISE 5.3. Let K = Q and L = Q(,/13, ,/17). Show that if x is a product of primes p such that
0
fi
=  1 (e.g. p = 2, 5, 7, 11, . . .), then
F7 * Hence S2, 72, 102, 112, 142,. . . are some examples of numbers 0 which are local norms everywhere from Q(,/13, ,/17) but are not global norms. Of course, not every such number is a square; for example, 142 is the global norm of 3(7 + 2&3 +,/17), and comparing with the above we see that  1 is a local norm everywhere but not a global norm. EXERCISE 5.4. Suppose now that our global Cgroup extension L/K has the property that there is exactly one prime v of K where the local degree is 4: Let w be the prime of L above t’ and prove that fi ‘(G, L*) = 0, but &I
=
ExERcms
361
fi ‘(G, Lz) z Z/22. (Use the exact sequence near the beginning of paragraph 11.4. The map g is surjective, as always when the 1.c.m. of the local degrees is the global degree. And the map g: R ‘(G, Jd + R(G, Cd is also injective, because of our assumption that the local degree is 4 for only one prime.) Let A, resp. A,, be the group of elements in L*, resp L$ whose norm to K (resp. to K,,) is 1, and let A be the closure of A in L$ It follows from the above that A = (L*)p(L*)yL*)y and that A = (L*w)p‘(L~y‘(L*,))‘’ is of index 2 in A,,,. Now, as is well known, there is an algebraic group T defined over K (the twisted torus of dimension 3 defined by the equation %/r&Q = 1) such that T(K) = A and T(K,) = A,. Hence we get examples which show that the group of rational points on a torus T is not necessarily dense in the group of vadic points (see last paragraph below). However, it is not hard to show that if T is a torus over K split by a Galois extension L/K, then T(K) is dense in T(K,) for every prime v of K such that there exists a prime v’ # v with the same decomposition group as v; in particular, whenever the decomposition group of v is cyclic, and more particularly, whenever v is archimedean. As a concrete illustration, take K = Q and L = Q(,/ 1, ,/2) = Q(t), where c4 =  1. Then L is unramified except at 2, but totally ramified at 2, and consequently there is just one prime, 2, with local degree 4. Let M = Q(i) where i = C2 = J1, and let L, and M,, denote the completions at the primes above 2. It is easy to give an adhoc proof without cohomology that the elements of L with norm 1 are not dense in those of Lz: just check that the element z = (2+i)/(2i) 8 M,, is a norm from L, to L,,, but that z(M,*)~ contains no element y E M such that y is a global norm from L to A4 and such that NM,o(y) = 1. Exercise 6 : On Decomposition of Primes Let L/K be a finite global extension and let S be a finite set of primes of K. We will denote by Spl, (L/K) the set of primes v 4 S such that v splits completely in L (i.e. such that L @ K, z KckKJ), and by Splk (L/K) the set of primes v #S which have a Kisomorphism L 3 G). equality holds if K is Galois, by the Tchebotarov density VIII, $ 3.)
silit factor in L (i.e. such that there exists a Thus Spl, (L/K) c Spli (L/K) always, and in which case Spl& (L/K) has density [L : K ] ’ theorem. (Enunciated near end of Chapter
362
EXERCISES
EXERCISE 6.1. Show that if L and M are Galois over K, then L = M2
SPJ, (M)
c SPl, VA
(Indeed, we have SPL (LWK)
= Spb W/K)
n SPL W/K),
SO
L c M + Spl,(M)
t Spl&)
=r Spl&M/K)
= Spl,(M/K) =s[LM:K]=[M:I+LcM;
where was Galoisness used?) Hence L = Mo Spl,(L) = Spl, (M). Application: If a separable polynomial f(X) E K[XJ splits into linear factors mod p for all but a finite number of prime ideals p of K, then f splits into linear factors in K. (Take L = splitting field of f(X), and M = K, and S large enough so that f has integral coefficients and unit discriminant outside S.) Finally, note that everything in this exercise goes through if we replace “all primes v 4 s” and “all but a finite number of primes u” by “all tr in a set of density 1”. EXERCISE 6.2. Let L/K be Galois with group G, let H be a subgroup of G, and let E be the fixed field of H. For each prime tt of K, let G” denote a decomposition group of 0. Show that v splits completely in E if and only if all of the conjugates of G” are contained in H, whereas v has a split factor in E if and only if at least one conjugate of G” is contained in H. Hence, show that the set of primes Spli (E/K) has density $P&~]/[G]. Now prove the lemma on finite groups which states that the union of the conjugates of a proper subgroup is not the whole group (because they overlap a bit at the identity!) and conclude that if Splk (E/K) has density 1, then E = K. Application: If an irreducible polynomial f(X) E K[X] has a root (mod p) for all but a