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(A 1_<;:>)),,-1. But this follows from Theorem q.3 because in the ring or integf'nl in thf' IInld of p-th roots of unity, the factorization of p is given by p = (1 - .,,)p-I t where E ill a unit. Another proof can be found in notes of Serre's COUI'lit' "HOlJloi
(k O : kO" ) in 8. local field k. Since this index is used at the end of ttM! proof IIf TRmma 2, Chapter VI, § 3 we outline the computation here. Let k be Il lIulI·ardlillmdean local field, let U be the group of units in k, and for each intr.gtlr , ~ I, let Vj . . 1 + p' be the group of units congruent to 1 mod pi. We let IIII' thl' "'''P f(z.) 1 for all Z E and 9(X) = ;t". Then
=
k.,
/lO,,.
(kO)' _ (kO : ''0") -
(k~: 1) .
We observe that the denominator is the order of the group in k. Now for any i,
flo,,,(k")
nr n th mots of unity
=Ilo,.. (k" IU)'lO,,,(U/ll.)flo ... (U,) :: qO, .. (Z)qO,n(finite
grOI1P)Ilo.,,(V,)
;;;: nqo,,,(Ui ). H i is large and n prime to the characteristic of k, thlm thr map z - zo.. mapa Ui isomorphically onto Ui+m where m ord~(n), as one IIfM'fI for IIxlUnple by the binomial series for (1 + t)" and (1 + t)l/". Hence
=
qO,,,(Ui) ;;;: (Ui : UHm) = (0 : p)m = (0 : no) where
Inlk denotes the normed a.bsolute value of n W :k
O " ) :::
I~~ (k~
in k.
1/1"1.
nllll!
finally:
: 1)
and in particular, if the n-th roots of unity are c.ont&lnlWi In ~, tlNm
(kO : k O") ::
I:'•. ~
These formulaa ca.o be checked directly in the art'.hinmdl!AI1 c''''', A: - • lUI" II: = C (recall tha.t in the latter case, the normed ab:;oII1t.I· valul' ill tI~ 1M11IQff. "' tIll' urdinary absolute value). They hold formall,. if the chlU'A("tnilltir of k dlvld.. n, bnc-AII8e
then
Inlt = 0 and (k"
: kO")
= 00.
•
PRELIMINARIES
4. Local Class Field Theory We now tWll to loca.l class field theory. Let k be a local field, and n its algebraic closure. We axe to show that the formation (Go/}" n·) is a class formation. If k is archimedean this is completely trivial, so we assume k non-archimedea.n. For any notmallayer KIF we have Hl(GK/F,K*) = 0 by Hilbert's Theorem 90, i.e. our formation is trivially a field formation in the terminology of Chapter XIV, because it is a formation of (multiplicative groups of) fields. Probably the best way to prove the Second Inequality
(F* : NK/FK*) t;; [K : F]
Ie the method of Chapter XI, because the detailed study of the norm mapping car~ ried out there is necessary for the ramification theory, and must be done sometime anyhow. On the other hand, there are short cuts if one wants only the inequality. As explained in Chapter XIV, § 3, one needs only establish our inequality for cyclic
layers of prime degree. Let KIF be cyclic of degree n. Then for any Bubmodule V of finite index in the group of units U of K, we have h 2 / 1(GK/F,W)
= h2/1(GKIF,Z)~fl(GK/F' UIV)h2 / 1 (GKIF, V) = nh2!1(G K / F , V)
and we shall prove this is equal to n by constructing in any normal layer KIF, cyclic or not, a subgroup V such that HT(G, V) = 0 for all r. Indeed, let {oa} be a normal basis for KIF. Replace 0 by 7r i (j where 7r is a prime element in F, and where i is sufficiently large so that if we put
M= ~OF~ ueG 7r0 K • Finally, put V ='1 + M. Then it is easy to that V is an open G-submodule of U, and moreover V is filtered by subgroups
we hil.Ve ]t..(Z C 7r M and M C !lee
~
= 1 + 'lfi M such that for each i, the module
~/V;+1 ~ M/lrM
is G-regular, and hence haJ3 vanishing cohomology. Now we can apply the following elementary lemma whose proof we leave to the reader.
Let A be a complete topological group and G a finite group operating A. Let A = Ao ::> Al ::> A2 :J ... he a decrea&ing sequence of subgroups invariant under G, and which shrink to the identity in the sense that for each neighborhood U of 1 in A, there is an index i lUck that Ai C U. If HT(G, At/AHd = 0 for all i and some r, then Hr(G, A) = O. LEMMA.
continuotlSly
011
(In characteristic 0, one could avoid the preceding construction by taking a sufficiently small neighborhood of 0 in the additive group of K, and mapping it onto a neighborhood of 1 in K* by means of the exponential fWlction.) For cyclic layers KIF of prime degree p different from the characteristic, one can also deduce hZ{l(GKjF, K*) = P from Theorem q.4 above, using the equations p . p /}(J,p(K·) = Ipl" and qo,p(F*) = lPl"
4. LOCAL CLASS FIELD THlOkY
•
obtained in our computation of the power index (XO : KO,) aWvt!. Finally, one could ignore the second inequality ('Olllplpl,·ry hy provin,; directly that every 2-dimensional class has an Wlramified splitt.ing fif'hl, or ""hAt is the same, that the Brauer group of the maximal unramififlcl I'xtmll,ifln of k III trivial (cf. for example [15)). To complete the proof that our formation of mult.iplirl\livr gmuJlII of local fields is a class formation, we must establish Axiom II' of ChApt.pr XIV. For this, we consider the unramified extension KIF of dej!;r«:c~ n. Sinn' tllC' FI'lIiduc dass field is finite, the Galois group G K/ F is cyclic, with a ca..nollica.l K,'ul!ralor, lhe Ftobenius automorphism q; = 'PKIF' For any normal layer K / F, unramified or not, the ex",,:l lI«'Iquence 0-+ UK
-+
KO
-+
Z -+ 0
yields, on passage to cohomology, po = K oG -+ Z
-+
HI(GK/P,UK) ..... H1(GK/I',K·) - 0,
from which we see that HI (G K/ F, UK) is isomorp hie 1.0 the c:olwrncl of FO ..... Z, i.e. is cyclic of order equal to the ramification index r.K/F, b('('aullt' Z hl!re repreoents the value group of K. Thus, for our unramified K/ F, M: haY\' II I (G K IF, UK) = O. On the other hand, we have HO(GK1F,l/K) :: lhlNI\/~IJK - 0 allio. This follows in various ways: either a direct refinement p1'OCI~ "h()win~ that every unit in F is a norm of a unit in K, or from the fact that IIll I (/I,..) . I. 1\Ii Wlili shown in course of proving hZ/l(K*) = n above. Thllll for IInramitll1d I
H 2 (G K IP, K*)
- / G K F
:::::
1 -Z/Z. n
Composing these two we obtain an injection
invKIP: H 2(G K/ P ,KO) - Q/Z. For any a E F", and X E G, the class HO U ~x E 1/2((;K"" J(.) III repnwmtlld by the 2-cocycle aDx(O'.r). Taking ordinals, i.e. applyiD& tbe lDap XO - i Z, we get the 2-oocycle (ordK a)6X(u, T) which represents the cl~ ordK aU6X 6(xo,d,.. ") ill H 2(OK",Z). TIIIJII M have
=
(**)
invK/F(xa U OX) = Xo'd K "(!p) . X(1pord, .)
since ordF a = ordK a, our ex:tecsion being wlrluililicd. TbiI allUWI ,baL for unramified extension K/F, we will have
(G, K/F)
all
",II.·"
:: q;1\/~'
(In some classical texts, tbe oppOIIite sign ifl r.hOltlln.) F'lIrthMmnnl, IIII1t'l1'! OK'1" Is cyclic, every 2-dimemdnnal d"'~N i~ of the f(lrm Hfl U c\\ 1\1111 wr f'All 111M' t.lu~ rule (... ) to (ltjtablillh the requin,'(j prolJCrtiCli of iIlY,./". Jo'ilHtur all, lhft IIIVlU'ilUlt docs not change under inHation to a biggl'r unr8J11ifitlCI t!XlfOlllliun 1./10' wit.h L .) K :> F,
PRELIMINARIES
10
because 'PK/F is the image of 'PL/F under the canonical map C LIF ...... GK/F, and inflation of xa U oX amounts to viewing a character X of C K / F , as a character of GLI F by this same canonical map. Hence
fI2(./F)
=
U
H2(CK/F,KO)
KIF uoramified
the subgroup of the Brauer group H2 (*/ F) consisting of the elements coming from unrarnified layers. We obtain an isomorphism invp: fl2(./F) ..... Q/Z (surjectivity because there exist unramified extension of arbitrary degree). To complete the proof of Axiom II', we must show that the invariant multiplies by the degree IE: F] under restriction from F to E. This follows from (**) when one takes into account that ordE :: e OrdF, where e is the ramification index, and that, under the canonical map C KEjE .... CKIF the image of IfIKE/E is 'P~/F where I is the residue class degree. Hence the invariant multiplies by ef = [E: FJ. This just about completes our introductory comments. Concerning the existence theorem, we have given in Chapter XIV, §6 an abstract discussion which shows that the existence theorem follows in abstracto from Axioms lIla-lITe. In both global and local class field theory, these axioms are all trivial to verify except b IIId. The proof of this axiom in the global case is carried out in Chapter VI, §5. In the local case, it is not covered in these notes, but would follow readily from the theory of the norm residue symbol in Kummer fields. Chapter XIII and Chapter XV are not needed for the remaining parts, but note that there is a proof of the principal ideal theorem in Chapter XIIi We hope that the preceding remarks will to some extent reduce the inconvenience which the reader will suffer from the missing portions of the notes, and other imperfections occurring in them.
CHAPTER V
The First Fundamental Inequality 1. Statement of the First Inequality In this entire chapter, k is a globallield and K/k a cyclie cxtCRlliOIl or degree n with Galois Group G. We let J = JK be the idetCli of K, n.nd C = CK be the idhle classes of K. Then G acts on J and 0, and the fixed groupe are JO -" Jk, C G =Ok. We let hi and h z denote the orders of the first and soomrl cohnmology groups. ~/l a.bbreviates h2 /h 1 • We wish to determine the order '12 (G'. C) of 'H,2(G, C), and it will be shown in this chapter that h2(G, C) ~ n. In fo.ct, we prove THEOREM 1.
Let k be a global field and let K / k be a qclie ~'c""ion oJ degree n
with group G. Then
or in other fDonLs, h2/ 1 (G,.CK)
= n.
To simplify the notation we omit G and write h.(C) Ill1Iiend ur h,(G, C) whenever G is the group of operators. We shall prove this inequality first in function fields, bcc-AUHC (lOn..dderable simplifications occur in this special case. Afterward:;, we Hhall Kive .. ullifi,~1 proof for all global fields. We shall make constant use of the pmpertieN of thf! Inrln "'all tlf!vcloped on pages ~7 (Section 3 of "Preliminaries"), and recall here the Lhree RlOit important properties for the convenience of the reader. PROPEIrrY 1. The index ~/l is multipli('.ative. In othr.r wont., If A is aD a.belian group on which G acts, and Ao is a subgroup Invarlllilt undor G we have
~/l(A)
== ~/l(A/Ao)~/I(Ao), in the sense that if two of these quotients are finitfl, then II) II ,he third, IIld the relation holds. PROPERTY
2. H Ao is a finite group, then hall (Au) • I, and hem'e hall (A)
=
"-2/1 (AI Ao). PROPERTY
3. If A s;:s Z is infinite cyclic Md r: Optll'IltM trl"'ally, t.hf"n ~/I (Z) ~
n is the order of C.
2. First Inequality In Funrtlnn FINet.
,f"
We rmppose here thFLI. k is " fundiOIl field W" \col U - II" be the unit ideles of K, and Jo ;;: ~ the itlclt:s of volutlll'l I of 1(. 1.11. t~ IIWlIfII 0 IUch that II
u
V. THE FIRST FUNDAMENTAL INEQUALITY
n!ll/I!I'.J1 = 1.
Then Jo J U obviously, and Jo J K" by the product formula.. Hence
J o :J UK·, The muitiplicativity of h2/1 gives
and it will come out that all three quotients on the right are finite.
To begin with, J I Jo is G-isomorphic to the additive group of integers Z with
trivial action under G, via the degree map. Hence
Since the number of divisor classes of degree zero is finite, Jo/U K· is a finite STouP. Hence
1VJ./l (Jo/U K·)
= 1.
The factor group UK· /K" is G-isomorphic to u/(Un KO) and hence
h2 / 1 (UK" /K*)
= h 2/ 1(U/(U n K·» = h2/ 1 (U)(h2/ 1(U n KO)r 1,
Here we use the multiplicativity in reverse, and it will be shown that both ~/l(U) and ~/l (U n K*) are 1. We know that U n K* == is the multiplicative group of the constant field of K, and is finite. Hence h2/1 (U n K*) = 1. We contend finally that h2/I(U) = 1. Indeed, we can express U as a direct pr.oduct,
Ko
U=
IIp (11 U!lI) 'PIp
where each component 11'Plp U'P is semilocal, and invariant under G. For each p let U, be one of the groups Uq!, and let G p be the local group, leaving Up invariant. The 13emilocal theory states that 1(' (G,
II U'P) ~ 1f'(G
p, Up)
!PIp
and we have 1nU);:::
II 'Hr(G
p , Up).
p
We know from the local class field theory that
h1(G p, Up} ::;: h2 (Gp , Up) =
=
e,
where e p hi the ramification index. But ep 1 at almost all p. This shows that h 2 (U) and hI (U) are both equal to ep and therefore that h2/1 (U) = 1, as was to be shown. If we piece together the information just derived, we get the desired result:
11,
3. FIRST INEQUALITY IN GLOBA.L PmLos
13
3. First Inequality in Global Fields , We treat now the two cases simulto.neoWily. The (:xi~tfll1ce of archimedeo.n primes prevents us from giving the same proof for number fields thlLt was given for function fields in the preceding section. Using Ha.a.r me8liUrI', and a generalized Herbrand quotient for Haar measure, one could indeed giw. all argnment in number fields which parallels completely that of function fields. Sill<;e we wish to avoid the use of Haar measure, we give below a modified version of our preceding proof. We let k be a global field and S a finite non-empty set of primps (If k including all archimedean primes. The subgroups K', Js, KO Js of.J art' mvariant under G. (Recall that S also stands for the set of all the primes of K dividing those of k which are in S.) We have therefore
h2 / 1 (J/W) = ~/l(J!JsK·)h'/./l(J8K· /K"),
and it will be shown that both indices
00 the right are finite. From the finiteness of class number theorem we know that J/JsKo is a finite group, and consequently hZ/l (J/JsKO) 1. JsK* /K* is G-isomorphlc to JsjJs n K' Js/K.~. Consequently
=
h-J/l(J/KO)
=
= h2/l(Js)(h 2/ 1(K;W 1,
and it will be shown that both quotients on the right are finite. We can write
Js ==
(IT U~) x (11 K~) ~ES
q:lIlS
where each factor is invariant under G. By an argument llimila.r to that used in function fields, we have h2 / 1 (I1op~s Uop)
= 1.
We can decompose the finite product
IT K.p = n(IT 'Gr).
!pES
pes q:llp
Let K; be anyone of the fields K",. and let G, be its local grDUp. The aemilocaI theory
ShoWll
h2/1
that
(G, II ~eS
K~) = 11 h2/1 (0, pES
nK~) = IT
~Ip
"2/1 (G,. K;).
PES
From the local theory we know that h2 (G,. K;) = np is thf! InrAl df!gIa, and h1(Gp,K;) = 1 is trivial. Hence
ha/l(JS) =
nnp.
pES
We shall therefore have completed the proof of the finJt inequality if ... 1IKlCIeed in proving the following LEMMA. Let S be a fini.te 8et 0/ prime. 0/ k including all arrhimMMn primes and let K/k be a cyclic extension 0/ degree n unth group G. Tlv.n
~/l(Ks) = (11 np )/n pes
where n, (., the local degree [K" : k,).
14
V. THE FIRST FUNDAMENTAL INEQUALITY
PROOF. 1 Let s be the number of primes !p of the set S in K, and let JR' be the additive group of Euclidean 8-Space. Let {X'.Il}'.Iles be a basis of IRs and map the S-ideles of K into IR! as follows:
Cl -+ L(Cl)
=L
log IClI'.llX\ll. '.IlES This map is an algebraic homomorphism of Js into R'. We shall make it a Ghomomorphism by defining a suitable action of G on R". Namely we let X~ =X'U"
q E
G,
and extend q to R! by linearity_ Then G acts on IR", permutes the vectors X'.Il but JlOt necessarily transitively. We recall that la" l\}la = lal'.ll and using these facts we have L(u"} = log la"I'.IlXlP '.IlES
L
=
L log la"llPaX'.Il
o
lPes
= L log lal'.llX'.Il" '.IlES = L{a)". This proves that L is a G-homomorphism of Js into R'. The image L(Ks) is a lattice of dimension II - 1 in R', according to the Unit Theorem (see [16, p. 100J or [5, proof of theorem, p. 72J). This lattice is contained in the hyperplane R,-l of all elements £x
h:m(L(Ks» = h2/1(Ks) because "-2/1 of a finite group is 1. We now face the task of determining h 2/1(L(Ks»). We first extend the 8-1 dimensional lattice L(Ks) to an s-dimensionallattice M as follows. We let X = I:'.Il X'.Il' Then the vector X does not lie in the hyperplanelR8 - 1 spanned by L(Ks). We let M be the lattice generated by L(Ks) and by X, i.e. M:;;; L(Ks) + ZX
(Z are the integers).
Then M is a-dimensional, and spans R'. Since X" = X for all q E G, M is invariant under G and both L(Ks) and ZX are G-modules. The module ZX is
o
lThe key idea in this proof ill that for a fitrite cyclic group G and a finitely generated GJnodule M, the Herbrand quotient h2 / 1 (G, M) i. defined and depends only on the lR{GJ-module M ®l. R (_, for example, [5, Cb. IV, Sect. 8, Prop. 12j). In fact, if G is of order 11, genera.ted by v. and the characteristic polynomial of q acting on the vector space M ®1R is (x} = (x-1)"oI«x) with "'(1) '" O. then n" 1i:J(1 (G, M) = W{l)" For example, if 4>(x) = z" -1, then h:1./1 = n/r.
"
3. FIRST INEQUALITY IN GLOBAL FIELDS
G-isomorphic to Z with trivial action. Hence h2/l (ZX) =~. Furthermore, the above sum is c1elllly direct, and therefore ~/l(M) = h;z{1(L(Ks»' n.
The proof of our lemma will therefore be complete if we prove that
~/l(M)
= 11 np. ~es
For this purpose, we prove the following proposition. PROPOSITION. Given any s-dimensionallattice M in our R" that iI invariant under G, there eri3ts a su/}lattice oj finite index M which i3 InTJanant under G, and generated by basi3 elements Y!p (l:I) E S) on which the actlCJn vI G .., fIIfJen by ~
YJ =
y~..
(I
E G.
Before proving the proposition, we show how our IAmmA follows from it. Suppose we have found a sublattice M of M with the action de~(:rilll!d in the proposition. For each prime p, let Mp be the sublattice of it KI:III:raIA~1 hy /1.11 nlcments Y!p, \P I p. (In other words, break up Ai into transitivity dolUWI1lI UUUAr G.) Then each Mp is invariant under G and
..
M=EM, ill a direct sum. FUrthermore, e&::h pair (G,
Ai,> ill ..wlocllol.
By the eemllocaI
theory, we ha.ve for each
~/l(G,Mp>, = ~/1(Gqs,ZY,,)
Go,p acts trivially on the infinite cyclic group generated by Yo,p. COll8eQuently hz/ 1 (Mp) = np and ~/l(M) = Ilnp. Since if i:; of fiwl.e index 111 M, i~ foll~5
and
that
o
This proves our lemma.. PROOF OF PROPOSITION. For convenience
we define
IExo,pxo,p1 = max ~ o,p Ix., I
&
DOrm on R· by putting
z" £= R.
This norm is clearly invariant under the effect of q E G. Let b be a constant Buch that for any vector A (, R" there utll Y E M such that IA - Y! < b. Such a b exists since M is .9·dirnl'OIlLonru. For each prime PES let ji be one of the J.lcilUt:II \p I p. tAil Y, ( M be such
tba.t (1) Let Y., =
IY, -bX,1 < b.
1:pg=o,p .. eO Yf.
We contend that the
desired kind. We tint verify that
Wl.1.0TII
Y" poerate & IItlblattice of the
Ghall the proper efl',,'(:t.. Ltl. Y" ~ Y... Y,,;:: E Yt' == ,-",o,p
Indnod
u
V. TIm FIRST FUNDAMENTAL INEQUALITY
u was to be shown. All that remains to be done is to show that the vectors Y., are linearly independent. Given a relation
LX'IlY"=O
x.,
we shall prove that all = O. We note that the number of 0" E G such that p" = ql is exactly the local degree ",. Because of (1) there exist vectOrs Bp such that
Yp
= bXp +Bp
where IBp 1 < b. From the definition of Y., we get
L: bX; + jig='.P L~ = L: bX.,+C., jiO='.P
Y'.P =
jia=.,
=npbX'.P+C'.P 'where C'.P is a vector such that
(2)
0=
IC.,I < npb.
Substituting in the relation yields
L x'.PnpbX., + L x.,C1l. 'II
•
Let (C.,X'.P) be the component of C'.P along X'.l3' Since the vectors X\"jl are independent, the component of each X\"jl in (2) must be O. It is
X., [npb + {C1l,X'.P>] anil the inequality 1C1l1 < n"b shows that npb + (C". X.,) :/:., = 0, and the Y., are linearly independent.
cannot be O. Hence
This concludes the proof of the first inequality.
0
4. Consequences of the First Inequality We note that in Theorem 1, we have proved more than an inequality. Namely, we have actually proved that h2 (G, C) is equal to n· hl(G, C). This win be used at & later stage of the theory. However, at this point we shall only give applications of the inequality itself. Since G is cyclic, we have
h2 (G,C) = (CO: NO)
=(GoI:: NK//CCK)
or in terms of ideJes,
h2 (G,C)
= (J,,: k*NK/"h).
'Ib simplify the notation, we abbreviate NK/k by N, and locally write N'.P for
NKll / k ,. The first inequal1ty implies therefore that the nOrm index (J" : kN J K) of a cyclic extension is greater or equal to its degree. In particular, if this index ls I, then the cyclic extension is trivial, a.nd [K : kJ = 1 also. This remark is used to prove all important consequeIlce of the first inequality.
t. CONSEQUENCP,s OP THE FIRST JNeQUAI.ITY
1'1'
THEOREM 2. Let K be a normal extension 0/ a global field k. 1/ K '" k then there exi8ts infinitely many primes II of k which do not ..plit r:omple!ely in K. Phrased another way: If K'.j} = kp for all but a finite "um&t:r of primes P. then K=k.
PROOF. First let K/k be cyclic. Suppose K., shall prove that any idele a E Jk can be written 11
= o.N"K
11K E
= k, at almoet ail primes.
We
JK.
Let S be the set of primes where Kll '" kp • We eM find () E Ie suC'.h that 0.- 1" is so close to I at all PES that o.-lllp is a localnurm: U-1Qp = N<prt
t two monomials, this proves our lemma. 0
e-
Xl. HIGHER RAMIFICATION THEORY
IN
Let K/k be normal of degree I, and complete.ly ramified: p be the different. Then S(\ll~) = pr where r :=; [(n: + s)/lj.
LEMMA 9.
Let f)
= \lIm
PROOF.
= p~.
Let S(\J}')
= ~.
We must determine r. We have
p-"S(lPS )
= S(p-rlPs) = S(lP
8-
cD.
lr )
By the definition of the different, we see that \lIB-if' c f)-I and 1) C \lIlr-,. This me8llS that m ~ ir - 8, or equivalently, r ~ (m + s}/I.
Conversely, we have S(~)
rt pr+l => p-(r+1) S(l3') ¢. D ~
rt D rt 11- 1
S(1,l"-l(r+l»
~ ~-t(~+I}
rt
~ f)
~(r+l)-B
+ l)i - 8 > m + 1 > (m + 8)/1..
~ (r ~r
This yields r
o
= [(m + s)/t] as desired.
We are now in a poIIition to investigate the cyclic extensions of prime degree. Let K/k be cyclic of prime degree l, with group G. Since G has no subgroups except itself and 1, the pa.ttern of the ramification groups of K/k will be:
= Vo = ... = \It lit+l = lit+2 = ... = I,
G = V~1 where t is an integer
~
-1. The order of the different is
OrdK
iJ
=L
(#v. -1) = (t + 1)(£ -1),
i=O
according to Theorem 1. K/k is unramified if and only if t and unramifted cases are as follows:
:=;
-1. The graphs of ",(x) in the ramified
Unramified
Ramified
,p(x)
",(x)
slope t slope 1 -------~r-------~
-------,I'-----x
1) = (t,t) t/I(x)
= x for x ~-1
t/I(x)
= {:+I(X-t)
•. GENERAL LOCAL CLASS FIELD THEORY
THEOREM 8. Let K/k be cyclic oj prime degree
t. For an integer i
~ -1
VIe
have 1. NK"'(i) C k i and NK"'(i)+l C kt+l' I iIi I-t 2. (k; : k;+1 NK,,(i» ~ { f. .. ill = t where t is the integer such that Vi. I- Vi.+1' PROOF. We distinguish several cases. i = -1. Then.p( -1) = -1. The inclusion statements are obvious, and so is the index which = 1 if K/k is 1'Imlified (t ¥ -1) and = f. if K/k is unramified (t = -I). i = O. Then !/J(O) = O. Obviously, NKo C /ro. We also have NKl C kl and we eee that the inclusion statements are obvious. If t = -1 (K/k unramified) then every unit is a D01'IIl by Lemma 1, and
(kc : k1NKo) = 1Let - denote .residue clasaes (mod ql). We have
(ko: kl NKo) = (ke,: k1NKo)(kl : ~ nklNKo) = (k: klNKo) ~ (k: kt).