finite number of primes p, or even for a set of primes p of density 1, then it has a root in K. This statement is false for reducible polynomials ; consider for example f (X) = (X2 a) (X2  b) (X2  ab),where a, b, and ab are nonsquares in K. Also, the set Spl’ (E/K) does not in general determine E up to an isomorphism over K; cf. Exercise 6.4 below. EXERCISE 6.3. Let Hand H’ be subgroups of a finite group G. Show that the permutation representations of G corresponding to H and H’ are isomorphic, as linear representations, if and only if each conjugacy class of G meets H and H’ in the same number of elements. Note that if H is a normal subgroup then this cannot happen unless H’ = H. However, there are examples of subgroups H and H’ satisfying the above condition which are not conjugate; check the following one, due to F. Gassmann (Math. Zeit., 25, 1926): Take for G the symmetric group on 6 letters (xi) and put
EXERCISES
363
H = (11 (Xl X,)(X, X4), (Xl x,>w, X,)9 (Xl X,)(X, x3>> H ‘= (1, (Xl x,>w, X‘s), (Xl X,)(X, X,)9 (X3 &)(X5 X,>~ (H leaves X5 and X6 fixed, where H’ leaves nothing fixed; but all elements # 1 of H and H’ are conjugate in G.) Note that there exist Galois extensions of Q with the symmetric group on 6 letters as Galois group. EXERCISE 6.4. Let L be a finite Galois extension of Q, let G = G(L/Q), and let E and E’ be subfields of L corresponding to the subgroups H and H’ of G respectively. Show that the following conditions are equivalent: (a) H and H’ satisfy the equivalent conditions of Exercise 6.3. (b) The same primes p are ramified in E as in E’, and for the nonramified p the decomposition of p in E and E’ is the same, in the sense that the collection of degrees of the factors of p in E is identical with the collection of degrees of the factors of p in E’, or equivalently, in the sense that A/pA zz A’IpA’, where A and A’ denote the rings of integers in E and E’ respectively. (c) The zetafunction of E and E’ are the same (including the factors at the ramified primes and at co.) Moreover, if these conditions hold, then E and E’ have the same discriminant. If H and H’ are not conjugate in G, then E and E’ are not isomorphic. Hence, by Exercise 6.3, there exist nonisomorphic extensions of Q with the same decomposition laws and same zeta functions. However, such examples do not exist if one of the fields is Galois over Q. Exercise 7: A Lemma on Admissible Maps Let K be a global field, S a finite set of primes of K including the archimedean ones, H a finite abelian group, and cp: Is + H a homomorphism which is admissible in the sense of paragraph 3.7 of the Notes. We will consider “pairs” (L, a) consisting of a finite abelian extension L of K and an injective homomorphism a: G(L/K) + H. EXERCISE 7.1. Show that there exists a pair (L, a) such that L/K is unramified outside S and q(a) = a(F,,,(a)) for all a E Is, where FLIR is as in Section 3 of the Notes. (Use Proposition 4.1 and Theorem 5.1.) EXERCISE 7.2. Show that if q(u) = 1 for all primes tr in a set of density 1 (e.g. for all but a finite number of the primes of degree 1 over Q), then cp is identically 1. (Use the Tschebotarov density theorem and Exercise 7.1.) Consequently, if two admissible maps of ideal groups into the same finite group coincide on a set of primes of density 1, they coincide wherever they are both defined. EXERCISE 7.3. Suppose we are given a pair (L’, a’) such that a’(FrIK(u)) = q(u) for all o in a set of density 1. Show that (L’, a’) has
364
EXERCISES
the same properties as the pair (L, a) constructed inExercise 7.1; in fact, show that if L' and L are contained in a common extension M, then L’ = L and a’ = a. (Clearly we may suppose M/K finite abelian. Let 8, resp. O’, be the canonical projection of G(it4IK) onto G(L/K), resp. G(L'/K). By Exercise 7.2 and Chapter VII, § 3.2 we have a O0 OFMIEL= a' o8' o FM,=. Since a and a' are injective, and FM,, surjective, we conclude Ker 8 = Ker 8’, hence L = L’, and finally a = a’.) Exercise 8: Norms from Nonabelian Extensions Let E/K be a global extension, not necessarily Galois, and let M be the maximal abelian subextension. Prove that NE,, C, = NM,,&,, and note that this result simplifies a bit the proof of the existence theorem, as remarked during the proof of the Lemma in Chapter VII, 3 12. [Let L be a Galois extension of K containing E, with group G, let H be the subgroup corresponding to E, and consider the following commutative diagram (cf. Chapter VII, 3 11.3): 82(H,Z)# Hub2;C,/N,,&, z! l?'(H,C,) car 1 1 car 1 NE/K *1 fi 2(G, Z) # Gab? CK/NLIRCL x fi’(G, Cll). Since G“"/bJ(Ho">z G(M/Q this gives the result.]