If t = 0 then K/k is tamely ramified and I ¥ p. By Lemma 5 the index is Et t. If t > 0 then K/k is strongly ramified and l p. Since k is perfect, the index is 1. This settles the case i O. We shall now treat the cases where'i ~ 1. We distinguish several values of t, and use Lemma 9 constantly, m (t + l)(e -1). t -1. We have ¢(i) i. Therefore SqltP(i) pi. Also, we see that ~~(i) and ~(i)+1 are contained in pi+l We have trivia.Ily N\j:rI'(i) = pti C p'+l. Hence by Lemma 8 we have
=
=
=
=
=
=
N(l + ql",(i) == 1 +pi
(mod f)i+l)
which gives NK"'(i) . ki+1 = k., as was to be shown. t ~ 0 and i ~ t. We have ¢(i) t + l(i - t) = -(t - l)t + ii. Therefore "'(i) + (t + 1)(l -1) ti + (£ - 1). From this and Lemma 9 it follows that
=
=
SlP",(i)
Furthermore,
SlP211>(i)
= p'
and
SlPI/;(il+l
= pHl.
C pi+!. Since t/J(i) ~ i, we have N!,JrI'(i) C p'
and
Nql",(i)+1 C
pHI,
The inclusion statements follows from these remarks and Lemma 8. To get the index we consider two cases. i > t. Then 1j>(i) > i. and NlIl'p(i) C pi Hence
N(l
+ ~(i»
=: 1 + pi
(mod pHl)
and "'+tNK",(.) = ko. i = t. We have already considered i = 0, 60 that we may assume i = t > O. We are in strong ramification. We have ,p(i) = i = t. By the inclusion statements, there exists an element Q E qlt, Q ;. lilt+! such that Sa ¢ plH and Sa = brr t ,
Xl. HIGHER RAMlFlCATION THEORY
b ¢ p. Then No == 71'(a, where a t 0 (mod pl. For any integer x of k we have by LemmaS: N(l
+ xo) == 1 + xSa + 7;1 No (mod pHI) E 1 + '1I't{axl + bx) (mod pHI)
where ab ¢ 0 (mod p). We have considered the polynomial I(x)' = axl + bx in Lemma. 6, and found the index (k : f(k» ~ p. Multiplicatively, this gives (k t : kt+1NKd ~ p, as was to be shown. t~Oandi
< t. Therefore ,p(i} + (t + 1)(£ - 1) = i + (t + 1)(£ - 1) > i+(i+l)(t-l) = li+l-l. This proves that !./J{i)+(t+l)(l'-I) ~ l(i+l). Hence by Lemma 9, SI.P"Ci) C pHI, and a fortiori, ~(i)+l and S~2"'(i) C pHI. For the norm, we are in the ramified case, and therefore N!p"'(i) pi, and N':P"'(i)+1 pHI. We have !./J(i) == i
=
=
Using Lemma 8 proves the inclusion l;tatements. Furthermore, N(l
+ \13>/1(;» = 1 + p'
and this me8.DS multiplicatively ~+lNK1/J(i) == our theorem.
(mod pi+1) ~,
thereby completing the proof of 0
Having settled the case of a cyclic extension of prime degree, we can treat the general normal extension by showing that the statements we wish to make are transitive. We recall the notation: ~(x) and tbHx) are the right and left derivatives of 1jI(x). We denote by \f.>~/f(x) the q?otient 1jI~(:t)N~(x). THEOREM 9. ~ -1
an integer i
Let k be a general local field, and Klk a normal extension. For we have:
1. NK",(i) C k., and NK.p(i)+1 C 2. (k;: ki+1N K",(i») ~ "'~/l(i).
ks-tl'
PROOF. The two statements have been proved in Theorem 8 in a cyclic extension of prime degree. We know that Kjk is solvable, and it suffices therefore to prove that the two statements are transitive. This transitivity is essentially a trivial consequence of the transitivity of the nonn, and of the 'I/J function (proved in Theorem 7). Let K ::> E ::> k be two normal. extensiOllS and assume the theorem for K IE and Elk. We have
By assumption, NE¢(i) C '"
NE~(j)+1
C
kHI
and NK;j;(~(i» C and
E;ji(i)'
NK~(~(i»+l c E~(i}+l'
The first statement is DOW trivial, because NK,,(i)
= NNK",(.) C NEiJ(i}'
To prove the second statement, we write
(/0.;: ki+INKtjJ(j» = (k.;: "'+lNE~(i»)(k.;+1NE~('l : ~+INNKJ(~(i)))'
4. GENERAL LOCAL CLASS FIELD THEORY
We can insert the group E~(i)+l in the last index because of the inclusion N E~(i)+l C kHl whim we have just proved:
I\;+lN[E,ji(i)+1NK~(~(i»] ... l\;+lNNK~(..p(i}l· Hence our index becomes by induction
Et if;~/l(i)(Eifo(i) : E~(i)+lNK~(~(i))) ~ 1fr~/l(i)~/t(if;(i)) = 1/J~/l(i). This final :3tep follows from the chain rule for right and left differentiation which bolds because our functions are strictly monotone. 0
If we take i very large in the preceding theorem, we get ~/t(i) =< 1. This means that for all sufficiently large i, (I\; : k;+lNKtfJ(i)}
=1
and from this index we see that every unit of ki iB congruent to a norm from KtfJ{i} mod PHI' Such a Wlit can therefore be refined to a norm by all obvious argument. COROLLARY. Let k be a general local field. Let K/k be a normal extension. Then k; C N K for surne integer i.
The conductor of K/k iB the least power p" of P such that k" c NK, and we see that the conductor exists. We denote it by f K/,,, or briefly by f if the reference to the field is clear. We may now write the norm index as '" product:
(k: NK) = (k: koNK)(koNK: kiNK) .. . (k._lNK : k.NK). If IS is big enough, k. C N K. Hence DO
(1)
(k: NK)
= II(kjNK: ki+lNK). -1
Furthermore, by an elementary isomorphism theorem, we have (2) (koNK: I\;+1NK)(~ n NK: ~+lNK",(O n NK) = (I\; : ki+lNK1/>(i»
:;;;; tP~/£(i) (by Theorem 9).
From (1) and (2) we obto.in (3)
(k : NK) ~
co
IT t/!~/t(i) Et n -1
where n ;;: [K : k]. Indeed 'I/I~(i) ~ ~(i + 1) whence l&~)l) ~ 1. t/!H-l) = 1/1, and .,p~(oo) = e. Hence the product is"; ef = n.
FUrthennore
The index inequality of (3) is the second inequality. The claBS field theory in general local fields may now be developed in exactly the same way as the theory in the classical case. k; a consequence of the second inequality, we can prove that hz(K/k) ~ [K : kj. and then use the same method as that of Chapter XIV. Instead of the FrobeniU::l Substitution, we select in the Galois group of the algebraic closure of k an element it whim operatea non trivially on every finite extension of
XI. HIGHER RAMlFICATION THEORY
98
k. The effect of it on each finite extension is to generate the. Galois group, which is cyclic. The automorphism (j has a corresponding a.utomorphism a in the Galois group of the maximal unramified extension of k, which can play the same role as the Fl'obenius Substitution. It has not been canonically selected, but this does not make any difference to the proofs, as long as we develop only a local theory. It is only in the global theory, when the local theories were pieced together, that it became important to choose the proper automorphism in the local fields. Having done local class field theory, we obtain all the results of Chapter XIV. In particular: (0
THEOREM 10. Let k be a general local field. Let n be its algebraic closure, and the Galois group. Then (~,n·) is a class formation.
We may now return to the indices computed to prove the second inequality. We know that in abelian extensions the norm index is equal to the degree. Thls means that the doubtful indices of Theorem 8, 9 and (2), (3) above for abelian fields are no more doubtful, but are actually equal. THEOREM 11. Let k be a general local field, and let K/k be a finite abelian extension. 1. ~ ("I. NK = k'+lNK.,i) n NK c ki+1NK",(i). 2. (ki : k'+1N K,p(j») == (kiNK: ki+lNK) == ""~/t(i). 3. All breaks in'l/.>(x) occur at integral arguments. In other worr.b, Vf../l(x) "" 1 if x is not an integer. 4. kz C NK if and only iIV~(=.V,p{:t» ::: 1.
PROOF. Statements 1 and 2 arise from the equality in (2) and (3) above. 'Th prove 3 we note that yp(x) has a finite nUIIlber of breaks. If we take the product n'l/.>;/l(x) over the numbers :r: for whlch there is a break, we get ef = n. But ~ ¢~/l(x) is > 1. Since the product over integers already yields n, this means that the integers give all the breaks. Using the equality of the indices in 2 we have:
k.. C NK
~ (k j : kHiN K,p(j» '= 1 for all j <=> l/l~/l(j) = 1 for all j ~ 2:
~ 2:
<=> 'I/.>~(x) = ypHx) = yp'(oo) = e <=> (#Vo : 4W",{o:)} = (#Vo : #V".,) <=> V,p(z) = V"" = 1 <=> VO: = 1
o
thereby proving 4.
THEOREM 12. Let k be a general local field, and w the norm residue mapping, into the Galois group 1!5 of the maximal abelian utension of k. Then w(k.,) is everywhere dense in !?)x.
PROOF.
Let K/k be a finite abelian extension. K is left fixed by
=1 <=> k., c NK <-> w(k",) leaves K fixed
!l)'" <=> Vi/I<
S. THE CONDUCTOR
o
and this proves the theorem. COROLLARY.
Let K/k be afinite abelian eztension of IJ gtmemllolXl.lfield. Then
w(A;) :::; Vi. 5. The Conductor Let k be complete under a discrete valuation with perfect re;idue class field k. We shall investigate the conductor of a normal extension of k attached to a character of its Galois group, and defined formally in terms of the ramification groups. Most ()f our discussion will be valid in the field k, and it will be explicitly stated when k is to be specialized to a general local field. Let K/k be normal with group G. We assume known the representation theory of G over the complex numbers. Let f «(7) be a function of G into the complex. Let H be a subgroup of G. Then I is also a function on H by restriction. We let
k
I(a) cW ==
E~~(u) .
The integral is an average over H. Let X be a character of G. Its re;trlction to H is a character of H. On H, we can express X as a sum of irreducible characters:
X::: L;PtXi. We call IJ.i the multiplicity of X, in X. It follows from representation theory
that LX(U)=p.#H H
where IJ is the multiplicity of 1 in X. (We refer to the irreducible character X(u)
for all u as 1.) We have
=1
L
x(a) dtT = IJ
and the integral is therefore an integer ~ O. A character X is linear on the group ring. We have
L
X(l- u)du == dim X -I'
IH
where IJ is again the multiplicity of 1 in X. From this we see that X(l- 0') dO' is an integer ~ O. Given a character X on G, we define a real number veX) as follows:
fI(X) =
17 (JV'
X(1 -IT)
dtT) cit.
Then yt = I for large t, and X(1 - 1) = O. Hence Iv' X(I - a)du = 0 for large t. The irmer integral is always ~ 0 according to the preceding remarks, and therefore II(X) is a well defined non-negatlve real number. We define formally the conductor belonging to this character to be p"(xl = fx' The function I(t) = x(l-u) dcr is easily seen to be a step function. Suppose that G is abelian and that X is an ordinary character of G (Le. a homomorphism into the unit circle), X oF 1. It follows from repte5entation theory that 2:v. X(u) "" 0 if
Iv.
Xl. HIGHER RAMlFlCATION THEORY
100
and only if X is nontrivial on V t • Under these circumstances, f( t) grapb. of f(t) is 88 fonows:
:
-1
= 1.
Hence the
Xo
Let Xo = inf{x: f(x) = a}. Then we see that vex) ==:1:0 + 1. Furthermore, X is not trivial on V"O but is trivial on ~o+6 for fJ > O. Suppose in particular that k is a general local field. By definition,,p(xo) must have a break. According to Theorem 11 of section 3 the breaks in ,p(x} occur only at integers in the abelian extensions of genera.llocal fields. This proves THEOREM 13. Let
k be a generallorol field and K/k an abelian extension. Let = 0 trivially.)
X be a character of G. Then II(X) is an integer. (If X == 1 then II(X)
Let w be the mapping of the norm residue symbol. Then X . w is a homoIl'iorphism of k into the unit circle. From the existence of a conductor for K/k (Theorem 9, Cor.) we see that X· w is continuous in the class topology on k. To get precise information we use the Corollary to Theorem 12. We know that w("=i) = Vi. Combining this with the preceding dL'lcussion we obtain THEOREM 14. Let k be a general Weal field and K/k an abelian extension. Then II(X) is the least integer i 81J.ch that X . w("') = 1. COROLLARY.
Let f be the conductor of K/k. Then
f:::= l. c. m. h. We see that the formal conductors obtained from characters of the Galois group are the same as the conductors arising in the class field theory. Having shown tha.t the conductors coincide when cla.s5 field theory holds, we shall return to the abstract case, and investigate the formal properties of v(X). We assume therefore tha.t the residue class field of k is perfect. THEOREM 15. Let
K/k be normal and let Xl, X2 be two charncter8
0/ G.
Then
II(X1 + X2) ~ vex)~ + II(X2). PROOF. Immedia.te from the definition of v(X), because characters are additive.
o THEOREM 16. Let K ::> E::> k be two normal extensions. Let X be a character of GI H = G, so that X may be viewed as a character X of G also. Then
II(X) :::= II(X). PROOF. Let = Uc'7(vt n H). We know that Vt =: VtH/H, and the it are na.turally elements of V t , and X(u) = X(u'Y) for all 'Y E H. It suffices therefore to show that
Vt
f X(l-u)do'= ~ X(l-u)dO'. lv· lv·
5. THE CONDUCTOR'
Note that #vt == #Vt· #(yt
l"X(l-O')da=
101
n H), and hence
EVi;~t-O') = #(~~H)~X(1_iT)= lv_.:W-iT)da.
0
v' THEOREM
17. v(x)
PROOF. Since
yt =
= ~E~oEv.x(I-O'). we have
V"'(t),
v(X) = .
1
00 ((
Jv,,'t)
-1
X(I - 0')00) dt.
The function f(t) "" J,v..".) X(l- 0)00 has only a finite number of discontinuities, and we may take the sum of the integrals in the intervals where no discontinuity occurs. Let 8 = ¢(t) and t = ,(8). Then dt q/(8)d8 and by the chain rule:
=
=
17 (!v.
=!
(00
e J-l 1 00
X(I - 0')00)
:~ ds
(2: X(I -(1») ds V.
= - LLX(l-O') e 0=0 V. because
Vi+
= Vi+! for 0 < 0 < 1, &ncl
l'
'+1 (
2: X(I - (1) )
ds
= 2: X(l -
(1).
o
v,+l
V.
The formula giving veX) in the preceding theorem might have been taken as a definition. We have selected the integral because the important Theorems 13, 14, 15, and 16 were immediate consequences of our definition. THEOREM 18. Let K/k be norma.l with group
G. Let H be a subgroup, and E
the fixed field. Let ¢ be a character of H, and X the induced character of G. Then
PROOF.
h: = D~W NE/kf",. (D = discriminant.) Let G = UHe. The induced character X has the value X(T) =
L¢(crc- 1 ) c
where ¢(11) = 0 if 11 ¢ H. Starting with the expression derived for veX} in Theorem 17 we have therefore:
XI. HIGHER RAMIFICATION THEORY
102
But ~f-l =
Vi because Vi is normal. We can therefore take ~ut the c- 1: =
1
00
1
00
eLLLtP(l-o) o v."
.
= eLLL 1/;(l-o) o
c
V;
"" = -1 L(efLW(l-o»). e
Since e =
ee and 1/;(a} = 0 unless 0 E Vi n H,
= ! f: (#ViW(I) e
But
v,
0
V. n H =~.
1
#(Vi n H)1JJ(I) +
0
L
,p(1 -
a»).
V;nH
We may add 1 and subtract. 1, thereby giving
1
00
eo
LL ,p(1-0').
1""
00
"" -;- L(#Vi -1)1JJ(1} + -;- ~)#v. -1)1/1(1) + f-;eo
e O j i,
Using Theorem 1 fur the order of the different, and Theorem 17 giving the expression for 11(1/;):
= !1f;(l)[ord K !J -ordKD] + jlJ('I/J). e =
But ordK!J - ordK'; = ord K 3. Also, lordK ordE. Furthermore, ordkNe/k = jordE . Combining these remarks, and writing the formula multiplica.tively, we have
fz: = p"(x) where Delle
= D~WNEI"f",
= N E /,;;; is the discriminant.
This proves our theorem.
0
=
Ilf'e
COROLLARY 1. Let,p 1 and let the induced character X == E; Jl.sXi where Xi the irreducible characters 0/ G. Then
DE/i: PROOF.
= fx == II ~.
Imw.edia.te from the additivity of the characters.
COROLLARY
o
2. Let K/k be abelian. Then DK/k
= ITfx x
where the product is taken over all the ordinary characters o/G. PROOF. Let E = K in the preceding corollaxy. The character X induced by 1 is the character of the regular representation. The irreducible characters have dimension 1 50 ~i = 1 in the product. The irreducible characters are simply the ordinary characters of G when G is abelian, and this proves our corollary. 0
The preceding theorems giving the formalism of &I(X) are valid in a complete field with any perfect residue class field. We shall now specialize to a general local field to obtain one more statement concerning the conductor:
103
Ii. THB CONPUCTOR
THEOREM 19. Let k be a generalloc4ljield, and K/k a normal extension. Then v(X) is an intege,.. PROOF. A character is a. linear combination with integer coefficients of irr~ ducible characters. Using the linearity of v(X) we see that it suffices to prove the theorem for irreducible characters. Suppose next that X is I-dimeosional. Such a character will be called more briefly an abelian character. It is a homomorphism of G, and its kernel Go is a normal subgroup. FUrthermore, G = GIGo is cyclic. Let X = X on G. Then II(X) = D(X) by Theorem 16, and by Theorem 13 we know that v(X) is an integer. This proves the theorem if X is abelian. Let 'I/J be an a.belian character of a subgroup H. Then iI(fjJ) is an integer by the preceding remarki>, and f", is integral. Let X be the induced character. The formula
f)C = D~W NEllcf", of Theorem IS shows that h: is also integral, i.e. that v(X) is an integer. Brauer has proved that every character is a linear combination with integer coefficients of characters induced by abelian characters. In view of the linearity of v(X) it follows that v(X) is an integer. We shall give here a proof independent of Brauer's Theorem, by reducing the problem to p-groups. Indeed, for a p-group, it is shown fairly easily that every irreducible character is induced by an irreducible abelian character, 2 and the preceding argument can then be applied directly. We may of course assume Kjk completely ramified. e == eopr = #Vo. We must show that e divides E.':o Ev. X(I :- u). We begin by treating eo. Let -'(X) = ev(X) be the double Bum. Then OQ
-'(X)
= E[X(l)#Vj - Ex(u)] Yo
;=0 OQ
=L:>[X(l)#V. - LX(u)] -
.=0
Vj
OQ
~)i-l)[x(l)#Vj - ~x(O')] V-
;=0
= Ei[X(l)#Vi - Ex(u)] - ~ i[X(l)#V;+l- E x(u)] 00
""
V,
i=l
v.+ 1
i=-1
"" =X(I)#Vo- LX(U) + Li[x(l)(#Vi-#V;+l)Vo
;=1
L
xCu)].
V.-V'+1
But eo I #Vo and ~v. X(u) = #Va· ~ where ~ is the multiplicity of 1 in X. SO the first term is divisible ~>y eo. To handle the sums, we recall Theorem 5 and the discussion following it. The sum EV;-VH-l X(u) breaks up into a sum over equivalence classes and we have
L \';-V.+l
xCv)
= l: ,..,X(u) f1
where the second sum is taken over a representing the equivalence classes. (We can do this because a character depends only on the conjugate class of a group element. ) ~See th~ corollary of Theorem 3 in the appendix which
follows.
104
XI. HIGHER RAMIFICATION THEORY
We know tha.t EH x(a) is a rational integer for any subgroup H of G. Hence ~v.. '-" .- .. + 1 X(o') = • X(o') ,+1 X(U) is II. rational integer, sa.y mi' We also
v..
know that
Lv..
eo I iTa.
Ev..
Hence
'
ms = eo ~)iTC7/eo)x(O').
=
But L.,.(ira/eo)X(O-) mo/eo is an algebraic integer, and a rational number. It is therefore an integer, and this proves that eo divides the questionable sum. The preceding discussion is valid for any character X. Taking X = 1 yields the term i(#Vi - #Vi+1) which is thererore also divisible by eo. All that remains to be shown is that veX) is divisible by p". We have >.(X) = 2:=l 2:v, x(l- a) + I:vo X(l- 0'). The sum 2:vo X(l- a) is == Vi n Vl divisible by #Vo ;: e. Let E be the fixed field of VI. We know that and therefore
Vi
00
00
LLx(l-O') = LLx(l-O') i=1 V,
i=l
where Xis the restriction of X to Vl. We have in Eby
V.
Vo = Vt.
Our sum differs from l:(X)
and this sum is divisible by #Vi = pro Hence it suffices to prove tha.t X(X) is divisible by p' and reduces therefore the theorem to the case of the p-group Vl. We saw already at the beginning of our proof that the theorem follows in full, if we use the fact that every character of a p-group is induced by an abelian character of a subgroup. 0
This concludes our discussion of the higher ramification in general local fields.
Appendix: Induced Characters For the convenience of the reader we shall develop here the theory of induced cha.ra.cters, used in the preceding section. We begin by recalling basic notation. Let G be a finite group, F a field, algebraically closed and of characteristic O. All the spaces we deal with will be finite dimensional F-spaces. This will not be mentioned explicitly any more. We denote F-spaces by U, V, W•.... Let V be a G-space. G acts on V linearly, and is represented by linear transformations of V. If a basis of V is selected, then the trlWSformations may be given by matrices and we have a homomorphism
a .... M a of G into a group of matrices. The character X of the representation is the function on G given by
x(O')
= SCM,,)
where S is the trace (sum of diagonal terms). The fundamental theorem states that the character is an invariant of the representation, and that in fact it characterizes
APPENDIX: INDUCED CHARACTERS
105
it: Two representations of G are defined to be equivalent if their spaces are Gisomorphic, and two representations are equivalent if and only if they have the same character.
Induced representation. Let H be a subgroup of G. For each coset c = H 0', let c be a representative element, so that G = U c- 1H, disjoint union. Let V be a G-spa.ce, Wand H-space, and i : W -+ V an H-homomorphism. One says that V is induced by W via i if i is an isomorphism into and V :;: $c- 1iW, direct Sum. For each W there does exist such a V, for example, V = F[G] ®F[H] W, with i(w) = 1 ® w. For any such V it is easy to check that if V' is another G-space the map J t-+ f Q i is an isomorphism Homo (V, V') ~ HomH(W, V'). From this it follows that V is uniquely determined by W up to a unique G-isomorphism f such that f 0 i = i'. Therefore we usually view i as an inclusion, don't mention it, and write simply V
= Vw.
THEOREM 1. Let G:J H::> I, and let IV be an I-space. Let Uw be the G-space induced by Wand Vw the H -space induced by W. Then Uw ~ UVw' [IWl and W2 are two I-spaces, then Vw,+w. ~ Vw, + Vw. where + mt4n.s the
direct sum. PROOF. The theorem is an immediate COJI8equence of the Wliquenees of the Wduced representation. 0
Let G :J H, and let W be a.n H -space. Let t/J be the character of the repre&entation of H in W. The character X of Vw is called the induced character. The next theorem gives a formula which allows us to compute the values of X in terms of the values of t/J. THEOREM
2. Let G
= UHc.
Then
X(u) ""
:E ""(CdC
I)
c
where tve let 1/;(7")
=0 unless l' E H.
PROOF. Let {1"",{n be a basis for W oveI F. We know that V = Let u be an element of G. The elements {cO'-l{dc,i form a basis for V.
Remark that 0'(CiT 1{) = Cl(cuCO'-l)~.
The action of 0' on this basis is therefore given by u(w-1e.)
= c 1 ~)CaCo'-I)1,.e,.
,.
= L(Co'CQ'-1)iU{C- 1 e,.).
,.
By definition,
But
CO'
= c if and only jf 001:- 1 E H. t/J(wc- 1)
Furthennore, = L(CdC- 1) ••• i
Ec- 1W.
106
XI. HIGHER RAMIFICATION THEORY
X(O') ==
L v(eoe
l)
(:
o
as was to be shown.
THEOREM 3. Let G be a p-group. Let V be an irreducible G-space. If V is not I-dimensional then there exists a proper stlbgroup H1 and an irreducible H1subspace W of V such that (G, V) is induced by (H1' W). PROOF. We begin by recalling that an irreducible representation of an abelian group is I-dimensional. This implies in particular that if V is not I-dimensional, then G is not abelian. We shall first give the proof of our theorem under the additional assumption tha.t V gives a faithful representation of G. (This means, if O'e = { for EV then 0' = 1.) It will be easy to remove this restriction at the end. Let H be a normal subgroup of G which is abelian and contains the center properly. (Proof of existence of H: G has a non-trivial center Go. Let C == G/Go. Let 11 be an element of period p in the non-trivial center of (j and let fI be the subgroup of G generated by U. Then fI is normal in G. Its inverse image H in the natural map G -+ (j is normal in G, and is generated by an inverse image 0' of ij, and by Go. Furthermore, 0' '/:. Go so H ::) Go properly. Finally, 0' commutes with its powers and with Go (the center of G) 50 that H is abelian.) We denote the elements of H by'Y. AE. an H-space, V is a direct sum of irreducible H-spat;eS which are I-dimensi-
all,
anal.
Let,
E V generate a I-dimensional H-t;paee. Let '"
be its character, so 'i'e =
tP(-r)e. where Ibb) E F. If 11 gives an equivalent representation, then "'If! == "'bY"I.
If a, b E F then 'Y(a{ + 1Tq)
"'b)(a{ + brj}. Hence the vectors of V giving rise to the same irreducible representation of H form an H -space W. We contend that V i- W. Assume V == W. Let e be any element of W V. Let 0' E G. Then 0'-1 t is a I-dimensional H-space by assumption, and has character f/J.
=
=
Hence 'Y(a-1t)
= "'("'I)O'-l~
= uf/J("'I)u- 1e;:;: "'('Y)e. This shows that -y
center, "'I
i- "fu
for some 'Y E H,
0'
E
is not contained in the
G, and we have contradicted the fact that V
is faithful. We may therefore assume that V - WI + ... + Wr where each W, is the space generated by all the equivalent 1-dimensional H-spaces having the same chal'BCter The sum is direct, and r > 1. We shall prove that G permutes the Spat;eS Wi transitively. For definiteness, we consider the effect of G on one particular say WI. Let 0' be an element of G. Then O'W1 C Wi for some i.
"'i.
W.,
PROOF.
Let {E WI. For any 'Y E H,
'Y(O'e)
= u(O'-I'YO'){ = U,p('Y"'-l l{ = "'u('i')(O'el
APPENDIX: INDUCED CHARACTERS
lOT
where the function t/J" given by t/J,,(,) = "'h,,-I) is a character on H and depends only on u. For a fixed u, we see therefore that 1/1" = ¢i for some i. Since Wi contains all the spaces with character tP;, it follows that uW1 C Wi. 0
u is an isomorphism of WI into Wi' By symmetry, u-1Wi must be contained in WI, and since Wj and W; are finite dimensional, this implies that their dimensions are equal. Hence uW1 = Wi. The preceding argument proves that G permutes the spaces Wi. All that remains to be shown is the transitivity. Let V' = GWI . If V' I V then GV' "" V' and this contradicts the irreducibility of V. Hence GWI = GW; ::: V. This proves that the spaces are permuted transitively. Let HI be the stabilizer of W1 . Then HI :::) H, and HI :F G because r > 1. Furthermore, (H), WI) is a local constituent of (G, V). We contend that WI is irreducible for H). This will finish the proof of our theorem (in the case that V is faithful). Suppose WI is not irreducible for HI' Then there exists a. space U C Wi> U i= WI such that H 1 U c U. Let G UHlc. We know that V = L c C I W1 • Let V' L c-1U. Then VI f:;. V, and we shall prove that V' is a G-space. thereby contradicting the irreducibility of V.
=
=
We have
uarlu = C 1(wcu I )U c ciU because (wen-I) E HI. But CO'runs through all cc.x:>ets as c does, and this proves that uY' c V'. Hence V' is a G-space, contradiction. Suppose now that V is not faithful. Let Go be the normal subgroup of G con.isting of all u such that == ~. for all { E V. Then V is an irreducible space of = GIGo and gives a faithful representation of If V is not I-dimeIlllional, then G is not abelian and there exists a proper subgroup H of (; and an irreducible fI-space W such that (fI, W) induces (a, V). Let H be the inverse image of if in the natural map G -+ G. Then H J Go. and W is naturally an irreducible H-space. FUrthermore we contend that the stabilizer HI of W is H. Certainly, HI :::) H. Suppose u E Hlo a f/ H, then uW C W. iJ ¢ fl. Since a has the same effect as u on W, this coutradicts the fact that iI is the stabilizer of W. This proves that (H, W) induces (G, V) and concludes the proof of our theorem.
a
ue
a.
o
COROLLARY. Let G be a p-group. Let X be an irreducible character. If X is not I-dimensional then X is induced by a I-dimensional character.,p 0/ a su.bgrou.p H o/G. PROOF. Let V be the irreducible space of x. A subgroup of a p-group is a p-group. We may apply the preceding result step by step using Theorem 1 Wltil we cot a. subspace W which is I-dimensional. The character 1{J of W will then induce
X.
0
CHAPTER XII
Explicit Reciprocity Laws 1. Formalism of the Power Residue Symbol The global norm residue symbol was obtained from local ones. The definition of the local symbol was obtained indirectly from a non-constructive proof of the fact that all cocycles have an unrarnified splitting field. We are therefore faced with the following unsolved problem: To determine explicitly the effect of the norm residue symbol on totaJIy ramified extensions. (On the unramified extensions, it is the FrobenilUi Substitution.) We shall treat special cases of Kummer extensions. We need an auxiliary algebraic statement. LEMMA 1. Let F be a field of characteristic p ~ 0, containing the n-th roots of unity, P f n. Let a E F. Then -0 and 1 - 0 are norms from F( a l/n ).
PROOF. We distinguish two cases. [F(a l/ ") : F] = n. Then x" - a is irreducible and its roots generate F(a 1/"). We have . x" - 0 = (x - (nol/n)
n C,.
where Cn ranges over all n-th roots of unity, and (n0: 1/" are the conjugates of olIn. Putting x = 0 shows -a is a. norm. Putting X = 1 shows 1 - a is a. norm. [F(a l !") : F] = d < n. By Kummer theory, din. Let ct"· = f3" where din is the period of a (mod Fn). We have F(ol/n) = F«(31/d) and by selecting a.1l n and (3l/d. suitably (Le. by multiplying them by a suitable root of unity) we may assume a lln = f31/d.. The conjugates of al/" are therefore (d.al/n where (<1 ranges over all d-th roots of unity. The polynomial
z" - a
= 11(x (
.
("a l / n )
is not irreducible, but for each factor (x _'naIl") it contains also all its conjuga.tes. Putting x = 0 and x = 1 as before shows that - 0 and 1 - a are norms. 0 We now let k be a loea.! field, p the characteristic of its residue class field if k is not archimedean. p is the prime of k. We assume that k contains the n-th roots of unity, p f n. Let (J = (0, k). On each abelian field K/k of finite degree, (7 = {a, K/k}. In particular, on k{(3l/n) we ha.ve (J
= (a, k({3l/n)/k).
Define (0,(3) to be ({3I!n)a-l. Then (o:,f3) is an 7'I-th root of unity and does D.Ot depend on the n-th root f31/n selected. If k is complex then (0:, (3) = 1 for all 109
110
XII. EXPLICIT RECIPROCITY LAWS
a, fl. If k is the reals, then n == 2. The explicit values of (a, (1) will be discussed in section 3. The symbol (a, fJ) satisfies the following formal properties which are in most CIUieS immediate coW3eQ.uences of the properties of the local symbol u = {a, k}. Propertiesl of (a, (3). 1. (0,(1) = 1 if aud only if a is a norm from k(f31/"). 2. (ala2,(1) = (al,(3)(az,p). 3. (a, (311h) = (a,pl)(a,~)
PROOF. We have
because the norm residue symbol is consistent. 4. (-a,a) "" (1- a,a)
o
= 1.
PROOF. Use Property 1 and Lemma 1.
o
5. (a, (1)({1,a) = 1 (inversion theorem). PROOF. We have by the preceding property: (a, (3)((3, a)
= (a, ,8)( -,8, (1)( -a, a)({3, a) = (-ap, ,8)( -a(1, CIt) = (-uP, a(3) = 1.
o
6. (a, (3) = (a, a+(1)(a+(1, (1)( -1,a+8) (provided, of course, that a+(1 1: 0). PROOF. Let 7 = u
+ p.
Then
1 = (1- cry-l,u-y-l) = (fJ-y-1,a-y-1)
= ({3, a) «(1, ,,(-1 ) (,,(-l, a)(,),-1, ,,(-1). But (,-1,..,.-1) = (_..,.-1,..,.-1)(-1,..,.-1) = (-1,7- 1) by Property 4. The property now follows by transposing the suitable terms, and taking the exponent -1 outside ~esymbcl.
0
7. The symbol (a, fJ) is continuous in both arguments.
PROOF. It is continuous in the first argument because if Q is close to 1 then a is a norm, and Property 1 can be UBed. It is continuous in the second argument by the inversion theorem. 0 lIf F is a field aod A an abelian group, a map ~ x F" .... A with properties 2,3, and 4 II celled a Steinberg BYmboi &lid can be interperted as a hoIPomorphisro iC2F .... A.
111
2. LOCAL ANALYSIS
2. Local Analysis Let k be complete Wlder a discrete valuation with perfect residue class field k, and integers o. We shall specialize k Ia.ter to an ordinary p-adic field. We recall some elementary facta. A series L~=o a" converges in k if and only if lim a" == O. A product lla" converges in k if and only if lim a" == 1. Let o{t} be the ring of formal power !ieries J(t) ::: E:'o a"t" with a" Eo. The map
I(t)
-+
J(tr}
is a homomorphism of oft} onto o. This is dear for the polynomial ring and extends to the power series by continuity. Let Flk be a finite, completely ramified extension. Then F == k. Let II be a prime in F, and !.p(t[ = !nell, k, t). Every a E D can be written a = E a"rrv where a" E 0 because F = k. The map
f(t) -+ J(ll) where J(t) E oft} is therefore onto D. We contend that its kernel is the principal ideal generated by !.p(t) in oft}. Indeed, suppose
o= f(ll) :: ao + aln + a2fi2 + ... with all E o. Then IT I ao and hence 11' I ao. Since !.p(t) is an Eisenstein polynomial with lw>t coefficient divisible by 1f exactly, we have, for suitable do
J(t) - clo!.p(t) = bIt + ~t2 + ~t3 + ... and 2::'1 b"t" is in the kernel. Hence by the same argument we can find d1 E ., such that
88 before, 11' I b1and
J(t) - (do + dlt)!.p(t)
= c2f + cst3 + .... Repeating this argument shows that I(t) = g(t)",(t) where get) tended. Let a E 0, a derivative,
= I(ll) == E:'oa"n" a' :;
E oft}, as con-
where I(t) E oft}. We may take the
L llaoIT,,-l.
Then a ' is not well defined in K. However, we know that any other expression for a as a power series is of type a =0 J(II) + g{II)tp(II) where get) E o{t}. Taking the derivative shows that 0:'
== t(rI) + g(rI)tp'(ll).
But
=
We now specialize the field k: SUppOlle that k is a. completion at a finite prime
of au algebraic number field. In other words, k is an ordinary p-adic field. We shall consider special series in Ie, analogous to the log and exponent. Let pip. By ord we shall mean ordp. IT p - p' then ordp = e ordp.
112
Xli. EXPLICIT RECIPROCITY LAWS
THEOREM
1. The series log(l +x)
=:
x -x2/2+~/3- ...
convf1Ye6 for all x E k such that ord x > O. If 000 x > 1/(p - 1) then ord x" /n > ~ 2, and old x = ordlog(l + x).
ordx for n
PROOF. Let pr ~ 11 < pr+l. We have ordxR/n = nordx-ordn ~ nordx-r. An elementary limit computation shows that ord x"/n -+ 00 Wi n -+ 00, provided that ord x > O. This proves the convergence statement.
Suppose ordx > l/':P - 1). We must show that ordz" /n equivalently, ordx"/n- ordx > O. But
= (n -
ordx"jn -ordx
l)ordz - ordn
'> (n -l)/(P - 1) ==
> ordx for n ~ 2, or
r
(because n ~ 2)
«n -1) - r(p -l»)/(P -
1)
~O.
To justify this last step, note that it suffices to show r(p - 1) ~ n - 1. Since p" ~ n it suffices to show r(p - 1) ~ p" - 1. If r = 0 this is clear. Otherwise, dividing by
P - 1 shows that the inequality is equivalent to r ~ 1 + p + ... +pr-l, which is true because there are r terms on the right. This concludes our proof because of the strict inequality in the second step. 0 REMARK.
If 0;;;; 1 (mod p) we define log a
THEOREM 2.
Let a and {3 be == 1
(mod
pl.
=log(l -
(1 - a».
'Then
loga.8 = logo + log,B. PROOF. The identity in formal series is known, and all the series involved 0 converge by the preceding theorem. COROLLARY.
Let ( be a p. root of unify in Ie. Then log ( is defined and = O.
PROOF. We certainly have 1 - , == 0 (mod p) aod hence log(l - {l converges. By the functional equation,
0= log 1 = log (p'
Heuce log (
(»
= p·log(.
=O.
o
Although we shall not need the exponential function in the sequel, we give it
here anyway [or completeness. THEOREM
3. The series
exp x = 1 + x + z2/2! + x3 /3! + ...
Jor all x 1l!JCh that orr! x> l/(P -I). In that case, ordzfl./n! > ordx for 2, and ordx = ord(expx -1).
COtWe7ye8
n
~
PROOF.
Write n in the p-adic scale: n- ao+alP+ ... +a.,.pr
2. LOCAL ANALYSIS
113
where all are rational integers, 0 :S;; au :S;; p - 1. Then
[nip) = 0.1 + (J.2P + ... + (J,rp,.-l [n/p2) = ~ + ... + ar P,.-2
We clearly have p.
= oed n! = [nip) + In/p2) + ... + In/prj = =0.1 + (1+p)a2 + ... + (1 +p+'''+ p"-l)ar
and therefore (P - l)p.
= (I -
1)0.0 + (p - I)al
+ (p2 - 1)0.2 + ... + (pr - 1)a,.
=n-8" where
8"
= Go + 0.1
+ ... + 0.,..
This gives
p.
Now
= (71 -
sn)/(P - 1).
ordx" 1711 = nordx - p.
=71 (ordx- p~
1) +s,,/(P-l).
Since Sn is positive we see that ordx"'ln! -> 00 when ordx > II(P - I), and the series converges. To get the second statement ~ have to show that ordx"/n! - ordx > O. We have ordxn/n! - ordx = (n -l)ordx - (n - sn)/(P -1)
=(n -I>(ordx - p-l _1_) + (sn -l)/W - 1). If n ;?; 2 the result is obvious because Sn THEOREM
;::
1 always.
o
4. Ilordx and ord y > Ij(P - 1) then exp(x + y) = (expx)(expy).
PROOF. The formal identity is known, and all the series converge by the previous theorem. 0 THEOREM
5. Ifordx> l/(P-l) then explog(1 + x) logexpx
= 1 +x
=x.
PROOF. The formal identities are known, aud all the series converge by the previous theorems. 0
Let" = [e/(p-l)j+l. The log and exponent give mutually inverse isomorphisms of 1 + p6 onto pol. We leave it as an exercise to the reader to prove the converse of the Corollary to Theorem 2, i.c. the kernel of the map for
Q
!!!!
Q -logo 1 (mod p) is exactly the p"-th roots of unity contained in the field k.
XII, EXPLICIT RECIPROCITY LAWS
114
3. Computation of the Norm Residue Symbol in Certain Local Kununer Fields We treat the simplest cases of Kummer fields, and begin with the infinite prime. Let It be the reals, 0, b E lit. We may interpret the results of Section 1 with II = 2 over the reals. We obtain
(a, b) PolO -_ (_ 1).ia".A-I.'JD!-1 .
e
Next we consider finite primes. Let be a prime and k == Qt((t) where Qt is l-adic completion of the rationals and (t is a primitive f-th root of unity. We shall determine (0:,11) in the field k({Jl/t) explicitly. Since the symbol is continuous and multiplicative in both arguments it will suffice to determine it for a multiplicative basis of k. We first consider t' = 2, k = Qz. A multiplicative baBis2 of (h is given by 2, ~e
-1, and 5. THEOREM 6. Let
a,
U2 be ~nits (2,2) :;; 1
bE
of~.
Then
= (_1)(4 -1)/8 (a, b) = (b,a) = (_l)e(4)c(b)
(2, a)
2
where c(a) ;: (a - 1)/2 {mod 2}. PROOF. We see that (a,b) == (b,a) from the inversion theorem and the fact that (a, b) = ±1. We prove that the exponent c(a) is multiplica.tive. We have
al - 02 - 1 ==
111a2 -
(a1 -
1)(Cl2 - 1) !!!! 0 (mod 4)
'.,
and therefore
Thus
e(ala2) = e(al) + e(a2) (mod 2).
For the other exponent we have (0102)2 -
and therefore
o~ -
af -1 == (af -l)(a~ -1) ;: 0
(mod 64)
(al a2)2 - 1 = ai - 1 + ~ - 1 (mod 2) 8 8 B
as was to be shown. Both sides of OUI equations a.re multiplicative. It suffices to verify the sta.tements for a basis of k. We note that: Q2(i) is ramified (because (1 + &)2 = 2i). Cl2 (~) is unramified. Indeed, let 0 = ( J5 - 1)/2. Then 0 generates Q2 (vrs) and satisfies the equation 0 2 +0:-1 = O. Read mod 2, it is the canonical equation for an extension of the residue class field. Hence Q2( Q) Q2 (v'fi) is unra.mified. Now:
=
2Here 8J\d in the next four pagea, "multipliC4tive ba8is for" meana ~a. set; of genera.tors for a d _ subgroup of" •
11Ii
3. COMPUTATION OF THE NORM RESIDUE SYMBOL
(2,2) = (-1,2) 0: 1 becaUBe -1 = N(l + J2) is 8 norm. . (2,5) = -1 because the prime 2 is not a norm from the unramified extension 4~MJ5)· . (-1.5) = 1 because ~he unit -1 is norm from Q2(.;5). (5,5) = 1 for the same reaBon. (-1,-1)::: -1 because -1 is not a. norm from fJ2(v'-i). (Otherwise, -1 = x 2 +11 which is impossible mod 4.) One verifies directly that the values we have just found ooincide with (_l)£{a)t{b) in each case, and this proves our theorem. 0 We suppose from now OD that i is an odd prime and that k = Qt«") where, is primitive loth root of unity. Then (k : Qt] = e- 1 and k is completely ramified over Qt. Let ). = 1-(". Then ). is a prime in k, and e '" ).t-l. It is actually easy to determine £1>.1-1 (mod >.). Namely: X I - 1 + ... + 1 == n~-:,~ (x - (") and therefore f = rr!-:'~(I - (") We get II.
£/>. 1-1 = 1I(1-(") =- IT(I+(+ ... +("-1) 11(1 - ()
,,=1
.
But (== 1 (mod ).). By Wilson's theorem, we see that
£
>.1-1
== (£ - I)! == -1 (mod >.).
We shall abbreviate the modulus, and write (>.) for (mod .>.). Let I1i = 1_>.i, t;;r: l. LEMMA 3. The 'Ii form a multipticative basis of the 'Units 0/ k == I().). A full multiplicative basis is given by the 11i, the (£ - l)-th roots of unity, and the powers of a prime. PROOF. The residue class field of k is Zt = Ql because k/Ql is completely ramified. The (£-1) roots of unity lie in Gt and obviously any Wlit can be multiplied by such a root of unity to make if == 1(>.). Every unit e == 1(>') can be written e = 1 + alA + a2,\2 + ... where a" are rational integers. We can obviously solve formally
(1 +alt+a2t + ... ) = (l_t)-0.1(1_ t 2 )-b,(l- t3)-c3
for integers
b:.J,
•••
e3, by a recursive process. The power series obtained by putting 0
t = >.. will converge, and this proves our lemma.
We are now interested in computing the symbols ('Ii, >.). We note that
(1'],,>")
= (1- ).i,>.i)
0:
(1- ).i,>.)i
=1
by Properties 1 and 4 of the symbol. This shows that if because ('Ii, >.) is an loth root of unity. LEMMA
4. Let e be a unit of k, e
tfi
then (l1i,).)
=- r().t+l) for ,orne:r E k.
=1
Then e iI an
t-tiI. power in k. PROOF.
that
Suppose e =:; r().") with" ;;r: t
+ 1.
We try to refine x. We contend
116
XU. EXPLICIT RBClPROCITY LAWS
Indeed, the remaining terms of the binomial expansion are of type
f) ""
I (~) the intermediate terms are divisible by t ( >.2"-1+1 and hence are O{>,II+1). Furthermore Alii It; >''' (>.,,(1-1)-l(i-1)). Also, II-I. ~ I, and £-1 ~ 2. Hence the !.a.st term is also E O(N'+1). This proves that we can solve for y such
Since t E
that
xl + yx' - 1 >'" == c( >. ..+1) and by a standard refinement prOce!lS, we can find an t-th root for c, as contended.
o
THEOREM
7. For all i '" f. we have (1'Ji, >.) = l.
e
PROOF. If i < t then t i and we have already considered this case. If i ~ t + 1 then '11. == I(.\t+l) and "Ii .., (i by the lemma. Hence ("Ii, >.) (0, .\)l 1 thereby proving the theorem. 0
=
=
It will be much harder to prove the following results. THEOREM 8. ('1t,>')
= (-1, or equivalently, (A,f'll) =(.
PROOF. The inversion follows f;om the inversion property for the symbol. We shall deal with (A, '111), because we enjoy the advantage of the following statement: k('1i/l)lk is unra.mifted. PROOF. Let A = fJ;/l. Then At - 1 +).1 = O. Let B A = >'B + 1 and therefore B satisfies the equation
= (A -
1}/>.. Then
0= (>.8 + 1)1 -1 + At
== >.1 Bf +
G)
>.1-1 B'-1 +.... + tAB + A'.
But l"" )"t-l. Divide the equation by ).t. Since
t I (~
we see that B satisfies
,.,t + (tA/>.l)Z + 1 == 0 (>.). We have seen a.t the beginning of our discussion that (lA/>.I) equation becomes therefore
E
-1 (A) and our
Zl - X + 1 == 0 (>.). Since k/Qt is totally ramified, the residue class fields k a.nd Ql are equal and are simply the prime field Zt. Hence the equation above is irreducible in the residue 0 class field and this proves that k(A)/k is unramified. We conclude that u = (A, k{A)/k) = therefore prove that AI"-l = (.
I(J
is the Frobenius Substitution. We must
3. COMPU'l'ATION OF THE NORM RESIDUE SYMBOL
We know that lJ'P on Awe have
== Bt (>'), and therefore B'P == B -1
11'1
(~). To get the effect
A'P- 1 = (1 + >'B)'I' /(1 + >.B) = (1 + >'B'I')(I-.\B + >.2& - ..• ) iii 1 + >'(B'I' - B) (>.2) i!! 1 _ >. (A2)
==,
(>.2).
But if(" = '" (>.2) then ("(1_("-") =0 (>.2). Since (1-,,,-,,) "" >. unless (mod t) we conclude that A",,-l = ( thereby proving our theorem.
II;;;
P 0
Our next problem is to determine (1]" f/j) in terms of ('Ii, >.). We begin by deriving a certain functional equation for ('1};,1/i)' We see immediately from the definition that
+ ),i 1]; = fli+j'
fli
=
Note that {-1)l -1 because t is an odd prime, and hence (-l.tJ) = 1 for all {3. From the addition theorem we get
(fli. >.11];)
= (flj, '1Ii+j)('1'+1, >.3'1;).
We use the wu]tlplicativity of the symbol and ('1i' >.i)
(1)
to get
= (1 -
>.i. ),i)
=1
(fli' f/i) = ('1;', 'h+i )(f/i+j. '1/i)(1/i~i' A)i
and invert everything:
(fl., '111)
(2)
= ('1/i, flHj ){11i+,.. 1'Ii)(1/i+j. >.)-i.
PROPOSITION.
('h, '1j) =
IT (17ri+'i. >,)-(1'O
iHoi)
r,.~l
(r,.)=l
where r. s are positive integers, relatively prime, and for each pair (r. s) the pair (ro, so) is one solution of the equation TSo - STo = 1. PROOF. We must first show that tile expression is well defined, Le. does not depend on the choice of (ro, so). Indeed, it is easy to verify that any other solution (Tb sd is given by rl = TO + rt and S1 = So + st where t is an integral parameter. This implies that the exponent changes by (Ti + sj)t. Let 11 = ri + sj. From (1) we know that (1)1-<' >.)1-< = I and therefore our expression is well defined. Let [11;, "Iil be the expression. We prove that ['1}" 1)j] satisfies the functional equation (2) II.'l a function of i, j. We consider different values of r. s. r = s. Then r = s = 1 because (r, s) = 1. We may choose ro = 0 and SO = 1. The term gives the contribution ('h+;. >,)-i. T > s. Let $ = s' and r = r' + s'. This gives a 1-] correspondence between (r',s') and (r,s). The product over such terms gives the contribution
IT (·,r'o+.'(O+J ,,, n.
where r',s' ~ 1 and (r',s')
= 1.