Author
Index
Numbers in italics indicate the pages on which the references are listed.
G
A
Gaschiitz, W., 241 Gassmann, F., 362 Godement, R., 51,67,262,265,308,347 Golod, E., S., 232,241, 249 Grothendieck, A., 253,265
Artin, E., 45, 56, 70, 82, 91, 143, 173, 176, 220, 225, 230, 272, 273, 277, 306,347,356
B Bore], A., 262, 263, 264 BoreviE, Z., I., 360 Bourbaki, N., 51, 67, 119, 127, 138, 143, 161 Brauer, R., 160, 225,230 Bruhat, F., 256, 264 Brumer, A., 248, 303, 303
H Hall, M., Jr., 219, 230 Harder, G., 258, 265 HarishChandra, 262,263,264 Hasse, H., 204, 211, 230, 232, 249, 269, 271, 273, 274, 275, 278, 293, 296, 299,303, 351 Hecke, E., 205, 210, 211, 214, 218, 230,306,347 Herbrand, J., 275, 278 Hilbert, D., 267,268,269,270,278,351, 355,356 Hochschild, G., 197
C Cartan, H., 139, 308, 347 Cebotarev, see Tchebotarev Chevalley, C., 180, 205, 206, 230, 250, 252, 253, 264, 265, 275, 276, 277, 306,347
I
D
Iwahori, N., 256,265 Iwasawa, K., 249 Iyanaga, S., 273,278
Dedekind, R., 210,230 Delaunay, B., 267,277 Demazure, M., 253, 265 Deuring, M., 293, 296
K E
Kneser, M., 255,264,265 Koch, H., 249, 302, 303,304 Kronecker, L., 266,267,278
Eilenberg, S., 127, 139 Estermann, T., 213, 230
L
F
Labute, J., 300, 304 Landau, E., 210, 213,214,230 Lang, S., 51, 168, 193, 214, 225, 230, 346 Lubin, J., 146
Franz, W., 273, 279 Frobenius, G., 273, 277 Frohlich, A., 248, 249 Fueter, R., 267, 277 296, Furtwiingler, Ph., 270, 273, 277, 356 365
366
AUTHOR
M Magnus, W., 273,278 Mahler, K., 68 Mars, J. M. G., 262,265 Matchett, M., 306, 347 Matsumoto, H., 256, 265 Minkowski, H., 357 Montgomery, D., 127
N Nakayama, T., 197, 199 Noether, E., 22
0 O’Meara, 0. T., 360 Ono, T., 261, 262,265
R Ramachandra, K., 296,296 Raynaud, M., 161 Rosen, M., 248 Rosenlicht, M., 250, 251;26.5
S SafareviE, I. R., 200, 232, 241, 249, 303,304, 360 Samuel, P., 4’ S&mid, H. d9, .?79 Schmidt, F (I> ?75, 279 Scholz, A., 249, 273 Schumann, H. G., 273,279 Serre, J.P., 41, 87, 89, 90, 168, 187, 198, 249, 253, 255, 257, 265,297, 300, 304, 351
INDEX
Shimura, G., 165, 168, 293,296 Siegel, C. L., 264, 265 Speiser, A., 225, 230, 267, 279 Stalin, J. V., 366 Steenrod, N., 127 Steinberg, R., 255, 265 Swan, R. S., 22, 161
T Takagi, T., 267, 270, 272,279 Taniyama, Y., 168, 293, 296 Tam, J.T., 51,70,143,146,173,176,199 Taussky, O., 249,273,279 Tchebotarev, N., 273,279,355, 363 Titchmarsh, E. C., 211,213, 230 Tits, J., 256, 264
V Vinberg,
E. B., 241
W Wang, Web&, 279. Weil,’ 258, Weiss,
S., 176 H., 210, 230, 266, 267, 268, 296 A., 51, 67, 70, 173, 200, 253, 262, 264,265,296, 346, 347 E., 86, 229, 230
Z Zariski, O., 41 Zippin, L., 127