.
') \)-«~o-Bo)H.o(i+j»
This is precisely [11i.'1/i+j)'
ll8
XII. EXPLICIT RECIPROCITY LAWS
r < s. Let r = r' and s = r' +S', Again we have a 1-1 correspondence between (1", S') and (r, s). The product over such terms gives the contribution
IT (·,r'('+J>-'J' '" n
. ,)-(roli+i)-(.o-ro»
. .
where r', s' ~ 1 and (r',8') = 1. Thill is precisely ['7i+it'1i1. We have therefore shown tha.t [ni,1/j] satisfies (2). Taking the quotient we
obtain
(7].,7]j) _ ('1i, 'h+i)('1i+j, 1/j) ['1'" '1j] - {lJi, 17i+jJ['7i+i, '1j] .
(3)
For la.rge 71 and p. we know that 1/... and 1/1' are i-th powers and therefore (,,.,'1.,)
= ['1i,'1.,J = 1
('1p, 'Ii)
= [1}I"
lJi]
= 1.
Using (3) recursively to increase the indices we see that the quotient is 1, thereby 0 proving the proposition,
We have proved in Theorem 7 that (771" A) == 1 unless p. == t. Hence in the product only terms with ri + sj == will give a contribution. From Theorem 8 we
e
get ('Ii.,,;) =
IT (rui+-.,j (r •• )
where the product is taken for rela.tively prime positive integers r, s such that ri + sj = If no such integers exist, ('1/ij'7/j) "'" 1. II ri + sj = then r and s < i and
e.
e
roi + so;
= TOr (ri + 8i) + tr == jfr
where 11r is the integer inverse to r mod t.
(mod t)
We have proved
('h,1}j) == nCr,s) (l/r where the product is taken over all integer" 1 such that ri + sj == e. jfr is interpreted mod i, and (T/i, 'li) = 1 if there exist no (r, s) such that ri + sj = f. THEOREM 9.
r, s
~
Having found ('1;' '1j) we may now get (a, [3) for a.ll a, [3 E k. By factoring out powers of A, we see that it suffices to compute the symbol for units, and >. aeparately. REDUCTION
1. Let a,
13
be unita. Then
a l - 1 == (1-1 and (c;l-l,pl-l)
= {a,,6}.
;:
1 (A)
This reduces our problem to uni" == 1 (>.).
REDUCTION 2. If Q 5 1 (A) then we can write Q == 1 + p.>. (>.2) where J.' i& a rational integer (because k = Qt = 'I.e). Since (" == (1 - >.)1' == 1 - p.A (>,2) tile obtain a(" == 1 {>,2). This reduces the computation o{(a,(J) to the case where Q == 1 (>,2), and to «,13) where {J == 1 (>.). REDUCTION
3. We hare «(,() == (-(,()(-l,() == 1. This reduces the compu-
tation of «, {J) to the case where {J == 1 (A2 ).
S. COMPUTATION OF THE NORM RESIDUE SYMBOL
119
Before computing the symbol for the above cases, we make a definition. Let {J be a unit of k, {3 = ~:'=O a,,>." where a" are rational integers: Define D log {3 to be {J' /fJ where (3' is the derivative of the power serie:;, as discUS$ed in section 2. Then Dlog(3 is well defined mod )"t-2 according to Lemma 2. Indeed, the different of k/Qt is )"t-2 because k/Qc is tamely ramified. We have obviously Dlog(a(3)
== D log {3 + D log a (>/-2).
Note that log.B may not be defined since {3 is not necessarily == 1 (A). Even in this log {3 may not be a unit, 50 D log fJ is not necessarily (log (3)'.
C8Be,
REMARK. The good radius of convergence for the log in ~ 2, or equivalently, if ord>. x> 1 then ordx >
Indeed, if ord>. x
k is precisely >.2. 1/(£ - 1) which is
the number of Theorem 1. We let S denote the trace from k to Qt. THEOREM 10. (a,{3) is determined
1.
0'
== 1 (A2)
(0', fJ) 2. a
== 1 (A).
=(is((log aD log p) •
Then
«(,0') 3.
according to the following scheme.
and {3 == 1 (A). Then
= (~S(Iog",).
0'== 1 (>.). Then (a,>.) = ,tS(t log,,>.
The previous statements hold in tlie sense that the exponents involving log a and D log (3 are well defined mod and hence ( can be raised to such exponents.
e
PROOF. We begin with the first formula and must prove that the exponent is well defined mod F}om a =: 1 (>.2) we get log a == 0 (A2) by Theorem 1. We know that Dlog{3 is well defined mod At-2. Hence logaDlog{3 is well defined mod >.2.V-2 = >.t = l>.e where e is a unit. We see that
e.
1
[8(£A7)
= 8(>'7) = 0
(mod l)
where "y is an integer. This proves that the exponent is well defined. From the functional equation of log and D log we see that the exponent is multiplicative in CI and (3, and that it is continuous in both arguments. It suffices therefore to verify our theorem for a multiplicative basis, i.e. for (f);, '7i) according to Lemma 3. In fact, since CI == 1 ().2) we may assume i ;;?: 2. Considering Theorem 9 we must prove that
E
j/r ==
~S«log'l1.Dlog'7j)
(mod l)
r.$~l
"+8j=£
We have
logf/i = log(l- ).i) = _jAi- 1 Dlogl1j = ---. I-Ai
"" /r LAn
.=1 ""
.
= - '" jA81- 1 • L... .=1
120
XII. EXPLICIT RECIPROCITY LAWS
Multiplying the two series:
ls(
~>ti+'i-l) = L ~. ~. S(0.'">+8j -1).
r,4~1
Let m
r1s~1
= ri + 8;. Let f!' I r but £,,+1 f r.
We shall prove that
_1_s().m-l) == {I (l) tr+l 0 (t)
if m:::; t,
II =:
0
otherwise.
This will establish the fact that our two exponents are congruent mod £. We haw tv ~ r and 2ev ;,;; m because i ~ 2. Also, 2r ~ m = ri + sj. Case 1. 2 ~ m ... and 1/ = O. We expand Am - 1 and take ( into the sum to
e
get
~s«>.m-l) = ~s (~ (m; l)(_l)~(~+l) = r~ (m -l)(_l)~S«I'+l). #=0
But m , t implies
1.10
+ 1 ::::: t.
1.10
We have
8«1'+1)
== {-I
e- 1
if J1. + 1 < t if J1. + 1 = £.
Indeed, if J1. + 1 < t then (1'+1 1= 1 is a primitive loth root of unity satisfying :1:1- 1 + x f - 2 + ... + 1 = 0 over Qt. Hence 8«1'+1) = _(coeff:l;l-2) = -1. If 1.10 + 1 = then (I = 1 a.nd 8(1) = 1. We note that
e
e-
0= (1_1)",-1 =
'E (m 1'=0
Hence the sum is 0 except when m
!S«A t
m - 1)
=
_!
'f (m - 1)
tJo'=o
1.10
-1)<_1)1'. 1.10
= £. In that case we obtain
(m - 1)
(-1)" + ~ (-I)"'-1(t- 1) £ m-l
1 (1) t =--2:: m(-1)"+J1. £ ",-1
£1'=0
=1.
This concludes our proof.
e.
Case 2. m > We show that our expression in (4) is
== 0 (t). Taking e-(v+1) inside the trace
gives
tV~l s(om-l) = S«e),m-1-(V+l)(1-1») where € is a. unit. It suffices to prove therefore that m-l- (11+ 1)(£-1) > O. This is a hypothesis for (I = O. We may therefore assume II ~ 1. We have trivially m - 1 - (II + 1)(t - 1) ;?: U V
-
1 - (II + 1)(£ - 1)
=/(11)
)
3. COMPUTATION OF THE NORM RESIDUE SYMBOL
and we shall prove that
/(11 + 1) - /(1.1)
lev) > O.
We have
121
•
= 2f.,+l - 1 - (II + 2)(t =21"(1.- 1) - (£-1)
1) - (2l" -1,- (II + 1)(l- 1»
= (2l'" - 1) - (l- 1)
>0 for II > 1. But J(l) = 2t -1 - 2(l- I}
=
1. Hence f(lI) > 0 for all v ~ 1, as was to be shown. ThiB conclude!! the proof for the first formula of the theorem. We shall obtain the second as an immedia.te corollary. We first consider the case a == 1 (>.2), and must show
«,a)
= (lS(logct l .
Since, == 1 (A) we can use the preceding result on (a, ~). We ha.ve ( = 1 - A, (' = -1 and Dlog( =('/( = -l/(. This gives
log aDlog (
= -loga
and by the preceding result,
(a,() = ciS(loga,. The inversion property now giVe5
«, a) =(a, ()-I = ,is{loga) as desired. . We ha.ve remarked in Reduction 2 tha.t if a == 1 p.) then a(I-' = 1 (,\2) for a. suitable power of(. We know that «,0:) = (l- ,\,,\) = 1, and we also know that log( = 0 by the Corollary of Theorem 2. Hence
(a,()
= (0:(1',()
and
log{a()
= loga.
This pr0Ve5 that we may assume 0: == 1 (A2), and that the formula. is valid for 0:
== 1
('\). We now come to the third formula involving the prime A. The same argument concerning 0: that has just been made shows that we may assume a == 1 (,\2). We must prove (a,,\) = (lS( tIoga). The exponent is defined because loga == 0 (,\2) according to Theorem 1 and hence
II S{t log a). It suffices to verify the formula. for the multiplica.tive basis '1i (i ~ 2). It has been proved in Theorems 7 and 8 that
;':F l I ('1i, A) = { (-I , = l.
(5) Our exponent for (' giVe5
~S (~ log(l- Ai») = - ~ :tS(A·,.-l). We refer to the computa.tion carried out previously, i.e. to the result (4), with m = rio The only term giving a non-zero contribution will be the term with ri = f.
XII. EXPLICIT RECIPROCITY LAWS
=
and II = O. Since t is a prime Ij,nd i ~ 2, this combination can occur only if r 1 and i ;:; f.. In that case, we g t 1 (mod l) from the SWIl, and the minus sign in front. These values are seen t coincide with those of (5), and this concludes the proof of our theorem. 0
4. T e Power Reciprocity Law We make first some re arks on finite fields. Let F be It. finite field with q elements. Then x q - 1 = 1 D all x E F, x f O. Furthermore, m I (q -1) if and only if all m-th roots of unity Ii in F. Suppose this is the case. We have x q- 1 - 1 =
II (x(Q-l)/m - () (
where ( are the m-th roots of unity. The mapping z _ x(Il- 1){... %('1-1)/'" = 1 if and only if z ;:; '11m for some y. We let k be a global field, po It. finite prime, q == Np. P f m, and we suppose that a primitive m--th root of unity lies in k. From this it follows that m I (Np - 1) and that the preceding remarks all finite fields can be applied to the residue class field ofp. We recall the elementary fact that (" == (~ (mod p) if and only if II == ~ (mod m). Suppose p f a. We let (~) be the m-th root of unity defined by
bas (q - l)/m elements in its kernel and
(*) ==
a(N.-l)/m
(mod p).
The existence and uniqueness of the root of unity follows from the preceding remarks. It follows immediately that
(01;2) ;:; (~l) (~2). THEOREM 11.
(i) == 1 if and only if a = pm
PROOF. Immediate
(mod p) for some p.
from the definitions and the remarks on finite fields.
0
Let A = a 1/m be one of the m-th roots of a. From Kummer theory we know that k(A)/k is unraInilled at p. Let I{Jp be the Frobenius Substitution at p. Then
Arp,-l ==
AN . -1
(mod p) (mod p)
== a{N,-ll/m (mod p). The congruences a.re written mod p because they hold for each \1J I p. For {3 satisfying the conditions 1. {3 is prime to a, 2. {3 is prime to m,
~
TBB POWER RECIPROCITY LAW
123
(tJ~) = (~)(:a). We wish to investigate the number (~) (~) -1. We let ({3, a:)p be the local symbol (tJ, a) discWJSed at the beginning of this chapter. (Note that
, = Poo.)
Before proving the theorem which gives the oonnection between our various symbols, we make a useful remark: Let (7, T be two automorphisms over k. Then A,,-l = (" and A.. - l == (.. where (" and (T are two roots of unity. We have Aer.. = « .. A)" == ("(TA. This proves that
A"-l A .. - 1 = A"T-I. This remark will be used in the course of our proofs. THEOREM 12. If P f mpao then'
(~)(~):l ==(p,a:)p. PROOF. If pta then
(~) (~):l. = (~)p == AonI.p(ap-1l. But ~(A)/kp is unramified and therefore
(71'
= ({3, kp(A)/kp) == I()~' fl. By defini.
hl~
({3, a:)p = Aer.-l = Aord pfJ('Pp -1) by the remark on automorphisms. This proves our theorem, if II f Q. If II I Q then l' t f3 by IlBSUIUption, and
(~) (~rl == (~rl. o
The rest follows by symmetry. THEOREM 13.
Let 0,
tJ E k.
Then
II(tJ, a)p = 1 p
where the product is taken aver all primes (including the infinite primes!).
(f)
3For a prime p which divides Q, 80 that is not delined, the eJ
(.6, a), when! i
= onI, <Jr. and j = Old p p.
=(
(_1)ij ",) p
<Jr.,
/3 E k;, is
/3-') ,
124
XIl. EXPLICIT RECIPROCITY LAWS
PROOF. We know tha.t
0'1'
= (fj, k,(A)/k,).
Our theorem is an immediate
CODSequence of the product formula. for the norm residue symbol:
n,
O'p
=1 o
and of the remark on automorphisms.
COROLLARY. Let a, {3 be relatively prime. Then
(~) (~) -1 =
n
(!3,a)p
II (a.J3),.
=
ptmp~
IJlmpoo
The computation of (~) (~) -1 is reduced to a computation at a finite number of primes, namely the primes pimp"". We must still consider the specio.l primes dividing m. Let). E k be divisible oo1y by primes of m. Define
(~) =
n (~rd.a.
PfmP""
P
Then by the same arguments as before we have THEOREM 14.
Let). be divisible only by primes dividing m. Then
n (a,'\)" = n (.\,a)p.
(~) =
pimp..
\li'mpOo
EXAMPLE 1. Quadratic Reciprocity Law. Let k = Q be the rationals, m = 2. Let a, b be two relatively prime odd integers. Since (a, b)p = ±1 we have (a, b)p = (b, alp. Using Theorem 13 and its Corollary, &Ild the formula. found in Theorem 6, we obtain
(~) (~) = (a, bh(a, b)ao
= (_l)~¥(_l)·i"'rl~. The complementary result coming from Theorem 14 and Theorem 6 is, for odd a,
(2) = (2,ah=(-1)-e-. .2_1
~
EXAMPLE 2. Let k = Q(C) where t. is an odd prime, and ( a primitive l·th root of unity. Take m = t. '" .\l-l, where .\ -= 1- (. Only one prime divides e, namely A. All Poe are complex so they will give no colltribution in the power residue syulbol. Suppose a := 1 (,\2) and J3 == 1 (.\). Then by combining Theorems 10 and 13 ~get
,I
(~) (~) -1 == (a, .8). =
S<\ log ctD 1ogj1).
Noting that (~) = 1 because <" is a unit, and that
(~) (~rl = «,ah.
t. THE POWER RECIPlWClTY LAW
we also obtain
(!) == (NQ-l)/t = (!S( i jogQ») (where N is of course the absolute norm). The complementary statement for ~ is
(~) = (.\,a).\ == CisH log,.).
CHAPTER XIII
Group Extensions 1. Homomorphisms of Group ExtensioDII This section concerns the elementary facts about group extensions and homomorphisms of them. Since these things are quite well known, the treatment is sketchy throughout, many details being left to the reader. The main results are summarized in Theorems 1 and 2 at the end of the section. Let G be a group, not ne<x::ssarily finite, and let A be an abelian group. A grovp extension 0/ A by G is an exact sequence of group homomorphisms
{l} -+ A .!.. u L G -+ {I}. Then U is a group containing a normal subgroup iA which is isomorphic to A, such that the factor group U/iA is isomorphic to G. Since i is an isomorphism it is no essentiallOISS of generality to IIBSUme that i is an inclusion, and we will habitually do BO. Then we have A c: U and U / A ::::: G. Associated with such an extension there is a natural operation of G on A which can be described informally as follows: Since A is a normal subgroup of U, the elements of U operate on A by conjugation, and since we have assumed A commutative, the elements of A operate trivially on Ai therefore the factor group U/ A ::::: G operates on A. We shall write A multiplicatively, and we will denote the effect of an element (1 E G on an element a E A by afT. According to the discussion above we have
aa
=a.... = uaau;\
is any element of U such that ju" == u. To analyze the structure of an extension U one selects for each u EGa definite representative element U a E U such that ju" (1, i.e. such that uaA is the cooet of A in U corresponding to the element u E G. Then each a element 'U E U can be written uniquely in the form where
Ua
=
u = au",
(1
E G, a EA.
Indeed, the u and a are given explicitly in terms of u by (J = ju, and a For each pair of elements u, T E G, there is an element aa, .. e A such that UutLT'
= uu;1.
= aU,TuUT'
These elements a",.. :0 u.,.u,.u;; constitute what is called a. factor set. The multiplication of elements of U can be described in terms of the multiplication in A and in G, the operation of G on A, and the factor set a", ... Indeed, if u = au". and v = bu.. are any two elements of U we have UlI = au"bu.. = au"bu;luaUor = ab"uaUor = abaa",.,.uaT • 127
XIII. GROUP EXTENSIONS
The 8S5OCiativity of the multiplication of elements in U i:{I1poBes a CODdition on the factor set aQ',.,.. On the one hand we have
and on the other hand {upU" )u .. = ap,O'''"""," = (lp,O'a/XI,,.'Upa-r-'
Comparing the two expressions we find
or p
-1
-1
°tT,'f"apu,.,.a p ,(11'a p ,a
= 1.
The left hand side of this last equation is the multiplicative form of the cobound· a.ry formula for standard 2-cocha.ins (d. the discussion of H2(A) in §2 of Ch. II). Therefore, our associativity condition means simply that a.,.,T is a standard 2-cocycle of G with values in the G-module A. Working backwards, we can easily prove that, given any mode of opera.tion of G on A, and any 2-cocyde aU,'T of G in A with respect to that operation, there exists a group exteIlSion U from which they come. We define U to be the set of all symbols a * u.. , a E A, a E G, with the multiplication rule
(a * 'UO')(b,. u T )
=:
(ab"a",.,.) * ('1'0' .. ).
The associativity of this multiplication is an immediate consequence of the coboundacy, relation (oa}p,.,.,T" 1, and of the usual properties of the exponentiation a". For fixed a and {j E U the equatiom
=
aE, = {j fiG: = f3 are easily seen to be solvable in U, and it follows that U is a group, i.e. has a unit element and inverses. They are explicitly given by
1 = ai}toUl
(a
*
-1 Ua )
-1 -1 = a_,,-1 a"_I,,,a 1 ,l ... '11.0'-1.
(Various identities which arise in this connection, and which are of course direct consequences of the coboundary relation are:
"cr,Q'-lal,l
= af,l"~-l,.,..)
Putting j{a ... ucr) = a we obtain a. homomorphism of U onto G whose kernel is the !let of all elements of the form a ... '11.1. This subgroup ill isomorphic to A under the COl"respondence a .... (aal.~ .. 1£1)' We may therefore identify A with this subgroup, i.e. put a = aal,~ ... '11.11 and if at the same time we put'll." = 1 ... U", we find au" = (aal,t ... ud(l u"a U,,1L,.
* ucr) :: auiJal,,, '" '11.0' = a ... '11."
= (1 * uu}(aal.t '" '11.1) :: a"al,laO',l '" 'U = a~ * u.,. = aO'u/7 = (1 '" 1.1.. )(1", 'U = a.., .. '" '11.".. = ao-,ruQ'"t. o'
r )
Therefore, the operation of G OD A, and the 2-cocyc1e aa,.,., with which we began the construction are those derived from the group extension we have constructed.
129
1. HOMOMORPffiSMS OF GROUP EXTENSIONS
A homomorphism of one group exteDSion into another is a collection of homomorphisms I, F, and 'fl such that the diagram
(1) -
(1) -
A ~ u -..L. G -
'1 . Fl
B -..!..., V -
j
(1)
¥!
(1)
H-
is commutative. Since we are viewing i 811 an inclusion, this means in particular that f ill the restriction of F to A. Now suppose f and 'fl are given. Under what conditions does F exist, and how many F's are there? To study this question we select representatives 11.0' E U and VA E V. Supposing for the moment that an F is given, we have for arbitrary 11. = GUO' E U
Fu:: F(auO') = (Fa)(Fu a) = (fa)(FuO'). Hence to de8Ctibe F completely it is enough to give the elements FUaj and since
j(Fu a) = rpjucr:: tpf1 =j(v¥"), there are elements Ccr E B such that
FUcr
=Ccrv~.
Thus, F is described by a function (J .... Ccr from G to B. If P is to be a ~omorphism it is furthermore necessary that
F(uaau;l) = (Fucr)(la)(Fucr)-l and F(u",tI..r)
= F(Ucr)F(~).
Writing out these conditions we find that they amount to
aeA,
rrEG
where aa,... and bA,1' are the 2-cocycles 8IISOCiated with our two extensions and our selected representatives 11." and V).. To interpret these conditions, we must view B as a G-module under the operation b'l"" which is induced by the homomorphism cp: G -I- H. Then the first condition means simply that f is a G-homomorphism of A into B; and the second condition means that the 2-coeylc1es fu a ,.. and b¥".'P'I" of Gin B differ by the coboundary of the l-cochain CO' of Gin B. Thus, necessary conditions for the existence of F are that f be a G-homomorphism when G operates on B through fj), and that fa r7 ,'r and b'l"u,<pr be cohomologoWi 2-cocycles of Gin B. ConveJ"lleiy, it is easy to check that these conditions are sufficient; if Ccr is a l-cochain satisfying (*), then 8. suitable F is obtained by putting
F(au g ) = {Ja)c,,1J~. In order to investigate the number of different F's for given I and 'fl, we consider
the totality of l-cochains CIT = (Fu" )v~ which describe them. If F and pi are any two F's, then, as we have seen, the corresponding c and have the same coboundary, so their quotient
c
do- =C.,C;l = (F'ua)v;;v¥"(F'Ucr)-1 = (Fu.,)(F'Ua)-1
130
XIII. GROUP EXTENSIONS
is a l-cocycle of G in B. This I-oocycle is independent of the choice of representatives u" because upon replacing Uu by au." a E A, we find that da chang8s into (fa)d,,(fa)-l = d".. Therefore there is a natural operation' of l-cocycles d = d." on the set of homomorphisms {F}, defined by (d. F)(uO') = d,,(Fuu ); the rule d'(dF) = (d'd)F is obvious. From the above cOlllliderations it is clear that this operation is transitive, and without fixed points, i.e. if Fo is one fixed F, then the correspondence d ..... dFo is a (I-I)-correspondence between the d's and the F's. It is natural to ask what is the significance of the l-coboundaries, dq = (6b)" = b.......6- 1 , b E B, in this connection? We have «6b). F)uu
= b-1 b'l"7(Fu = b- 1 (Fu u)b. q )
Therefore, (6b)· F is obtained by following F by the inner automorphism tI __ b- 1 "b of V. We shall call two F's equivalent if they differ by such an inner automorphism. Then it follows from what we have proved that the l-dimensional cohomology classes of G in B operate tJ:ansitively and without fixed points on the equivalence
claases of F's.
. Putting all these results together and consideriDg special cases we obtain the following theorems: THEOREM 1. Let us call two group extensions U and U' of A by G equivalent if there exists an isomorphism F: U ::::: U' such that the following diagram is commutative:
A--U-G I<mrtity
!. 1 1 F
identity
A-U-G. Then the equivalence classes are in natural (I-I)-correspondence with the 2-dimensional cohomology classes of G in A, lor the variow modes 0/ operation of G on
A. THEOREM 2. Let U/A ~ G and U'/A' ~ G' be two group extensions. Then, if homomorphisms I: A -+ A' and cp: G ..... G' are given, there exists a homomorphism F: U ..... U' such that the diagram
A-U-G
It
F!
~l
A'-U'-G'
is commutative, if and only i/: 1) f is a G-homomorphism, G operating on A' through cp, and 2) 1.0: = cp.a', where 0: E IJ2(G,A) is the class of the extensiml U, and 0' E IJ2(G',A') 18 the clal;s of U'. (Here f. and cp. stand lor the homomorphisms induced by (1,/): (G,A) ..... (G,A') and ('P. 1): (G',A') ..... (G,A'), respectively.) Furthermore, if we call two such F's eqUivalent whenever they differ by an inner automorphism of U' by an element a' E A', then the group Hl(G,A') operates transitively without fixed points Oil the set of equivalence classcs. In particular, if H1(G, A') = 0, then all F's are equivalent.
2. COMMUTATORS AND TRANSFER IN CROUP EXTENSIONS
131
2. Commutators and 'ftausfer iD Group Extensions
Let G be a finite group, A a (multiplicatively written) G-module, and let U be .. group extension of A by G: l .... A.!.U.4G_l. There is no loss of generality if we view A as subgroup of U, i.e. treat i as au inclusion. Then, if 110' E U are representatives of q E G, the elements of U are of the form au,., a E A, u E G. They are multiplied according to the rules where a."r is a standard 2-cocycle of G in A whose cohomology cl88S Q' E H2(G, A) is the class of the extension U. Let U" denote the commutator subgroup of U. We wish to describe the subgroup UC n A of A. According to the corollary of Theorem 16 of Ch. V, §4, a subgroup of an abelian group A can be characterized by the set of all chiIJ'3Cters (Z-homomorphisms f: A -+ Q/Z) which vanish on it, and this is the approach we shall use. It is clear that a charact.€r J of A vanishes on UC n A jf a.nd only if it can be extended to a character F of U. Indeed, if J vanishes on n A, then J can be viewed as a character of UC A/Uc by means of the parallelogram in the accompanying diagram, and can then be extended to U, by Theorem 16 of Ch. V, §4. Conversely, any character of U vanishes on UC because Q/Z is abelian. Given a character of A we IPust therefore investigate the question, whether it can be extended to a. character F of U. This can be viewed as a question about homomorphisms of UenA group extellsions as iII. the following diagram:
uc
t
A
,!
"U~G
Fl
!
....
"
Q/Z~Q/Z-O
Since the lower extension is trivial, we see from Theorem 2 of §l that necessary and sufficient conditions for the existence of F are: (1) f must be a G-homomorphism, G operating trivially on Q/Z, and (2) J.Q' = 0 in H2(G, Q/'L), where er E H2(G,A) is the cla.ss of our extension UtA ~ G. The first condition means simply that I must vanish on lA, the submodule of A generated by the elements arr-i. To transform the second condition we use the corol.la.ry of Theorem 18 of Ch. V, §4, which states that H2{G,Q./Z) is the dual of H-3{G.Z). It follows that I.Q' is the zero element of H2(G, Q/Z). if and only if it yields the zero character of H- 3 (G,Z), i.e. if and only if (J.o)( = 0 for all ( E H-3(G, Z). We have (f.Q')t; =: J.(er() because (fa)n ... J(an) for a E A, n E Z; hence, our second condition is equiva.lent to
(2')
J.(Q'() :: 0
in other words, to the vanishing of /. on the subgroup o· H-3(G, Z) of H-l(G, A). The effect of t. on H-l(G,A) is related to the effect of J on AN (= the kernel of
132
the norm map a ...
XlII. GROUP EXTENSIONS
11" a") by the following dnnmutative diagram AN
)K.
,1
H- 1 (G,A}
I.!
(Q/Z)N ~ H-l(G,Q/Z),
in which the lower horizontal arrow is an isomorphism. We see finally that our conditions amount to the vanishing of f on the suhgroup iK- 1(ex . H- 3 (G, Z» of A. Thus we have shown that the characters of A which vanish on U C n A are exactly those vanishing on iK-1(cr. H-3(G,Z», both r.onditions being equivalent to extendability to U. It follows that the two subgroups are equal, so we have proved: THEOREM 3. Let UtA ~ G be a group extension belonging to the class ex E H2(G, A). Then, UC denoting the commutator subgroup ofU, we have
U£ n A = iK-1(ex. H- 3 (G,Z).
In other words, if fA is the subgroup of A genemted by the elements a,,-l, and AN is the kernel 0/ the rwnn map, a -+ N a = fIuEG a" I then fA c
ue n A CAN,
and the factor group (Ue n A)/IA co1Tfsponds to isomorphism AN/fA ~ H-l(G,A).
Q.
H-3(G,Z) under the natural
It should be remarked that the two· inclusions fA c UCnA c AN can be proved directly. The first is obvious; for any a E A, u E G we have 0,,-1 = 'U"au;l a-1 E n A. The second is best understood by means of the group theoretical transfer from U to A which we will now investigate. It is a homomorphism
uc
VU,A: (UIU e ) .... (AlAe) == A
mapping the factor commuta.tor group of U into that of A, the latter being A itself since A is abelian. It is defined by
VU,A(UU C )
==
II u"uu;,l. "EG
where, for each given U E U, and each q E G, a = j(uau) is the unique element of G such that u"uu;,l E A. Being multiplicative, V is of course determined by its effect on A, and on the representatives u T • For these special cases we obtain
VU,A(aU C )
= II u.. au;l =
and VU,A(U ... U")
=
n
a"
= Na,
"
n u,,'UTu;;; = n
au, .. ,
6"I;G
"
where au,.. is the 2-cocycJe associated with the representatives 1£". Thus, in particular, the effect of V on A is that of the norm, and this explains the inclusion U C n A c AN, because V annihilates UC • V carries the representative tiT into the image of 1" under the Nakayama map 1" --+ fIT G" .... which we have studied in §5 of Ch. V, this being tbe other interpretation of the Nakayama map to which we alluded. there. Since the norm and the Nakayama map both have values in AG, we
• 2. COMMUTATORS
A.Nb
TRANSFER IN GROUP BXTENSIONS
133
see that the image of UIU e under V is contained in AG. This can aIoo be seen directly, for it is a special consequence of the following fact about transfer: If W is any subgroup of finite index in U, then
Vu.uWu-.(u')
= u(Vu,w u')u- 1
for all u, u' E U. Recalling that the Nakayama map can be used to compute the cup product a·(,. we see that we can a.bstract from the above considerations the following statement which is independent ()f the choice of representatives Ita. THEOREM 4. Let U / A ~ G be a group extension belonging to the class W(G,A). Then the following diagram is commutative 1
0'
E
where the right-hand ~rtical arrow denotes multiplication b1l 0', and N, i, and 1 denote the maps induced in the obviou.s way by N: A -+ NA, and by A ~ U L, G. COROLLARY. Let UI A ~ G be'a group extension belonging to the class W(G, A). Consider the following three homomorphisms:
Q
E
V: U/U e ...... AG Q-3: H- 3 (G,
Z)
-+ H-l(G, A)
Q-2: H-2 (G, Z) - t ~(G, A).
We contend tMt there exist exact sequences 0-+ H- 1 (G,A)!ImQ_3 -+ KerV -+ KerQ_2 -+ a 0-+ A G 11m V -+ HO(G, A)/IwQ_2 -+ O.
In particular, V 1S an isomorphism into if and only if Q-3 is onto and isomorphism into; V is onto if and only if 0'-2 is onto.
Q-2
is an
PROOF. Notice that the horizontal rows of the diagram of Theorem 4 are exact. Let us augment' that diagram by putting zeros above and below in the following
IThe diagram gives another interpretation of the Nakayama map 01, as the map indru;ed by tile Verlae;erung in the group extension corresponding to the class cr. In that version it is implicit in the proof of the Principal Idea.! Theorem due to Artin and Iyanaga which we give in Section 4 of this chapter. It was publiBhed in S. lyanaga (14). Iyanaga was in Hamburg at the time and states that the greater part of the proof is due to Artin, who geuerously let him publiah it under
his own name.
134
Xlll. GROUP EXTENSIONS
1DaIl1ler:
l ! 1
0-0--0-0--0
! ! !
0--*--"'--*--0
v!
R!
a_a!
0--*--*--*-0
! ! ! ! l !
"
0--0-0--0--0
Viewing the columns as chain complexes, we obtain an exact homology lII!Cluence, and the essential part of this sequence is:
o -+ Ker N =
AN I(uc iI A) -+ Ker V -+ Kera_2 -+
-+
0 - AG IImV
-+
ftl(G,A)/Ima_2 ...... 0:
Using the description of Uc n A given by Theorem 3, we see Aftll(UC H-l(G,A)/Ima:_3' This conclud~ the proof of the corollary.
3. The Akizuki-Witt Map
tI:
n A)
~
0
H2(G, A) -+ H2(G/H, AH)
Let A be a multiplicatively written G-module and let U be a group extension of A by G:
l-A~ULG-+l. If H eGis any subgroup and we put W = 1 (H), then W is obviously an
r
extension of A by H:
1 -+ A ~ W We might caJ1 W a Stlbe:.ctension of U.
J... H -+ 1.
If IX E H2 (G, A) is the 2-dimensional class describing U, then reBG ,H 0: E H2(H, A) is that describing W. Indeed, if U a , (7 E G, are representatives of G in U, then the elements Uh, h E H, will serve as representatives of H in W. Hence by restricting the 2-cocyc1e aa, .. =: UaU-ru;; to H, we obtain a 2-cocyc1e describing
W. The cosets of H in G correspond to the cosets of W in U, and if H is normal in G we ha.ve U/W ~ G I H. Therefore we can form the extension
l-W ~U --+GIH -+ 1. However to obta.in an extension of the type we ha.ve been considering in which the subgroup is a.belian, we must divide out by the commutator group of W:
1_ W/W c -+ U/W c _ G/H -+ 1.
1311
3. THE AKIZUKl-Wl'l"l' MAP ." Ha(G,A) --+ HOCG/H,AH)
This last extension may be called u. factor extension of U. Perhaps a lattice diagram will help in visualizing these various groups. Now let V VW,A denote the group theoretical transfer from W into A. It maps W/Wc into AH, and is eaaily seen to be a (GIR)-homomorphism. Consequently it induces a homomorphism V. of the (GIR, Wlwe) cohomology into the (GIH,AH) cohomology. (Needless to say, V. is not the cohomological transfer; it is simply an induced map.) The 2dimensional class {3 E H2 (G I H, W Iwe) associated with the factor extension is carried into a certain 2-dimensional class V•.8 E H2(GIH,AH). Obviously V.f1 depends only on the {l} equivalence class of the original extension UI A ~ G, that is to say, on the element Oi E H2(G, A) which characterizes the extension U, We have therefore constructed a map Q - V.f1 which we denote by
=
we
/
VG,(G/H): H2(G,A)
-+
H2(GIH,AH).
To review the definition of Va.(GIH): Given Q E H2(G, A) one constructs an extension UIA ~ G belonging to Q, goes to the factor extension (uIWC)/(WIWC) ~ G I H, and then takes the image of its 2-dimensional class (J under the homomorphism induced by the (GIH)-homomorphism V: WIWc -+ AH; the re.rult is VG,(G/H)Q. Following this prescription we can easily derive an explicit formula for the computation of VC,(C/H) in terms of standard 2-cocycles. Let a",T be a 2c:ocyde in the class of Q. Then a suitable group extension U consists of the elements au", a E A, U E G, with the multiplication rules u.,.a = aU'll" The subgroup W consists of all aUh with h E H. In order to compute a 2-cocycle br,8 E WIWc (for r,s E:' GIH) which describes the factor extension, we first select representatives f E G of the elements rEG/H. We may then use the elements Ur, r E GIH, as representatives of the cosets of Win U, and the corresponding 2-cocycle br" is determined by the equations U,U.
= br,a'tln (mod we),
Introducing the abbreviation 'Y == r irs 1 E H and comparing the equations ut Uj \
....".1 •.•
=llr, jUri
••.•••
u..,Ur,- = a.."n 'Ur.
1.
we find
bT ,. = ar,o a:;:}.. u.., (mod WC). Now we must transfer these eJements from W to A. Using the rules derived in the discussion preceding Theorem 4 of §2, we find
Vb.-,. = NH(ar,ia~,:")
IT ah,.., = II a~"a;.~ah,.,.. h
THEOREM
h
5. Let G be a finite group, H a normal subgroup, and let the elements = 1'H of H in G. Then the cohomology
f E G be representatives of the cosetB r map
136
XUI. GROUP EXTENSIONS
is induced b" the cocycle map 1lcr.1' -. (va)T •• ' which is (va)r .• := NH(o.r.i a:;.h) .
where "'f = "Yr.. = COROLLARY.
de.flne4 by
IT all.'Y' "
rsrs- 1 E H.2
The cohomology map v is multiplicative.
6. For a E lfl(G,A) we have
THEOREM
in!
(G/Hl.a
Va (GI9)Q •
=: Q(H;l).
PRoOF. If (J E G, we denote by it the representative of the coset in which lies; i.e. it = aH. Then inf vo: is represented by the 2-cocycle
(infva)",1'
=:
(VU)"H,1'H
q
= IT u~.f' a;~l7f-l.iR'all.iH'if'I'"'l. II
Using the coboundary relation to pull the his down from the exponents we obtain a. formula in which two pair of terms cancel:
II aM." a\k. all.u a;;-;f'll'P'-I.1n'~''" at~f,",-l f,..af'~-l. h
Since it is in the same coset of H as r1 is, and since H is normal, we can replace the product over hO in the first factor by the product over (Th. Similarly, we can replace oof(f'f-l by h in the third uncancelled term, because iTf7ff- 1 E H. This yields:
(in! lIa)O'.1' Now the l-cochain Ca
== nil ah,a
=11 aah,f ah.u a;;~. h
has the coboundary
(OC)a,1'
=
na~.f ah.~
ah.it.
h
Dividing our 2-cocycle by this coboundary we obtain the following equivalent 2cocycle in the class inI va
e.,.,1'
-a = II aah.f ah,f" h
It consists of two factors of the ooboundary (6a);.~,i" and since this coboundary is 1 we can write
e"..,. =
n
-1 au,lIf aa,II
"
-1 == IT aa.1'h aa,II' II
2Multiplying our forn"ua for (lIa),. •• by 1 = It.EH(Sa),.."I.i'i OIH! finds another expr-.ion; (vu) .... NHQf',i TI" ah ~. In this form our map 11 was discovered independently by Witt and "h,!". Akilluki ill 1935 (see (31) and [1)}. Both these Quthors proved oW' Theorem 6. If the inBatioQreltriction sequence is exact in dimen.ion 2, e.g. if '}tl(H, A) = {OJ, then it is clear that there exists a unique map like 11 satisfying Theorem 6. and Witt's motivation was to find aD explicit fonnula. for it in terms of cocycles. Akizuki proved a result relating" to the N~a.ma map whlCll _ can expres5&S H '" (a.<;,,)NG/HAH. He used thiswitb H = G' to show that Nakayama's theorem about the injectivity of his map for abelian G implies with the 58Ille hypothesis that for arbitrary G the kernel is 0' (ef. footnote at the end of Section 2 of the Chapter "Preliminaries").
=
oo:·,..
4. SPLITTING MODULES AND THE PRINCIPAL IDEAL THEOREM
131
Using the coboundary relation to put the u in the first factor back in the exponent we find
e.....,.
n
= a;:ha.,.,.,,,a,,,.,.a;;,1 =
" (IT " -1
aT,haaf',hQ(1.h
)-1
(H:l) • AatT •
h
Since the quantity in the parentheses is the coboundary of the l-cochaiu e'" = fh a",h, we see finally "" a(H:l) e a ,1 ' · ~,.,.
In terms of cohomology classes this means inf va: = a:(H:l), as contended.
0
4. Splitting Modules and the Principal Ideal Theorem Let G be a group and A a. G-module. Let a: E Hr(G, A) be a cohomology clrulS of G in A. A splitting module for the class a: is a G-module B containing A, such that the image of Q under the homomorphism induced by the inclusion of A in B is zero. In other words, such that a cocycle in the class of a: becomes a cobounrlary when it is viewed in the bigger module B. Splitting modules exist for cohomology classes of all dimensions, but we will content ourselveti with a discussion of a special case, namely 8 canonical type of splitting module for 2-dimensional classes. This special case is the most important one becaUl:le of its intimate relationship with group extensions. Let a",.,. be a standard 2-cocycle of G in A, and for convenience assume a1,.,. = a",l :: O. (Any 2-dimensional clasS contains such a 2-cocycle; from the point of view of group extensions, the normalization al,r :: a."l = 0 corresponds to choosing the identity element of the extension U as the representative of the identity element of the factor group G.) We wish now to construct a G-module B containing A, in which there is a l-cocha.i.n x" such that
(6x)" ..,. = Ux'" - %".,. + z" = a",r' As the additive group of B we take the direct sum of A and of a Z.free module whose basis consists of symbols Zr, one for each element T E G except the identity, i.e. we put B = A+ 2:Zx.,. (direct sum). T~l
For nota.tional convenience we define Xl to be the zero element of B. We extend the operation of G from A to B in jUl:lt such a way that the 2-cocycle a."T becomes the coboundary of the l-cochain X r • This is done by defining for u, T "e G. In order to justify this definition, we discuss the general question of Z-homomorphisrns of B into an arbitrary Z-module C. A Z-homomorphism f: B -> C is obviously determined by its restriction fA to A, and by the values !(x r ) for T '" 1. Conversely, given any Z-homomorphism g: A -+ C and given arbitrary elements Cr E C for 7' :F- 1, there exists a homomorphism f: B -+ C such that fA = 9 and J(x T ) = ~. Now taking C to be B itself we can therefore associate with each q E G the unique homomorphism q: B --+ B such that (1A. is the originally given effect of u on A, and such that U:£.,. is given by equation (.) for
138
Xlll. GROUP EXTENSIONS
E 1. The equation (*) then holds automaticaUy for -r = 1 unit element of G acts as identity on A because
T
1x.. = :z;,. -
Xl
befause Go,I =O.
The
+ al,.. = x ...
= 0 and al.
T = 0.) The associative law for our operation follows from the coboundary relation for the 2-cocyc1e a..,.. , namely:
(&call that:z;]
p(u:z;.. ) = p(xa.. -:ta + a..,T) = Z(Jt1'.. - zp + Q",<7'T - zpa + xp - ap,(I + pa..,.. =ZPITT - x(Jt1' + (pa",,. + a p,.... - ap,a)
=- z(/><7)'" -
x(Jt1'
+GpIT,"
=- (pu)x ... We have shown tha.t the operation of G on B has all the required properties; B is a G-module. The mll.in properties of the splitting module B are stated in the next several lemmas. These lemmas will be used later to prove the principal ideal theorem. LEMMA 1. The factor module B/A is G-isomorphic to the augmentation idem 1. More precisely. the following is an exact sequence of G-homomorphisms
,
.
O..... A ..... B.41--+0, where i is the inclusion, and i is defined by jA
= 0, and by jx.. =(-r-l) jorT f= 1.
PROOF. Since the elements .,. - 1 for .,. I- 1 are a Z-basifl for I, and since the elements x.,. are a Z-basis for B/A, we see that j is a Z-bomomorphism of B onto I with kernel A, so the sequence is exact. To see that j is a G-homomorplili;m we DOte that the formula jx.,. =: T - 1 holds for all T, because it holds automatically for.,. = 1, and consequently
juxT = j(x a .. - Xa + aa,.. ) (uT-1) - (u-1) =ujx.,..
=
=
UT -0'
=O'(-r-l)
o
LEMMA 2. Let l' be an element of the group ring 0/ G. Then 1'B C A if and only if l' is a multiple of the trace, i.e. l' "" e(E" u) eS for some integer e.
=
PROOF. By Lemme. 1 we have B/A ~ 1. Hence 1'B C A # 1'1 =: 0 # "Y{T - 1) = 0 for all .,. E G. Writing l' = L:eau we see that thls means all 0 coefficients e.. are equal.
Now let us introduce the group extension U/A ~ G constructed with the 2cocycle aa,... When A occurs in this connection, we will write it multiplicatively, though we still preserve the additive way of writing A when we view it as a submodule of the splitting module B. Thus, U consists of the elements aUa with the multiplication rule
auvbu.r In particular, we have "'lUl = al,l'Ul al,l is the identity element of A.
=obIT a..,.,."'IT..'
= U1> hence 141 ::: 1, because we have assumed
4. SPLlT'I'ING MODULES AND THE PRlNCII"AL IDEAL THEOREM
139
The abelian group B/IB is isomorphic to the factor wmmutator U/Uc of U under the correspondence
LEMMA 3.
group
GUIIUc ~
a+x II + lB.
PROOF. Define a map log: U ~ BIIB by loga'UII the log of the product is the sum of the logs. Indeed
== a+:c..
(mod 1B). Then
rogauab'U.. = log(abuall,,.'UII.. ) where88,
Zogaull +logbu.. == a+x", +6+x.. , and subtracting these two expressions we obtain an element of IB, namely
ub - b+ (XI1'l'
- X'"
+ a"" .. ) -
x" = (O'-I)b + (0' - 1)x...
Since B/IB is abelian, our log homomorphism induces a homomorphism log: U/U c -- B/IB. On the other hand, we can define a homomorphism exp: B {
= 1.
U/UC bY putting
= aU e, aEA exp x .. = u ..U", 1';:' 1. expa
As usual, the second of these formulas holds for
u..
-4
'7
= 1 as well becauae :1:,. = 0 and
This homomorphism exp Vll.l1ishes on I B, because exp(u -1)0 = au-IUC = u".au;l(l-lUC =
and
exp(u -1)x.. = exp(xa.. -
X(7
uc
+ a..... - x .. }
== ~ ;; 1 (mod U"). 1£"1.1,,
Consequently, exp induces a homomorphism exJ): B / I B - U/ UC. A glance at the definitions shows that log and exp are mutually inverse maps; hence they are both
isomorphisms onto. This concludes the proof.
0
LEMMA 4. The transfer map VU,A: U/U c -4 A correspond.:!, under the isomorphism U/U" ~ B/IB, to the map S: B/IB - A which is induced. by the trace map S: B -+ B. (Notice that S carries B iuto A (cf. Lemma 1) and S vanishes ou lB.)
PROOF. For the representatives 1.1" we have
naa...
(viewing A in U)
=Ea. ..,.
(viewing A in B)
V(u,-U C ) =
a
fT
= LO'x", -
.
¢...,. +¢..
:: LUX.,.:: Sx.,.:: Slog 1.1.. , t1
XUl. GROUP EXTENSIONS
and for a E A we have
V(aU C ) = Na (in U) =Sa (in B)
= i(a+ IB) = sloga. Since any element of U is of the form au." this concludes the proof.
o
THEOREM (Principal Ideal Theorem). Let U be a group whose commutator Then the transfer map Vu,u.: U IUc --+ U C I (UC)C is the zero map. IlUbgroup U C is of finite index and is finitely generote.d.
PROOF. Dividing out by (UC)C does not affect the transfer map, so it is no loss of generality to lI$ume (ucy = 1. i.e. UC abelian. Furthermore, replacing UC by an arbitrary abelillll subgroup A containing UC, we see that it is enough to prove:
THEOREM 7'. Let U be a group whose commtdator subgroup U C is of finite index and is finitely generated. Suppose that A is an abelian subgroup of U containing UC. Then, if e = (A : UC), e times the transfer map VU,A: U IUc --+ A is the zero map.
PROOF. SInce A :) UC,A is normal in U and the factor group G = UIA is a finite abelian group. Let its order be n (U : A). Then (U : UC) = (U : A)(A : UC) = ne. Let B be a splitting module for Ii 2-cocycle coming from the extension UIA ~ G, as described in the paragraphs above. Then B is finitely genera.ted .over Z. because BIA R: I is a free Z-module on (n -I)-generators, Alue a. finite group, and UC is assumed to be finitely generated. By Lemma 3, the factor group B IIBis isomorphic to UIU c , and 'is therefore a finite abelian group of order ne. Let hi, b2 I • • • I bm be elements of B which are representatives, mod I B, of a basis for BIIB. Let ej be the period of bi mod lB. Then ne = ele2 ... em. Let bm+l, bm+2,"" b. be generators for I B (as subgroup of a finitely generated abelian group, B,IB is finitely generated). Put f:m+l = em+2 e. ;;: 1. Then we have achleved the following three things: 1) The elements bl , ~,". lb. genera.te B.
=
= '" =
.:,
2) e,b; E IB, i
3) n:=l e;
= 1,2, ... ,8 •
= net
=
From 1) we have B ~j=l Zbjl hence IB that there exist elements OJ; e I such that
=- ~j=llb;.
Therefore, by 2) we see
s
cib;
= L: 6;jb j,
i = 1,2, ... , •.
j=1
Putting "Yij :: e;a;i - Ojj, where O;j is the Kronecker delta, we obtain an 8 x • matrix (1';j) of elements of the group ring r of G such that B
(*)
E"Yi,ibj=O,
i=1,2, ... ".
j=l
Since G is an abelian group, the group ring r is commutative; hence the notion of a determinant of a square matrix with elements in r makes sense. If 6,;) is the matrix whose elements are the cofactors of the elements of the ma.trix ('Yii) we have
L ;YiJe"Y;j = 'Y • 5"i i
t
4. SPLITTING MODULES AND THE PRINCIPAL IDEAL THEOREM
141
where I = det "fi; and 0 ill the Kronecker delta. Multiplying (.) by ::rill: and summing
over i we obtain 'fbj =0 for all j. Since the bj generate 8 it follows that 'Y annihilates 8, and a fortiori, 'YB C A. Hence, by Lemma 2, I is a multiple of the trace, say"Y = tS. To determine the value of t, it is enough to compute the image of'Y under the ring homomorphism e: r ~ Z which is defined by <.u = 1 for all 0' E G, and whose kernel is I. On tile one band we have f."{ e(tS) = t(eS} = tn. On the other hand, going back to the definition of "Y, we have
=
f.'Y = edet('Yij)
= det(e'Yij) = det(eic)ii)
&
== II eo =en ;=1
becaUBe 'Yi = etOij - (Ji; == eibij (mod I). Comparison of the two equations above shows t = e, hence 'Y :;; eS. This proves the theorem, because according to Lemma 4, the transfer map VU,A is induced by the trace map S on B and, as we have already remarked, 'Y annihilates B.
o
CHAPTER XIV
Abstract Class Field Theory 1. Formations In this section we will introduce an abstract algebraic object called a formation. In order to motivate the definition and terminology we first consider an important special case arising from Qrdinary galois theory. Let k be a.ny field and let n be the separable part of the algebraic closure of k. Let G be the galois group of n over k, i.e. the group of all automorphisms of {} which are identity on k. (Of course G is usually infinite.) Let E be the Bet of all finite extensions of k in {}; we denote them by F,E, K, L etc. For each FEE, let G F denote the subgroup of G corresponding to F, Le. consisting of all automorphisms of {} which are identity on F. Then G can be made into a topological group by taking the family of subgroup> {G F } FeI; as a fundamental system of neighborhoods of the identity, and it follows then from galois theory that every open subgroup of Gis of the form G F for some FEE. Let A denote the multiplicative group of {}, viewed as G-module. For each finite extension F of k, let AF denote the multiplicative group of F. Then from galoiS theory we know that AF = AGI', i.e. A F , is the submodule of A consisting of all elements which are left fixed by GF. If F and K are two finite extensions of k, we have F C K if and only if GF::> GK, in which case K/F is a finite extension of degree [K: F] (GF: Gk), and AF c AK. The extension K/ F is normal if and only if GK is a normal subgroup of GF, and then its galois group GK / F , is isomorphic to the factor group GF/GK' In thi::; situation the finite galois group GKIF, operates on the multiplicative group AK of the normal extension Ki hence we have cohomology groups Hr(GK/F,AK)' It is this type of cohomology group in which we are Interested. However, if the fields K and F are finite algebraic number fields, the galois group GK/F operates not only on the multiplicative group of K, but also on the ideIe group of K, and on the idele class group of K i and in global class field theory it is essential to study the cohomology of all three situations. The ideles of the various finite algebraic number fields F can be assembled into one big group, the idete group of the field of all algebra.ic numbers, and the same goes for the idele cla.<;;ses. Thus, we will be able to treat each of these situations as special cases of the following abstraction:
=
DEFINITION 1. A formation (G, {GF}i A) consists of:
I) A group G, usually infinite, together with a non-empty indexed family {GF } FEE of subgroups of G satisfying the following conditions:
a.) Each member of the family {GF} is of finite index in G. b) Each subgroup of G which contains a member of the family {GF } also belongs to the family. 143
144
XlV. ABSTRACT CLASS FIELD THEORY
c} The interaection of two members of the family {G F} also belongs to the family. d) Any conjugate of a member of the family {G p } is also a member of tbe
family. e) The intersection of all members of the family {GF} is the identity:
n
GF=l.
FEE
2) A G-module A such that A = UFEE AGp, in other words, such that every element of A is left fixed by some member of the family {G F }. Notice th&t in this definition the symbol F has been reduced to the humble status of a mere index. Logically it would be more reasonable to drop it altogether and use the subgroups GF themselves 88 indices. However we retain the F for psychological purposes, and in order that our terminology will fit the a.pplications. The requirements concerning the family of subgroups {GF} could have been lJta.ted more briefly by our saying: G is a Hausdorff topological group in which there is a fundamental system of neighborhoods of the identity consisting of subgroups of finite index, and {G F } is the family of open subgroups of G. However, until much later there will be no need to consider G seriously as a topological group, so we prefer the explicit algebraiC description of the properties of the family {GF} to the more succinct topological one. For the time being, the only hint of the topological aspect Mil be our use of the convenient term open .subgroup as a synonym for ~a member of the family {GF}". . The terminology of formations which we now outline can be best understood if one keeps in mind the special example discussed in the opening paragraphs of this section, in which G is an infinite galois group and A is the multiplicative group of the infinite normal extension. If (G, {GF }, A) is a formation, we call G the galois group of the /Ormation, and we call A the formation module. The indices F are referred to as field:J. Corresponding to each field F we define a. submodule AF of A by Ap = A GF. We call these submodules Ap the various levels of the formation and we say that AI" is the F-Ievel. If F and K are a pair of fields, we write F c K whenever Gp :J Gx. and Bay then that F is a sub/ield of K, or K an extension of F. When this is the case, the F-level is contained in the K-Ieve1: Ap C AK. A pair of fields such that F C K is said to determine a layer of the formation, and this layer is referred to by the symbol K / F. AF is called the grotlnd level, and AK the top leve~ of the la.yer. The index (G F : GI'd, which is finite by assumption la) of the definition of a formation, is called the degree of the layer K / F and is denoted by [K : FJ. When GK is a normal subgroup of GF the layer KIF is called a normal layer. Then the finite factor group Gp/GK operates naturally on the top level of the layer, i.e. on AK = AG..-. This factor group Gpo/Gx is called the galois group of the normal layer and is denoted by GK/P' We have clearly
AF == AGl" = (AGK)GFIGK = A~KI"; in other words, the ground level of a. normal layer consists exactly of all elements of the top level whiell are left fixed by the operation of the galois group of the layer . .AB in galois theory, a. normalla.yer is called solvable, a.belian, or cyclic, if the galois group is solvable, a.belian, or cyclic.
145
1. FORMATIONS
By the cohomology groups 0/ a normal layer K / F we IQe&Il those associated with the operation of the galois group GKIF, on the top level ~-K' They are denoted briefly by W(K/F). ThWi we have by definition
Hr(KIF)
= W(GK/F,AK) == W(GF/GK,AGK).
Of course these cohomology groups are those of finite groups GKI F, in spite of the fact that the galois group G of the whole formation is usually infinite. Therefore a.1I of the theory which we have developed in the preceding chapters applies to them. In particular, we have at our disposal all the natural homomorphisms. We now list these, giving first the old notation, and following it by a new abbreviated notation which we are hereby introducing: Transfer1 and Restriction: If FeE have
c
K and KIF is normal, then we
1'
or, ¥CK/E),(K/F): If"(K/E) -+ B"(K/F);
and or, re5(K/F),(KIE): W(K/ F) -+ W(K/ E).
Conjugation: If KIF is normal and u € G, then we have
u.: If"(Gp/GK,A(CK»)
-+
H'(G'F/G'k,uA(GKl)
or,
u.: If"(K/ F) ..... H'(KV I F"), where KCT is defined by the equation GKa = G'k = rJGKu- 1. (This is pennissiple by assumption ld) of the definition of a formation, which states that the family of subgroups {GF } is closed under conjugation.) Inflation: If F eKe L and LI F and K/ F are normal, then we have for r~l
or,
inf
(KIF),(L/F)
: W(KjF) ..... H"(L/F).
The purpOllC of a formation is to facilitate the study of the cohomology groups of its normal layers. By putting these la.yers all together into one big object, the formation, we achieve the fullest possible freedom in studying their interrelationships, so that we can glean information about "difficult" layers from our knowledge of simple layers. A special notational device which we will employ is that of using K and L to denote the top field in normal layers and using E to denote the top field of nonnormal layers. For example, K / F and L / E are automatically understood to be normal, whereas E(F and EdE are not necessarily normal. Most features of ordinary galois theory carry over to abstract forma.tions. Two important examples of this general fact are furnished by the following propositions.
146
XlV. ABSTRACT CLASS FIELD THEORY
PROPOSITION 1. GWen any finite 8ft of ",yers over the 8ame ground field, Ell F, E2/ F, ... , EmlF, there exists a normal layer K / F containing all of them, i.e. such that F c ~ c K for i = 1, ... ,m. PROOF. Each of the subgroups G E, is of finite index in G F and has therefore
only a finite number of distinct conjugates GE• in GF. These conjugates are open subgroups of G by assumption d) of definition 1. Hence, by assumption c) of that same definition, their intersection,
is also open, i.e. is of the form GK for some field K. Since G K is obviously normal in GF, and is contained in GE , for each i, it follows that KIF is normal, and K::> ~ for each i, as contended. 0 PROPOSITION 2. Every subgroup of the galois group G K1F of a norma/layer
KIF is of the form GK / E for 80me E,F C E
c K.
PROOF. We have GK/F = GFIGK' Hence any subgroup is of the form HIG K for some group H such that G F ::> H ::;l G K. Since H ::;l G K there is a field E such that H = G E, by as:>umption b) of Definitioo 1. Then FeE c K and our original subgroup of GKIF is GE/GK == GK / E • 0
One difference between formation theory and galois theory proper is that in a formation one does not assume that the corre5pondence GK ..... AK
= AGI<
is one-tCH>ne. It is perfectly possible, according to our definition, that the galois group of a formation operates trivially on the formation module, in which case all levels are the formation module itself. The point is that the one-to-oneness of the correspondence plays no role whatsoever in the cohomologicaJ considerations, 80 there is no point in assuming it.
2. Field Formations. The Brauer Groups The main fact about the cohomology of a finite galois group opera.ting on the multiplicative group of a normal field extension is that the I-dimensional coh()mology group is trivial. For the convenience of the reader we insert here a formal statement and proof of this fact: PROPOSITION 3. Let K be a field. Let AK be the multiplicative group of K, and let GKIF be a finite group of automorphisms of K. Then Hl(GKIF,AK) == O.
PROOF.
Let
I: GK/F -. AK be a crossed homomorphism of GK/F, in A K : /(17r) = (f(r))" 1(17) U,T' e GK/F.
Select b e K such that the element
a
=
L:
b" I(T)
..eG",,,
is not zero, i.e. a E AK. The existence of such a b follows from the theorem on the independence of isomorphisms of a field (see, e.g., [3, Corollary in Section Fl, or [4,
2. FIELD FORMATIONS. THE BRAUER GROUPS
147
Chap. V, § 7, Theorem 3) or almost any modern textbook on Galois theory). Then the equations
a" f(u} =
E b"T(f(-r»" f(u) =L T
show that feu)
= al-II, i.e. f
.
baT f(UT)
"""
T
is a principal crossed homomorphism.
o
It is the vanishing of the 1-dimensional groups which gives the ordinary galois cohomology theory its peculiar flavor. It is therefore of interest to study abstract formations whose normal layers have the corresponding property, i.e. which satisfy
=
AXIOM I. Hi(K/F) 0 for each nonnallayer KIF. We will call a formation a field formation if it satisfies this axiom. All the formations considered in class field theory are field formations, although in the case of the formation of idhle classes in the global theory this fact lies very deep.
In order to establish wom I in the more difficult cases, it is necessary to knOw that it is a consequence of the seemingly weaker AXIOM I'. Hl(KI F) = 0 whenever KIF is a. cyclic layer of prime degree.
The fact that Axiom I' implies axiom I is a special case of the following lemma. LEMMA 1. Let
(G, A) be a formation in which all inflation-restriction sequences
Hr(KIF) ~W(LIF) ~W{LIK) are exact for a certain positive dimension r. Then in order to prove that a divisiflility of the form
(*)
order of W(L/F)
I [L: FJ"
holds for all normal layers L/ F, it is enou.gh to show that it holds for cyclic layers 0/ prime degree. .
=
=
REMARK. From the ca.se r 1, /.I 0 of this lemma., we see that Axiom I' implies Axiom I, because the inflation-restriction sequence is always exact in dimension 1. We will use the case r = 2, II = 1 in the next section.
PROOF. Let L/ F be a given normal layer for which we wish to prove the divisibility (*). By induction on the degree of the layers, we may assume (*) holds for all layers of degree < IL : Fl. Case 1. [L: F] is a prime power pt, We may assume t d1: 2; otherwise (*) is true by assumption. The ga.lois group of the layer, G LIF = GF/GL is of order pt, and therefore contains a proper normal subgroup different from identity, beca.use p-groups are solvable. This subgroup is of the form Gl./ K where F eKe L, by Prop. 2 of §1. Using the exactness of the inflation-restriction sequence we find
hr(L/ F) I h,.(K/ F) . hr(L/ K), where h,. denotes "order of Hrn. Since the layers KIF and L I K are of lower degree than L/F, (*) holds for them, by our inductive assumption, i.e. hr(K/F) I [K:
FJ"
and
hr(L/K)
IlL: KJ"·
Combining these divisibilities with the above and using the obvious multiplicativity of the degrees, [L: FJ = (L : KJIK : FI, we obtain
h,.(L/ F) I [L : F]"
146
XIV. ABSTRACT CLASS FIELD THEORY
which is the divisibility we are trying to prove.
Case 2. IL: F] is not a prime power. For each prime p dividing IL : Fj, let GLIE be a p-sylow subgroup of G L / F . Then, using the fact that the restriction map from a group to a p-sylow subgroup is an isomorphism into on the p-prirnary part of the cohomology groups, we find
h,.(LjF)
I 11 hr (L/El')' p
Since IL : FJ is not a prime power, all the layers L I Ep corresponding to the Sylow subgroups are of lower degree than L/ F; hence (*) holds for them by our inductive assumption, i.e. hr(LIEp) IlL: El']" multiplying over p and using TIp[L : Ep) !L : FI (the order of a group is the product of the orders of the p-sylow subgroups for the various p) we obtain the desired result. This completes the proof of Lemma 1. 0
=
In the proof of Lemma 1 the exactness of the inflation-restriction lICQueDCe is used only in the form of the consequence
h,.(LIF) I h,.(KIF)hr(LIF). This consequence could a1so be obtained from an exact lICQuence in the other clil'eC1;ion such as the special O-dimensional sequence
HO(KIF)
t-
HO(L/F) ~ HO(L/K)
mentioned at the beginning of §5 of Ch. N, and from this we get a similar sta.tement in dimension O. However we only mention this in passing, since we will never require the result. Throughout the remainder of this section we will assume that our formation is a field formation. The most important consequence of this assumption is the exactness of the inflation-restriction sequence in dimension 2. If F eKe L with KIF and LIF normal, then the sequence
0-+ H2(K/F) ~ H2(L/F)!!!. H2(L/K} is exact. This follows either from the Hochscbild-8erre spectral sequence, or by "dimensiona shifting", using the fact. that the I-diwensional cohomology group of the lay.er L/ K correllponding to the subgroup GL / K of GL / F , is trivial in our field formation. One consequence of this exactness is that for F eKe L, the map
H 2 (K/F) ~ H2(LIF) is an isomorphism into. We may therefore identify H2(K/F) with its image in H2{LIF), and view the inflation as an inclusion. If Pc K c L eM, then, by the transitivity of inflation we see that the direct imbedding of H2(KIF) in H2(MI F) is the same as the imbedding via H2 (L I F). Taking the injective limit we obtain a group which we xna.y denote by H 2 (*IF) limK H2(K/F), and which has the following properties: 1) For each normal layer KIF over the ground field F, H2(K/F) is a subgroup
=
Of H2(*/F).
2. FIELD FORMATIONS. THE BRAUER GROUPS
14.9
2) If F eKe L, then the subgroup H2(K/F) is contained in the subgroup Jf2(L/ F), and the inclusion map is the inflation, inf(K/F).(L/F)' 3) H 2 (*/F) is the union of these subgroups. The group H 2 (*/F) = UKH2(K/F) is called the Brauer group otler F ofthe field formation (G, A). It was first constructed by Richard Brauer in the special care of the fonnations in which the formation module is the multiplicative group of the separable part of the algebraic closure of a ground field F. He viewed it as the group of classes of simple algebras with center F, the multiplication of algebra classes being induced by the Kronecker product of representative algebras. The connection with the 2-dimensional cohomology groups arises from the possibility of writing simple algebras explicitly as crossed products, in which the multiplication rules in the algebra appear in the fonn of a standard 2-cocycle. This procedure had. been introduced by Wedderburn and Dickson in special cases, using non·standard 2-cocyles, 8.Ild was first "b'tandardized", and thereby systematized, by Brauer. To review the definition of the Brauer group over F: Any element of it is equal to an element a E F{l(K/F) If2(G K IF,AK) for some normal layer KIF. If 0.' E H2(K'IF) belong!! to another such la.yer. then a' is equal to Q in the Brauer group if and only if
=
inf
(K/F),{L/F)
u: =
inf
(K'IF).(LIF)
ex'
for a normal layer LjF such that L :::l K and L :::l K'. (Of course if this equality holds for one such L, it holds for all.) The sum of 0. and a' in the Brauer group is equal to the element
Info +infa'
L
K
/""""-/
K'
F
in H2 (L IF). In this way, all computations in the Brauer group are referred to computations in the layers. It is natural to denote the inclusion map of Ff2(KIF) into the Brauer group Ff2(*/ F) by the symbol inf(KIP),(.jP). and to view it as a symbolic inflation, since it is, so to speak, the limit of actual inflations, inf(K/F),(L/F) as L -+ 00. Our next task is to define a symbolic restriction map, resp.E: H2(*/F) -+ H'l (*/ E) between the respective Brauer groups of two fields FeE. Let FeE c K c L with KIF and L/F normal. Then the follOwing diagram is commutative:
F{l(K/F) ~ H2(LjF)
rea!
=!
H2(K/E) ~ H2(L/E). Therefore, if we view the two upper groups as subgroups of JI'l(*/F) , the two lower ones as subgroups of H2(*/E), and the horizontal arrows as inclusions, we see that the effect of the vertical restriction maps is consistent-the one on the right extends the one on the left. Taken all together therefore, for all K, L etc. which contain E and are normal over F, these restriction maps re6(K/F),(K/E) constitute a map of H2(*IF) into H2(*IE) which we denote by resF,E, and view as a symbOlic restriction. In case E is nonnal over F. the kernels of both vertical restriction Ill8.p6 in our diagram are the same, namely they are inf (H 2 (E/F». This follows from the
XlV. ABSTRACT CLASS FiELD THBORY
150
exa£tness of the horizontal inftation-restriction sequences
H2(E/F) ~ H2(K/F) ~ H2(K/E)
I!
iDf! .
mtl
/P(E/ F} ~ H2(L/ F) ~ 8 2 (L/ E) a.nd the commutativity of the diagram, in which we view the vertical inflations 118 inclusions. Consequently, the symbolic inflation-restriction sequence
o ~ H2(E/F) ~ H 2 (*/F) ~ H2 (*/E) is exact; the kernel of resP,E is the subgroup H2(E/ F) of Ef'J(*/F) in case ElF is normal. This fact is of the utmost importance because it characterizes the subgroups of the Brauer group H2 (*/ F) which belong to the individual layers; For an element 0 E H 2 (*/ F) to belong to H2(E/ F), it is necessary and sufficient that
O. Let us review how the image of an element a E H2(*/F) under the symbolic restriction map is explicitly computed in terms of the layers. The giVllD a will belong to some normal layer KIF, i.e. a E H 2 (K/F). Then we have resF,EO:::
resF,E a
= res(L,F),(L,E) inf(KIF),(LIF) a
E
H2(L/E}
e H 2 (*/E),
lOr any L containing E which is normal over F. A symbolic transfer map can also be defined which maps HZ( *1 E) into H2( */F) lOr any FeE. AB in the case of restriction, it is induced by the ordinary transfers in finite layers, whose consistency is guaranteed by the fact that the following diagram Is commutative:
H2(K/F) -.!!!.. H 2 (L/F)
vi
vi
H2(K/E) ~H2(LIE). We denote the resulting symbolic transfer by VE,F:
H 2 (*/E) ..... H 2 (*IF);
The important relation "transfer of restriction VE,FresF,E 0
=
FeE.
= multiplication by degree", i.e.
IE : Flo,
holds for these symbolic maps because it holds in the finite layers. Finally of course we have the isomorphisms between Brauer groups which are induced by conjugation with an element r1 E G:
a.: H 2 (*/P) ~ H2 (*lp a ) where Fa is defined by G(F"l = (GF)a. We leave their construction to the reader. 3. Class Formations; Method of Establishing Axioms In global class field theory the formation of idele classes plays a central role, analogous to tha.t of the field formation in local class field theory. If one analyzes the features common to these two formations one sees that they are both special C8SES of an abstract algebraic structure in which the constructiolls peculiar to class
3. CLASS FORMATIONS; METHOD OF ESTABLISHING AXIOMS
151
field theory can be carried out. We call this structure a class !tmlUltionj it is a field formation which satisfies the following additional axiom AXIOM II. For each field F, there is given an isomorphism a - invF a of the Brauer group f(2(*/F) into the group Q/Z of rationals mod 1, which satisfies the following conditions: a) If K/F is a normal layer of degree n, then invF. maps H2(K/F) onto the BUbgrOUp *1../'1., of Q/Z. b) For any layer E / F of degree n we have
invEresF.E = ninvF· The rational number mod 1 which we denote by invp a ill called the invariant of the element a E HZ( */ F). Since inv F is by definition an isomorphism into, these invariants characterize the elements of the Brauer group uniquely. The most difficult part of class field theory consists in proving that the formations of local fields and of global ideIe classes are class formations. In the next few paragraphs, we shall outline the method of proof that is used, carrying out here the reduction steps which can be done abstractly. In the later sections of this chapter, we will assume given an abstract class formation and derive consequences from that assumption. Notice that in the proof we are about to outline we do not assume at the beginning that our formation is a field formation. The proofs of Axioms I and II are intermingled, at l~t in the global theory. One first proves that the formation in question satisfies AXIOM 0'. In each cyclic layer' of prime degree, the Herbrand quotient h2/ 1 is defined and is equal to the degree.
The proof of Axiom 0' involves subtle arithmetical arguments, especially in the global case, but one has the advantage that the layers involved are of the simplest possible type, and that the Herbrand quotient has especially simple multiplicative propertie:s with re:spect to changes in the coefficient module (cf. q.l, q.2, q.3 in the first chapter of the book). Axiom 0' states that if K / F is a cyclic layer of prime degree, then
h2(K/F) = [K: Fjh1(K/F), where h..
= "order of H"
It.
Thus. when it is proved, one has at one's disposal the
"first inequality":
[K : FII h2(K/ F) in all cyclic layers of prime degree; and one knows that the necessary and sufficient condition for the equality h 2 (K/F) = IK: F] to hold in these layers is given by Axiom I' ofthe preceding section, i.e. HI(K/ F) = for cyclic layers of prime degree. On the other hand, another necessary and sufficient condition is obviously given by
o
AXIOM
I". The "second inequality"
h2(K/ F) I [K : F] holds in all cyclic layers of prime degree.
XlV. ABSTRACT CLASS FIELD THEORY
152
Therefore one prows next either Axiom I' or Axiom I", whichever is more convenient. In the local theory Axiom I' is immediate because the formation is that of the non-zero field elements and is therefore a field formation. But in the global theory, where one is dealing with the formation of idele-classes, no direct proof of Ax.iom l' is known. Instead, one proves Axiom I", making heavy use of the first inequality and of Kummer theory. Here one again has the advantage that the layers involved are of the simplest possible ty~cllc of prime degree. F'urthemlOre, in such a layer, the cohomology group in question, H2, is isomorphic to HO. which has a simple down-to-earth description: clements in the groWld level modulo norms from the top level. At this stage one knows that the formation is 8 field formation because, as we have seen in the preceding section, Axiom I' implies Axiom I. Hence, the infiatioDrestriction seqUeIlCe is exact in dimension 2, and we can use the case r = 2, " = 1 of Lemma 1 of §2 to show that the "second inequality" order of H2(K/F) I [K: F] holds in all normal layers K/F, siu.ce we know by Axiom 1" that it holds in cyclic layers of prime degree. Finally, by investigating the 2-dimensional cohomology groups of some special layers (corresponding to unramified extensions locally and cyclotomic extensions globally) one proves that the formation satisfies AXIOM II'. For each field F, there is a subgroup fI2(*/F) of the Brauer group H2(*IF), and an isomorphism invF of this subgroup into Qil. such that: a) For any layer E/F we have resF,E fI 2 (*/F) c fI2(*/E), and
iDVE resF.E a = [E:
FI inv Fa
tor all a
E Ji2(*/F). b) If there exists a layer ElF of degree n, then which is cyclic of order n.
fI 2 (*/F)
contains a subgroup
The point is that we now have enough information to prove that the subgroup
fi2(*fF) must in fact be the whole Brauer group over F. To do this we have simply to show, for each normal layer KIF, that J/2(K/F) is contained in fI 2 (*IF), beC8Ulle the Brauer group is the union of the groups H2(K/ Fl· Let n = [K: FJ be the degree of the layer in question. According to IT'b, there exists a subgroup T of fI2 (*I F) which is cyclic of order n. Using II' a we find invKresF,KT
= ninvFT =
invFnT
= iiiVFo = O.
Since inv K is an isomorphism into by hypothesis, it follows that and the exactness of the symbolic inflation-restriction sequence
resP,K
T
= 0;
0-+ H2(KjF) -+ H 2 (*jF) ~ H 2 (*IK) allows us to conclude T C J/2(K/F}. Recalling the second intlquality, which states that the order of 1/ 2 (KIF) divides the degree n, which is the order of T, we see that H2(KjF) = T and is therefore contained in ii 2 (*/F) as contended. Furthermore, its image WIder inv p is
iDvFH2(K/ F) = invp(T)
= !.Z/l. n
3. CLASS FORMATIONS; METHOD OF ESTABLISHING AXIOMS
153
because ~Z/Z is the only cyclic subgroup of order n of Q/Z. ,Thus the formation, together with the map invF. sati:;fies Axiom II. This concludes our discussion of how Axioms I and II follow from the seemingly weaker Axionis 0', I' or I", and II'. For the remainder of this chapter we assume that our formation (G, {G F }, A) is a class-formation and discuss the rather remarkable consequences of this assumption. According to Axiom II a), the 2-dimensional cohomology group of any normal layer K / F of degree n is cyclic of order n, being in fact canonically isoworphic to the group ~Z/Z under the isomorphism invF. Any rational number t which can be written with denominator n determines a unique element a E H2(K/ F) such that inv Fa == t (mod Z). This a is called the cohomology class with invariant t. If we are working with a complex X for the galois group G K / F of the layer, and I: X 2 --+ AK is a cocyc!e in the class a, we also say that f is a cocycle with invariant t. The class with invariant lIn has period n and generates H2(KI F). It is called the canonical class, or the fundamental class of the layer KIF. A cocyc1e f representing the fundamental class is called a fundamental 2-cocyc1e. PROPOSITION
4. Let FeE c K, with K/F fW7mal. Then the restriction of 0/ the layer KIF is the fundamental class of the layer K/E.
the fundamental class
=
=
PROOF. Let n [K: FJ and m [K: EJ. Then [E: FJ the fundamental class of KIF we have by Axiom lIb:
. m. n -;: 1 -1 lnVEresFEu = -lDVpUSi _. , n m n m
= n/m. and if a
(modZ) .
is
o
Since the fundamental class of a layer generates its two dimensional cohomology group, the following corollary is immediate: COROLLARY 1. Let FeE JP(K/F) onto H2(K/E).
c K,
with K/ F normaL Then the restriction maps
Keeping F and E fixed, letting K vary. and remembering that H2(*/E) =
UK H2(K/E) we see also
COROLLARY 2. Let E/F be an arbitrary layer 01/1 class formation. Then the symbolic restriction maps H2 (../ F) onto H2 (.. / E).
Corollary 2 is an important fact. For one thing, it shows that the isomorphisms for all fields E ::l F are determined by the single isomorphism inv F through the equation
mvE
invEresF,E
= [E: F] invF.
Furthermore, the onto-ness of the symbolic restriction allows us to prove 5. Let ElF be an arbitrary layer. Then 1) The symbolic transfer maps H2(*/ E) isomorphically into H2(*/F), preserving invariants in the following sense PROPOSITION
2) Conjugation by an element a E GF preserves invariants in the following
,ense
XlV. ABSTRACT CLASS FIELD THEORY
PROOF, 1) The homomorphism computation invF VE,FresF,E = invF[E: F) = [E:
F] invF = invETeSF,E
shows that invF VE,F == invE because reflp,E is onto. Hence the tJa.IlSfer preserves invariants and is therefore an isomorphism into. 2) An element u E GF acts as identity on H2(KI F) for all normal KIF, hence also on H 2 (*/F). Using these facts we make the computation invE~
u. re5F,E == invE" resF' ,Ea U.
== IE'" : F"'] invp.. u. = [E: F)invF = invE resp,lii'
This shows invEa u.
o
= invE because resp,E is onto. 4. The Main Theorem
The main fact about the cohomology of class formations is that the cohomology groups of a nonnal layer K / F depend only on the structure of the galois group
GK / P ' THEOREM 1 (Main Theorem). Let KIF be any norrnallayer in a class formation. Let Ct be tht; fundamental class oj the layer, i.e. the canonical generator of H2 (K/ F) H2 (G K/ P' AK). Consider the cu.p products associated with the natural pairing AK x Z ...... AI(. For each 'integer q, let O:q denote the map ( ....... 0:( which is obtained by multiplying the variable element ( E Hq(GKIF, Z) with a. We contend that these maps are isomorphisms onto for all q, -00 < q < 00:
=
Ctt: H9(G K / F ,Z) R: Hq+2(G K / F ,A K ) == H9+2(KIF). PROOF. We use an abstract cohomological theorem that was designed for just this purpose. 2 Each subgroup G' c G K / F , is of the form G' = GKIF' for some field F', F c F' c K, by Proposition 2 of §1. If Ct' denotes the fundamental class of the layer K / F', then, by Proposition 4 of § 3, we have 01.' res Ct. Let Ct~: Hq(GKIF/Z) -+ Hq+2(G K1F "A K ) be the ma.p obtained by multiplication with d. Consider these maps in the three successive dimensions -1,0, +1:
=
Hl(GKIF"AK)
= H 1(K/F'}
Ct~: HO(GK/Fr,Z) ..... W(GK/Fr,A K )
= H 2 (K/F')
a_I: H-1(GK/F"Z)
-+
a~: Ht(GK/F',Z) .... H 3 (GK/Fr,AK) =H3 (K/F'). In order to establish the hypotheses of the cohomologica.l theorem, we must show
that o.~l is onto, 0:0 is an isomorphism onto, and 0:1 is an isomorphism into. Then we can conclude from that theorem that a~ is an isomorphism 0Dt0 for ail q and our prE'Jlent theorem will be proved. a~l is obviously onto, because Hl(K/F') 0 by Axiom I. To show that ao is an isomorphism onto we have only to point out that HO(GK1F"Z) is cyclic of
=
2l-br .. very general vel15ion of this tileorem, see [21, Theorem B in Chapter IXZ, 581. See also Section 2 of our chapter "PrelimillMi.,.". where there is .. Bimple version, Theorem A. and Bketch of the proof.
':
.. THE MAIN THEOREM
.
1M
order (GK/FI : 1) == [K : PJ generated by xl, H 2(K/F') is cyl;lic ofthe same order generated by el, and 0o(x:l) = 0'(",1) = 1.a' = a'. Finally, a' is automatically au isomorphism into because H1 (G K/ 1"', '1.) = O. This concludes the proof oftbe main theorem. 0 The canonical isomorphisms which we have just established commute with restriction and transfer, according to THEOREM 2. Let F c F' c K, with KIF normal. Let 0: be the canonical class of the layer KIF, and a' that of K/ F'. Then each pair of vertical arrows makes the following diagram commutative:
Hq(G KIF, Z) ~ Hq+2(K/ F) reo
!t
V
rea
!I
V
Hq(GK/F"Z} ~ HH2(K/F'). PROOF. Recalling that a/ = res a, we see that this theorem is an immediate consequence of the identities relating products and restriction and transfer. Namely, for any (E Hq(GK/F"Z)
a'(res() = (resa)(res() "" res(o:(), and for 8J1Y (' E H"{GKIFI,Z)
a(V,') = V«res oX') = V(a',,).
o
The isomorphisms of the main theorem commute with conjugation, but since this commutativity is a special case of their commutativity with the isomorphisms induced by isomorphisms between cl&lS formations, we digress for a momeIlt to discuss this latter concept. If (G, {Gp}, A) and (G', G~" AI) are two formatiollS, it is natural to define an isomorphism of one onto the other to be a pair of isomorphisms
>.: G' R: G
I:
~A'
A
such that f({)'(l)a) = u'(fa), and such that), induces I). one-to-one correspondence between the subgroups {G p'} and the subgroups {GF}' Then for each field F' we candefindF' by G p,,..,) = )'(G p,), and it is clear that !(A).FI) = ApI, G),K'/).F" = >'(GK'IF')' etc. Consequently, for each normal layer K' /F' there is an isomorphism
().,j).: W().K'/),F')
R:
H"{K'/F /)
induced by the isomorphism of pairs
I:
A)'K' ~ A~,.
If one formation is a field formation, then so is the other, and the isomorphisms (>', f). on the 2-dimensional groups of the layers induce isomorphisms of the Brauer groups. If the formations are class formations, then one would also require that these isomorphisms between Brauer groups also preserve the invariants. When this is the case, it is clear that the two class formations are essentially the "same" algebraic structure and that any collBtruction carried out in one could be carried out in
XIV, ABSTRACT CLASS FIELD THEORY
ll;ti
the other with the "same" result. In particular, the isomorphisms constructed. in the main theorem would correspond, i.e. the following diagram would be commutative
Hq+2p.K' />.F')
Hq(G>'K'I>'F" Z) _ (>.,1).
t
(>',J).
t
H'l+2(K' /1"').
H9(Gk:'IF"Z) _
If (G,{GF},A) is any class formation, and .,. E G, then the "inner automorphism" .,-1: (ir = 1'(11'-1 _ r 1': a-+Ta
ill an automorphism of it in the sense we have described, because by Proposition 5 of §3 we know that conjugation preserves invariants. Thus we have proved THEOREM 3. Let l' E G, and let KIF be a normal layer. Let 0: be the fundamental class of the layer KIF, Then r.o: is the fundamental class OfTK/TF and the following diagram is commutative:
Hq(GK/F,Z) ~ Hq+2(K/F)
~.!
~.
!
Hq(G-rK/-rP,Z) ~ HQ+2(rK/TF). The isomorphisms of the main theorem do not commute with inflation in p0sitive dimensions. The correct rul,e is given by THEOREM
4. Let F eKe L with KIF and LIF normal. Let oKIF and
OL/F be the canonical classes of the respective layers. Then the following diagram is commutative for q > 0:
Hq(GK/P.Z) ~Hq+2(K/F) (£:K]
inf!
Olt.,,, Hq (GLIF,Z) -
PROOF.
iUf! 2
Hq+ (LIF).
The theorem is an immediate consequence of the general formula
(inf 0:) U (inf {3) = inf(o: U {3). This rule can be proved by a dimension shifting induction, and it is also an immediate consequence of the well known formula for the cup product of standard cochains, in positive dimensions, namely
(f U g)(O'1o (12.· .. ,O'p+q)
= f(O'l' (12, .... (11') u «(71, •.•
I
(7p)g«(1p+l .... , (1p+q)'
Granting this rule, the proof is immediate; we have only to observe that the inflation of O:KIF is [L : KjO:L/F' Namely, these two classes have the same invariant, because 1
[K : FI
:=:
[L: K] [£: F) ,
and inflation preserves invariants. (It is the inclusion in the Bra.uer Group!) HellCe, for any (E Hq(GK/F,'l.) we have inf(O:K/F()
= (in!uK/F)(inf() = ([L: KjO:L/p)(inf(> =O:L/ p([L : K) inf (l
BXERCISE
1lI7
o
and this was our contention.
In dimensioDB r > 2, the inflation map in a class formation is very weak. For example we have COROLLARY. If F eKe L, and if the degree [K : F] ditlides the degree (L: Kj, then the inflationfwm K/F to L/F is the zero map in dimensions r > 2.
PROOF. Sillce the horizontal arrows of the commutative diagram of the preceding theorem are isomorphisms onto, we need only show that the left hand vertical map, [L : K) inf, is the zero map. But this follows from the fact that it is applied to the group HT(GK/F'Z), in which every element has an order dividing (G K / F ; 1) = [K: FJ. 0
Combining the down-to-earth interpretations of the cohomology groups in low dimensions with the isomorphism of the main theorem we obtain the following speciaJ. results:
H4 (KIF) -;;:, H 2 (GK/F,'L)
r::.J
GKIF
H3(K/F} Rl Hl(GK/F'Z) =0 H2(K/F} -;;:, EfO(GK1F,Z} RlZ/nL HI(K/ F) -;;:, H-1(GKIF,Z) = 0 (AF/NK1FAK) -;;:, HO(KIF) Rl H- 2 (G K1F ,Z) -;;:, (GKIF/GK/F) «AK)NK/F/IAK);::: H-l{K/F) -;;:, H- 3{G K/ F ,Z). By far the most important of these special cases is the next to the last. It is the so-called reciprocity law Uromorphism of class field theory, and the whole of the next section is devoted to a detailed study of its consequences. The following exercise, with which we close this section, concerns the case of H4(K/F). .
ExerciBe Using the isomorphism 0: Hl(Q/Z) r::.J H2(Z) we see that the elements of H'(KjF) are of the form a:. 5)(, where X E Hl(G K /F,Q/Z) and a: is the fundamental class of the layer KIF. If X(O") is the standard l-cocycle representing X. then the map 0" -+ X(O") is a character of G, and we may identify X with this character as we have discussed. Thus the correspondence
X ..... a:. OX is an isomorphism between OK/F, the character group of the galois group of the layer KIF, and H4(K/F), the four dimensional cohomology group of the layer. a) Let FeE c K with Kj F normal. Show that the restriction from H4(KI F) to H4(KIE) corresponds to restricting the character X from GK / F , to the subgroup GK/E; and show that the transfer from H4(K/E) to H4(KIF) corresponds to the map of characters of the subgroup GKIE into characters of the big group GKIF, obtained by composing them by the group theoretical transfer. b) Notice that the restriction and transfer are weak maps in dimension 4; for example, if FeE c K, and the layer ElF contains the maximal abelian sublayer of KIF, then both maps are the zero map. (For the transfer statement, use the principal ideal theorem.)
tA
XlV. ABSTRACT CLASS FIELD THEORY
c) Let Fe K C L with K/F and LIF normBl. Show that the inflation map from H4(K/F) to H4(L/F) corresponds to the procedure of viewing a character of the factor group CK/F as So character of the big group CLIF, and raising it into the [L: K]-tb power. (Use Theorem 4.) d) Given a standard 4-cocycle f = !(O'l, (12,0'3, 0'4) of G K/ F in AK, representing a class (J E H4(K/F), show that the corresponding character X is given by
x(r) where /
*r
=invF(f * r),
is the standard 2-cocycle defined by
(f *T)(O'l,C'2) ""
L /(0'1. C'2,p,r). p
5. The Reciprocity Law Isomorphism
In order that our nota.tion correspond as closely as pOBSible to that which will be used in applications, we will from now on write the formation module A mUltiplicatively. The effect of C' on a is then denoted by a". If ElF is an arbitrary layer, the corresponding norm homomorphism N 1::/ F: AE ..... AF is defined by Ne{Fa ilia"', where the elements C'; are representatives of the left cosets ·of GE in GF: GF = U O',G E • It is the multiplicative analog of the trace; if FeE c K and K/F is normal, then the map NS/F is what we have previously called the trace from the subgroup G K / E to the big group GK / P • For each normal layer K/F, the main theorem gives us a natural isomorphism
=
HO(KIF.)::.; H- 2 (GK/P, Z). Both of these groups have aown-to-earth interpretatioIlB. n,O(K/F) is isomorphic ro AF/NK/pAK. the factor group of elements in the ground level modulo norms from the top level; the isomorphism is induced by the map x: AF -> HO(K/F) which is onro and has kernel NK1FA K . On the other hand, H-2(G K / F ,'Z) is naturally isomorphic to the factor commutator group GK/FIG'k/F of the galois group of the layer KIF. The isomorphism is induced by the homomorphismJ (f -> ( ... = 6- 1 )K (a - 1) which is onto and has kernel G'kIF. The special case of the main theorem mentioned above may therefore be interpreted as an isomorphism AFINK/FAK r:::.GK/P/G'k/F·
This is knoWll as the reciprocity law isomorphism, for historical reasons. Being an isomorphism between a factor group of AF and a factor group of GKIF. it is induced by a multivalued correspondence " .... CT, a E A p , (1 E GKIF between thol;e two groups. Then. by definition, we have
a +->
q
if and only if xa = a: . (cr,
where Q is the fundamental 2-dimensional class of the layer K / F. A dual description is given by 3Here IS: H- 2 (G,Z)
IG/ [~, IG/Fa.
We have H-l(G, T) :::: iIIomorpbism G/Oc::::
Z. H-l(O, I)Ls the connecting homomorpbiam of the exact sequence o .... fG .... Z[G) - Z -+ 0. and
88
Is well ktwwn, the map
(I .... ( 1 -
1 (mod I)~ induces an
159
Ii. THE RECIPROCITY LAW ISOMORPHISM
PROPOSITION 6. Let a E AI" and u E GK / P ' Then a
invF(xa· ox)
(.)
t-+
u, if and only
if
= xCu)
Jor all characters X o/GKIF. (On the left hand side of this equation, the chamcter X is to be interpreted as an element of HI (G, Q/Z) in the usual way, and oX is the corresponding element of H2(G, Z).) PROOF. Let Q be the fundamental class for the layer K/ F. If a ... u, then
invF(ka' OX)
= invF«(a. (a) . ox) = invF(a· «(a' cSx» = .!.O«a . X) = n
x(u).
This proves the proposition because the u's to which a corresponds are characterized by the values X(O') for variable X (00(: also pages 4-5). 0
Our next theorem concerns the commutativity of the reciprocity law isomorphism with various natural mappings between different layers. THEOREM 5. Using the symbol! (KIF) to denote the many valued COfTeSpO'II.dence inducing the reciprocity law isomorphism in the layer K/ F, we assert that the following dio.grams are commutative:
AI"
a)
inclU8ion
(KIF)l
AE !(KIEl
group theoretical
(GKIFIGK{F)
tr......f e r .
NEIP
b)
to
(FcEcK)
(OK/BlacK{B)
•
AI" ...E----'--'-- AE
t(KIB)
(KIF)!
GK / F c)
.. inclusion
(FCECK)
GK1E
AI" - - - -..' - - - -.... AFT (KIF)! GK/F
d)
AF (L/FJ
Jt
GLIF
lCK?lrl c:onjug"tion by ..
•
identity
G
(FCK, TEG)
KT/F"
• AF !(K/F)
lI&tllral hom. onto ta<:tor group
~
(F eKe L)
G
KIF
PROOF. To prove a) and b) we use Theorem 2 of §4 which states that the fundamentallsomorphisms of the main theorem COllllllute with restriction and tra.nsfer. We have only to recall how these homomorphisIml are obta.ined explicitly in cllBe of HO(AK) and H-2(Z). The inclusion ma.p AF -> AE induces the restriction from HO(K/F) to HOCK/E); while the norm map from AE to AF induces the transfer from (K I E) to (KIF). On the other hand, the restriction from H- 2 (G K / F ,Z) to H-~(GK/E'Z) is induced by the group theoretical transfer from G K / F to GK / E • while the cohomological transfer in the other direction is induced by the inclusion of GK/E in G K / F • Statement c) follows similarly from Theorem 3 of §4.
no
no
160
XIV. ABSTRACT CLASS FIELD THEORY
For d) we mUBt use another method because we have not introduced cohomology maps correspondi.ug to the natural map of AFINL/FAL onto Ap/NKjFAK, 8lId to the natural map of GLIF onto the factor group GK / P ' We use the duality criterion
of Proposition 6, which states that for a E Ap and
0'
E GLIF we have a I(LIF)~ u
if and ouly if
(*)
X(O')
= invp(xa. OX}
fur all characters X of CLIP. We must therefore show that if (*) holds for aJl X,
then (**) ?/J(O'G L / K ) = invp(Ka· o,p) for each character ,p of G K1P = CL/FIG L / K • To thi5 efi"ect, we let X
= inf01<'IP,GL(F!J.!.
Then X is the character of CLIF defined by X(u)
= 'I/J(UCL / K ).
Of course we have WI o'l/J = OX because inBation commutes with coboundarie.'l. Hence
=
t/J(uGL/K) = X(O') = invF(Ka' d"X) invF(xa· info,p), and since the cup product with the O-dimensional class Ka is given by an induced map (Theorem 10, §3, Ch. V), and therefore commutes with inflation, we can write this as !/J(uG L / K ) invF(inf(Ka· 01/;» invF(Ka· o,p). (The last equality follows from the fact that we have in! (xa . 01/;) = xa . o'l/J by the very definition of the Brauer group.) This concludes the proof of Theorem 5. 0
=
=
The reciprocity law isomorphism is of course induced by a homomorphism of AF onto GKIF/CK/F with kernel NK/FAK. This homomorphism is called the llOrm-re.'lidue map and is denoted by
a -+ (a, KIF),
a E Ap.
The symbol (a, KIF) is called the norm-residue symbol and is of historica.l. Origin. The reason for the name is that the value of (a, K / F) determines the residue class of a modulo norms from K, Le. modulo NK/FAK. Translating Proposition 6 and Theorem 6 into properties of this symbol we obtain: PROPOSITION 6/. Let KIF be a normal layer and let a E AF' Then (a, KI F) is the uniquely determined element afGK/pICCK/F such that
invF(xa. OX)
= x(a,K/F}
lor all characters X OICK/F' THEOREM
6. a) If FeE
c K,
(a, K/E) =
b) II FeE
then for a E AF we have
VGKrp,GK(E
(a,K/F).
c K, then for a E AI> we have (NEIFa, KIF) = (a, KIE)' G"K/F'
c) If Fe K and T E C, then lor a E AF we have (aT,l\" IF) = (a,K/F)1'. d) IfFCKcL, andaEAF. then (a,K/F) = (a,L/F)· GL/K'
6. THE RECIPROCITY LAW ISOMORPHISM
181
PROOF. These identities are simply restatements of the.commutativities as-
serted in Theorem 5.
0
The kernel of the norm residue homomorphism a -0 (a, K/F) is the subgroup NK(FA x of AF' We will call such a subgroup, and more generally, a subgroup of the form NE/pAs, for an arbitrary extension ElF, normal or not, a norm .rubgrvup of AF. Clearly, if FeEl c E2, then Ne.(FAE. C NBl/pAE) because N E./P = NE,/FNE./E)' Therefore, the map E -0 NE/FAE is a lattice-inverting map of the set of all extensions E of a fixed field F onto the set of all norm subgroups of A p . We propose now to discuss the features of this correspondence in detail. PROPOSITION 7.
Let FeE c K with K / F normaL Then lor a
E AF we have
a E NS/FAE <=> (a, KIF) E GK/EG"K/F' PROOF. If a E NE/FAE. say a we have
(a,K/F)
= NEIFb, with b E AB, then by Theorem 6b)
= (b,K/E)G"K/F E GK/EOCK/ F •
On the other hand, if (a,K/F) E GK/EGk/F then there exists an element b E As such that
(a,K/F)
= (b,K/E)OCK/ F ,. (NE/Fb,K/F)
because the map b ..... (b,K/E) maps AE onto GK(E/G"KIE' Since the kernel of the map a -0 (a, K/F) is NK/FAK it follows that there exists an element C E AK such that a NE/pb· NK/FC= NB/P(bNK/EC), Hence a E NS/FAE 88 COIItended.
=
o
If El and E2 are any two fields of formation, we define their compoaitum, El~' to be the field which is defined by the property G E) e. GEl n G e.. Then Er . ~ contains both Er Wld E 2 , and is contained in any field which contains both El and E2, because GE) n GE2 is contained in both GEl and GE., and contains any subgroup of G which is contained in both GE) and G E •• An extension M/F is called abelian if it is normal and its galois group G M / F , is abelian; in other words, if G M :) GF. From this latter criterion we see that if Mr/ F and M2/ F are two abelian extensions, then their compositum MI M2/ F is also an abelian extension. From this it follows immediately that any extension E/ F, normal or not, contains a maximal abelian subextension MI Pj that is, there exists a field M such that F C M C E, M/P is abelian, and any field between F and E which is abelian over F is contained in M. The following theorem concerns the properties of the correspondence between extension fields of F and their norm subgroups.
=
THEOREM 7. The norm group of an arbitrary extension E/F is the ,ame /18 that ofi,ts maximal abelian sube::ttension M / F, i.e. NE/pA E = NM/FAM. II {M} is the set of all abelian extensions oj F, then the correspondence
M .... NM/pAM is a one-to-one correspondence between {M} and the set 0/ all nONn subgf"OUpS 0/ AF, having the following properties: a) Ml C M2 ¢> NMI/pA MI J NM./FAM,; b) NM1M2/FAM,M. = (NMJ/FAM») n (NM2 I F AM.);
XJV. ABSTRACT CLASS FIELD THEORY
162
c)
1M: FJ == (AF : NMIFAM).
Finally, every subgr01J.P of AF which contains a norm subgroup i3 itself a norm IUbgrouP·
Proposition 7 shows that for the extensions ElF which are contained given normal extension KIF, the norm group NB/FAE depends only on the group GK/BGK{F" This group GK/ECCKIF is obviously the minimal normal subgroup of GK/F containing GKIB such thst the factor group GK/FIGK/EGK/F is abelian. Consequently the corresponding intermediate field M (defined by G K / M GK/EGK/P ) is the maximal abelian subextension of E. Thus, the norm group NE/FAE depends only on the maximal abelian subextension of ElF; and since the maximal abelian subcxtension of the maximal abelian sUbextension is that extenIlion itself, it follows that NE/FAE = NMIFA M • This proves the first part of the theorem, and shows in addition that every norm subgroup of AF is a norm group of an abelian extension. Let M/F be any abelian extension. The intermediate fields M1 , F C Ml C M, are in one-to-one correspondence with the set of all subgroups GMIM, of the galois group GMIF . Since the norm residue ma.p a ..... (a,MIF) maps AF onto GMIF, and has kernel NM/pAM, it follows that the subgroups GM/M 1 are in oneto-one correspondence with their inverse images under the norm residue map, and that these inverse images are those subgroups of Ap which contain NM/FAM. On the other hand Proposition 7 shows that the inverse image of GMIMI is the norm subgroup NMIIFAMI. Since any two abelian extensions MdF and M'lIF are contained in a single a.belian exteru;ion M == M 1 M2, it follows from whst we ha.ve proved that Ml = M2 if and only if NM1/FAMl == NM.lpAM2; hence the correspondence between abelian extensions and their norm groups is one-to-one. FUrthermore, since any norm subgroup of AF is the norm group of an abelian extension M, we hsve shown that any subgroup of AF which contains a norm 8ubgroup is itself a norm subgroup. All that remains is to establish properties a), b), and c). Concerning a) we know from the transitivity of the norm that the bigger the extension, the smaller its norm group; and in the case of abelian extensions we can say conversely that the smaller the norm group the bigger the extension, because of the one-one-ness of the correspondence. Now b) follows from a) by simply considering the lattice of norm subgroups of AF as compared to the lattice of abelian extensions of F. Property c) is just a special consequence of the reciprocity law isomorphism G M I F R: AF INM I FAM' This concludes the proof of Theorem 7. 0 PROOF.
in
8
=
If ElF is an arbitrary extension, the index (Ap : NE/FA E ) is called its nonn index. From Theorem 7 it follow~ that the norm index of an extension is equal to the degree of its maximal abelian subexteosion. Hence, the norm index of an extension always divides its degree, and is equal to the degree if and only if the extension is abelian. For historical reasons the top fields M of abelian extensions M/F are called class fields over F. According to Theorem 7, the class fields over F are in oneto-one correspondence with their norm groups in A F • If B is 8 norm subgroup of AF, the corresponding cla5s field M over F such that NMIFAM = B is called the class field. belonging to B. The galois group G M IF is canonically isomorphic to the
G. THE ABSTRACT EXISTENCE THEOREM
163
factor group AF/B by the reciprocity law. If ElF is an arbitrary extension, then by Theorem 7 we have M C E if and only if B:::) NE/FAE .•
8. Let B be a norm 5Ubgr'O'tlp of AI", and let M be the class fie.ld F belonging to B. Then a) For any T E G, M'" is the class field over FT belonging to B'I'. b) 1/ E is any field containing F, the group C = N;iF(B) is a norm stJbgroup of AE and the class field over E belonging to it is the compositum ME. PROpOSITION
otIe1'
PROOF. a) This follows from first principles: M1' IF" is abelian because M / F is, and we have
NM<,rAM< = NM·/F.(A~) = (NM/FAM)'I' = gr. Hence M1' is the class field over For belonging to B'I'. b) Let L denote 8. variable extension of E. Then, by definition, ME is the smallest L such that L ;) M. Since M is the class field over F belonging to B, we have
L::J M
~
NL/FAL C B #NL/EAL C C.
where C::;; NE}F(B). Thus, ME is the smallest L such that NL/EAL C C. This shows in the first place that C is a norm subgroup of As, because C contains the norm group of ME; and now it follows tha.t ME is the class field over E belonging to C because that is the smallest extension of E whose norm group is contained inQ 0
8. The Abstract Existence Theorem As we have seen (Theorem 7 of the preceding section), the lattice of abelian extensions M of a given field F is anti-isomorphic to the lattice of norm subgroups of the given ground level AI'" Thus, if we can in some way characterize tbQ8C subgroups of AF which are norm subgroups, then we will have gained an insight into the totality of all abelian extensions of F. In class field theory proper, both local and global, it turns out that a very simple characterization of the norm subgroups can be given, in terms of the natural topology in the levels AF-the norm subgroups turn out to be the open subgroups of finite index in AF' The aim of this section is to discuss a set of conditions on an abstract class formation, from which the abovementioned characteriza.tion of norm subgroups follows, and which are relatively easy to check in the case of class field theory proper. DEFINITION 2. A formation (G, {GF}, A) is a topological. formation if each level Ap is a topological group and if: a) In each layer E/ F, the topology of the ground level Ap is that induced by the topology of the top level AE; iu other words, the inclusion map Ap ..... AE is bicontinuous. b) The galois group G acts continuously on the levels; if U E G, then the map u: AI" ..... A(FO) is continuous for each field F.
It follows that the map u: AF -+ Ap" is bicontiuuous in a topological formation, because the inverse map, given by 0'-1, is also continuous. If E / F is an arbitrary layer of a topological formation, the norm map NeI F : AE -+ AF is continuous, and furthermore the ground level AF is a Cl08ed subgroup of the top level AE. To prove these statements, we imbed ElF in So normal layer KIF, and make
XIV. ABSTRACT CLASS FIELD THEORY
164
use of the fact that the topologies of All' and of AF are those they inherit as subsets of A K . Obviously the norm map a -+ NEIFa = 11. alT ' is continuous, because each of the isomorphisms a - aUi is a continuous map of AK into AK. FUrthermore, AF , being the set of elements of AK which are left fixed by the continuous operators 0' E GK/F, is closed in AK. !'tom now on in this section we assume that our class formation is a topological formation. In the course of studying the norm subgroups of a given level A F , it will be useful to consider the intersection of all norm subgroups. We denote this intersection by DF: DF:: NE/pAE. E-::;,F Thus, D F is the group of "universal norms" • elements which are norms from every extension. For any layer EjF we have NE/pDE C Dp, i.e. a norm of a universal norm is a universal norm. This is obvious from the transitivity of tbe norm and from the fact that every extension of F is contained in an extension containing E. Under certain topological assumptions one can prove the opposite inclusion, DF C NE/FD E . These assumptions are:
n
AXIOM IIIa). For each layer E / F
the norm group N E/FAE is a dosed subgroup
ofAF· AXIOM lIlb). For each layer ElF the kernel of tbe norm map, namely Ni}F{l), is a compact subgroup of All" PROPOSITION
have DF
9. In a topological/ormation satisfying Anoms IIIa) and b) we for each layer ElF.
= NE/FDE
PROOF. According to the remarks above, we have only to show DF C NE/FD E •
Let a E DF. For each field L containing E, let TL:: (NL1EAL) n (NEfF(a)) denote the set of elements of AE which are norms from AL and whose norm to F is a. Our task is to show that the intersection of the sets TL, over all L :J E, is non-empty, for an element of that intersection would be a universal norm in AE whose norm to F is a. The individual sets TL are not empty because, a being a universal norm in AF, we have for each L,
a E NL1FAL = NE/F(NLIEAL).
It follows that the sets TL have the finite intersection property, since n;=l TL, :> TL whenever L :J L, for i = 1, ... ,r. Therefore, in order to prove that the TL have a common point it suffices to show they are compact. This is where Axioms lIla) and b) come in; TL is the intersection of the closed set N L/EAL and the compact set Ni}F(a) and is therefore compact. This concludes the proof of Proposition 9. 0 In class field theory proper the forma.tions satisfy AxiOM IIIc). In each level AF the kernel of the map a .... a l , namely 11, is compact, for each prime number I.. AXIOM IIId). For each prime number l we have DE C A~ for all sufficiently large field:> E. (More precisely, there exists a field El such that DE C A~ for all E :> Et. Note that the required "largeness" is allowed to depend on the prime t.
tI. THE ABSTRAcT EXISTENCE THEOREM
For example, in the a.pplication to ordinary cla5s field theor)(, "sufficiently large" will mean "containing the £-th roots of unity".) PROPOSITION 10. In a topalogicallormation satisfying Axioms IlIa), b), c) and d) we have Dp = DF lor all natural numbers m and all fields F. PROOF. It is clearly sufficient to prove DF = D}, for each prime l. Using Axioms IIIa) and b) we know from the preceding proposition that Dp = NEIFDE for each extension E / F. Taking E to be "sufficiently large" for l in the sense of Axiom IIId) we have then
(*)
Dp = NE/pD E C NEIP(A~)
=(NE/FAE)l.
For each a E DF, let alIt denote the set of all elements is a. From ("') we see that the sets XE
01
v
J
AF whose t'b_power
= (NElpA E ) n (a l / t )
are non-empty. Therefore they have the finite intersection property. They are compact because NE/FAE is closed (Axiom IlIa) and alIt is compact (Axiom HIe). Consequently their intersection is not empty. An element of their intersection is an element of DF, whose eth-power is a. This concludes the proof of Proposition 10. 0 If our class formation satisfies one more axiom, then we can get information about the norm groups themselves rather than their intersection Dp. This final axiom is AXIOM IIIe). For each field F there exists a. compact subgroup UF of AF such that every open subgroup of finite index in Ap, which contains UF, is a norm subgroup. THEOREM 8. In a topological class formation satisfying the five Axioms IIIa)e), the norm subgroups 01 a level AF are just the open subgroups of finite index 'in AF' Their intersection DF is given by DF = n.':.l A p, and we have DF = DF for all natuml numbers m.
PROOF. The statements about DF follow Immediately from the preceding proposition. Indeed, for each field E :::) F we have
n AF 00
C AIJ!:Fl
=NE/FA p C NE/pA E
m=!
n:'=l
A~ C DF. Conversely, for each m we know from the preceding proposition that Dp = DF C A F. which proveB Dp c Ap. In a class formation the norm indices (AF : NE/FA E ) flIe finite {d. Theorem 7 of the preceding section}, and we have assumed (Axiom IIIe.) that the norm sui:r groups NEIFAE are closed. It follows that the norm subgroups are open. What remains to be shown is that conversely, each open subgroup B of finite index is a norm subgroup. Let N designate the general norm subgroup of AF • Then B contains DF = nN N because B :> A~F:B) J DF. In other words, in terms of the complements. the open sets A - N cover the closed set A-B. In particular, they cover the compact set (A - B) n Up, where UF is the compact subgroup of AF mentioned in Axiom IIIe). Thus there is a finite set of norm subgroups N 1 , N 2 , ••• ,Nh such that the sets A - N; cover (A - B) n UFo The intersection N = Nl nN2 n .. · n Nh is a nonn subgroup such that (A - B) nUF n N is empty,
which proves that
n:=l
166
XIV. ABSTRACl' CLASS FIELD THEORY
i.e. Up nN c B. Now consider NnB. It is open and of finite index in Ap because both N and B are. Multiplying it by UF we obtain a subgroup (NnB)UF, which is open, of finite index, and contains UF. Such a subgroup is a norm subgroup by Axiom lIIe). Since the intersection of two norm subgroups is a norm subgroup, it follows that N n (UF(N n B)) is a norm subgroup. This last subgroup is easily seen to be contained in B if one remembers that we have constructed N so that N n UFeB. Thus B contains a norm subgroup and is therefore itself a DOrDl subgroup. This concludes the proof of Theorem 8. 0
CHAPTER XV
Weil Groups In this section we shall apply the abstract theory of group extern;ions developed in Ch. Xlll, §I, 2,3 to the case of a group extension belonging to the fundamental class QKIF E H2(KjF) of a normal layer K/F of our class formation. In doing so we will gain a new insight into the reciprocity law isomorphism. For the sake of efficiency a.nd ultimate clarity, our discussion will be quite formal. We first define a certain type of mathematical 5tructure called a Weil group of the normal layer KIF. We then prove the existence and essential uniqueness of such a structure. Finally we discuss various further properties of the structure. DEFINITION 1. Let Kj F be a normal layer in a class formation. A Weil group (U, g, {fE}) for the layer K j F consists of the following objects:
1) A group U (called the Weil group by abuse of language).
2) A homomorphism, g, of U onto the Galois group GK/F' Having 9 at our disposal, we can introduce for each intermediate field E between F and K the subgroup UE g-l(GK/ E ). UE is the subgroup of U which is the inverse image, under 9, of the subgroup GK1E of G K / F • The final ingredient of the Weil group is:
=
3) A set of isomorphisms IE: AE ~ UEIU~ ofthe E-JeveIAE onto the factor commutator group of UE, one for each intermediate field E.
In order to constitute a Weil group, these objects U.g. and following four properties: WI) For each intermediate layer E' j E. FeE is commutative:
c
E'
c
{/El must have the
K, the following diagram
UE/U~
!V£IIB UEIIU~,
where the left hand vertical arrow is the inclusion map between formation level. and the right hand vertical arrow VE'iE denotes the group theoretical transfer (Verlagerung) from UE to UE'. W2) Let u be an element of U and put fT = g(u) E GKIF' Then it is clear that for each intermediate field E we have U1; = UB'" Property W2) states that the 167
XV. WElL GROUPS
lCiS
following diagram is commutative:
where the left hand vertical arrow is the action of a on the formation level AB and the right hand vertical arrow is the map of the factor commutator groups induced by conjugation byu: UB --> UUEU.- l = UBa.
W3) Suppose £1 E is a normal intermediate level, FeE map g induces an isomorphism
c L c K.
Then the
UEIUL ~ GKIE/GKIL == GLIE which we do not bother to name. Since AL is isomorphic to UL/Uf by h, we can view UE/Uf as a group extension of AL by G LIE as follows: (1) -+ AL
(*)
~ ULlUf .... UBIU'j,
-->
UBIUL
I':::
GLIE -+ (1).
The operation of G LIE on AL associated with this extension is the natural one, as one sees by applying property W2) to an element u E U1> having a prescribed image in G L / B (replacing the field E mentioned in W2) by our present field L). Property W3) requires that the 2-dimensional class of our extension (*) is the fundamental class (XL/E of the la,yer L/ E. W4) We finally require that UK == 1. This concludes the definition of a Well group. Fortunately it is easier to prove the existence of Weil groups than it is to define them! THEOREM 1. Let KIF be a nonnal layer in a class formation. t.Xists a Weilgroup (U,g, {fEll for the layer KIF.
Then there
PROOF. Let U be a group extension of AK by GKIF belonging to the fundamental class (XKIF of the layer KIF (Cf. Ch. XIII, §1, especially Theorem 1). Thus, U is a group containing AK as llormal subgroup, together with a homomorphism g of U onto G K / F1 with kernel AK' Choosing for each u E GKIF a preimage tia E U such that (J' = g(u a ) we have then au" = uuau;;l = aU, for a E AK. and furthermore au,~ == U,,'lJ.rU;: is a fundamental standard 2-cocycle of GKIF in AK' For each intermediate field E between F and K we put UE = g-l(GKIE)' In the two extreme caBeS E = F and E = K we have UF U and UK AK. In general we have AK C UE C U with UEIAK 1'>:J G K / E , the isomorphism being induoed by g. The 2-dimensional class of the group extension UE/A K I'::: G K/ E is the fundamental class (XKIE of the layer K/E, because it is the restriction to GK / E of the class O-K/F, of our original extension, and we know that Q.j(/E = re8QKIF' Let us now consider the group theoretical transfer map VUE •AK , designating it by VK / E for short. We have discussed this map for the case of an arbitrary group extension in §2 of Ch. XIII. It is shown there that VK / E carries Ue/U'i; not only
=
into
AK
but into A~KIE
= AE'
=
We therefore view VK / B as a homomorphism with
xv. WEaL GROUPS
159
values in AE: VK/E: UE/U'E
-+
A E.
In the corollary of Theorem 4, §2, Ch. XIII the kernel and cokemel of this homomorphism is analyzed in terms of kernels and cokerneis of the homomorphisms 0-3:
H- 3 (GKIS,Z) -+ H- 1 (G K1E ,AK)
and 0-2:
H- 2 (G K1E ,Z) -+ no(GKIE,AK )
which are effected by cup product multiplication with the 2-dimensional class of the group extension involved. In our present case this 2-dimensional class is the fundamental cla5s of the layer K / E, and by the maiD theorem of class field theory we know that the maps Di r : Hr(GK1E,Z) .... Hr+2(KIE) are isomorphisms onto for all T. It follows that the transfer map is an isomorphism onto, VK / E : UE/U;, ~ AE' We finish the construction of our Well group by defining the isomorphism IE: As ~ UE/U;, to be the inverse of VK / E. All that remains is to verify that properties Wl)-W4) are sa.tisfied, and this is not hard. Properties Wl) and W2) concern the commutativity of diagrams involving the isomorphisms /E. Replacing these isomorphisms by their inverses, VK/E, we see that WI) amounts to the transitivity of the transfer, namely
= VK/E'VEI/E(a). a E UE/U;" and W2) amounts to the rule, for q = g(u), (VK/E('»" = uVK/s(a)u- 1 = VKIE'" (uau- 1), aE UE/UB. VK/E(a)
which follows from the naturality of the transfer. 'lb verify property W3) we refer to Ch. XIII, §3, where the map v: H 2 (G K / E ,AK) -+ H2(GL/£,Ad
is defined. By the very definition of v we see that the class of the extension (*) mentioned in W3) is the image under v oftbe cl~s DiK/E of the extension Us/AK ~ GK/E. because the extension
UL/Ul -.. UE/uf
-+ GLIE
is a factor extension of the latter in the sense discUBsed in §3 of Cb. XIII, and the isomorphism ULIU'i ~ AL is given by the transfer VUL,AK' And since DiLlE = tJ0K/E it follows that uLIE is indeed the class of the extension (*). Finally we see that W 4) is satil:ified because by our construction UK AK and is abelian. This concludes the proof of Theorem 1. 0
=
Having defined the notion of Weil group, and shown the existence of Weil groups, it is natural to consider the question of isolnorphisms of Wei! group. It is clear how we should define isomorphisms, namely DEFlNlTION 2. Let KIF be a normal layer in a class formation. Let (U,g, {IE}) and (U',g'. U;".}) be two Weil groups for the layer K/F. Then a Weil isomorphism 'P from one to the other is an isomorphism 'P: U ~ UI with the follOWing two properties.
1'10
XV. WElL GROUPS
WI 1) The following is commutative
U""!'-G K1F
1 rp
!
ideut-ily
"
U'_G K / F • WI 2) From WI 1) it is evident that cp(Ufj) = U~, for each intermedia.te field E between F and K, and consequently cp induces an isomorphism CPfj: UE/Ui ::::: UE/UEc. Property WI 2) requires the commutativity of
AE~UE/U'i;
l~ti~
!rpB
AE ...l!..... U'E/Uli lor eacl! E. THEOREM 2. There exi3ts a Weil isomorphism cp: U ~ U' for any two Weil groups U and U' of a layer KIF. Furthermore, cp is unique up to an inner automorphi5m of U' effected by an element of Ui.
PROOF. If cp is a Wei! ilKlmorphism then the following diagram is commutative, by WI 1) and by WI 2} for the extreme case E = K:
(1) -
AI{
!
{g
UK
~ U~ GKIF ---- (1)
(l)-AK
1~.
1
Id.,·
rp
{s" Uk~UI~GK/F-(l)
Conversely, we contend that any homomorphism cp: U ..... U' which makes this diagram commutative ill a. Weil isomorphism. Indeed, let cp be 5uch a homomorphism. Then, from the exactness of the rows, it foHows that cp is an isomorphism of U onto U', and by the commutativity of the right hand square we have CP(UE) = Ui; for each intermediate field E. Thus cp induces an isomorphism CPE: UB/Uj; ~ UB/UFf and we can consider the following cube:
AE
Sd. j
loci
a
AK
~
j"{"
UE/IUk
V
!id.• UjI{
AE --,- ~ AK
'!:
Y'E
Ui;/Ui;c
V'
,
'-!.K ..
¥'K
Uk
The top and bottom faces are commutative by property WI) of Wei! groups. The back face is obviously commutative. The front face is commutative by the naturality of the transfer map V, because cP : U ~ U' is an isomorphism mapping UE on U and UK on Uk. The right hand face is commutative because the left hand square in the preceding diagram is commutative. Since the horizontal arrows are
e
xv. WElL
GROUPS
171
isomorphisms into, we conclude that the left hand face of our pube is commutative, and this shows that q; satisfies property WI 2) for all intermediate fields E and is therefore a Wei! isomorphism. • The problem of Wei! isomorphisms cp therefore boils down to the problem of middle arrows cp in the diagram at the beginning of this proof, that is, to the problem of homomorphisms between the two group exteIll5ions represented by the horizontal rows of that diagram. These two group extensions have the same 2-dimensional class, namely OcK/F' Consequently, maps cp exist, by Theorem 2, §1 of Ch. XIII. Moreover, since the I-dimensional cohomology group Hl(GK/F' AK) is trivia.i, it follows from the uniqueness part of that same theorem that I{J is determined uniquely up to an inner automorphism of U' effected by an element of U~ = f~ (AK ). This 0 concludes the proof of Theorem 2. THEOREM 3. Let FI C F e K e Kl with KdFl and KIF normal. Let (U,g, {IE}) be a Weil group for the big layer K 1/FI . Then (upfU'k,g, UdFCECK) is a Weil group for the small layer K / F, where 9 denotes the homomorphism of UF/U'k onto G K / F , which is indu.ced by 9 in the obvious way.
PROOF. This theorem is evident from the definition of WeibJ group. The lattice diagram at the right may help in visualizing the situation. 0
Now let KI/ FI be a fixed normal layer, and let (U,g, {fEl) be a Well group for it. In the next few paragraphs we suppose that all fields F, E, K, ..• under consideration are intermediate between K 1 and Fl' Clearly by choosing our fixed normal layer K)/ FI suitably large we can arrange that any prescribed finite set of fields F. E, K, ... are contained between FI and K I , and so are "under consideration". THEOREM
U~
"-Uk
"- (1)
4. Let E / F be an arbitrary layer. Then the following diagmm is
commutatwe: UF/U'f
.t (map
'.
induced by ) the inclusion UE C UF
UE/U,;; PROOF. This theorem follows from a certain simple propertyl of the group theoretical transfer when we analyze what it says. There is a minor technical difficulty arising from the £act that we do not aBSUnle E/ F normal. Because of this non-normality, we must first choose a K (e.g. K = Kl) such that E C K and K/ F IThe generaJ property of trllonsfer which is essentiaJly proved bel~ is a$ follow&: If U ::) Ul ::) A are group6 with A IIobelia.n and normal of finite index in U. then, vieweing A l1li a. (UI A)-module in the usual way. the tran5fer from V to A of aD element "I E VI is the nann from A.U,IA to AU/A of the transfer of "1 from UI to A.
172
XV. WElL CROUPS
is normal. We can then refer the things we are interested in to AK and UK IUK by meu.os of a cubic diagram of the following type:
AF
~ AK
Ina
t~
j '!:.
UF/U'}
V
A:=-l~ '-!:. i.
I
A:
~ UK/U'k
1
V
UE/U E
~
IN.
• UK/UK
Here we mlUlt explain Nl and N 2 • We choose Nl 50 tha.t the back face of the cube is commutative, going back to the definition of N EtF • Namely we write
GK/F
= UO"iGK/E
(disjoint union)
and put
Next, we choose N2 50 that the tight face of the cube is commutative. According to property W2) of Well groups, this can be done by choosing elements E UF such that g(v.)
v.
= CT, and defining
(mod UK) for
U
E UK (mod UK). Notice then that we have
UF
= UV,UE
(disjoint).
Now the top and bottom of the cube are commutative by property Wl) of Wei! groups. Since the horizontal arrows of the cube are isomorphisms into, the commutativity of the left side, which we want to show, will follow if we can show that the front face is commutative. This means that we must prove for u E UE that
VUy,UK(U)
= II Vi (VUS,UK (u»vil. i
To do tbis we write (disjoint). Then
UF = UViUE
= UViUKWj = UUK'lJiWj,
i
;'J
iJ
the last because UK is normal in UFo Now by the definition of the transfer we have
VUF,UK(U)
=
n
v.WjUWislV~l,
i ,;
where (i,j) ...... (i1.il) is the unique permutation of the pairs (i,j) such that each fa.ctor of this product lies in UK. But since UK is normal in UF we see that this
113
XV. WElL GROUPS
permutation is achieved by selecting first it so that il i. And now we are through because
=
WjUW;;.l
V
E K and then putting •
o COROLLARY.
IF and is induce isomorphism& AF/Ns/pAE :::;: UF/UBUP. N'E}F(l):::;: (UE nUF)/U~,
THEOREM 5. Let K / F be a nOTTnallayer. Then the reciprocity law isomorphism lor that layer is given by
. IF Ap/NKIFAK :::;: UFIUKU'j ~ GK/FIGK/p· where the right hand isomorphism is that induced by g. In other words, if
g.: UF/Up ..... GK/FIGK/F is the homomorphism induced bll g: UF -+ OK/F. then 9.1: AF -+ GK/FIGK/F is the norm re.sidue map. For each u E GK / F , select a representative u" E UF such that u = = iF-l(uaUf,-) E AF. We must then show that bO' corresponds to Wlder the reciprocity law map. From the commutativity of PROOF.
g(u,,), and let bl1 U
AF~UF/Uf,-
!;ncl'K
/1 v AK-UK Uk we see that
h(bO') == VIF(bO') "" V(Uu)
=
nuTUO'u;;Uk =iK(n a..,.,), l'
T
where we have defined elements Ur,,, E AK by a..,,, == IKl(uTtlau;~Uk)' Then a..,a is a 2-cocycle belonging to the extension
AK ~ UF/UK.!!. GKIF and is therefore a fundamental 2-cocycle for the layer Wei! groups. Consequently, from
II" =
KIF,
by property W 3) of
II aT,,, =Image of u under Nakayama map T
we can conclude that b" and u do correspond under the reciprocity law, as contended. 0
The theorem we have just proved shows that the entire theory of the reciprocity law is contained in the theory of Well groups. The reader will easily check that all the results of Ch. XIV, § 5, can be recovered immediately from our present theory. The reciprocity relationship between levels AF and galois groups G K / F becomes easy to visualize when one identifies AF with UF/U'F (by means of /F) and identifies GK/F with UF/UK (by means of 9). In this way, all the facts are wrapped up in
xv. WElL GROUPS
174
one neat non-abelian bundle, namely a suita.ble Wei! group U. From this point of. view we get one additional dividend. the Shafarevic Tbeorem.2 • THEOREM 6. Let F eKe L with K/F and L/F norma!, and L/K abelian. Then we may view G L / F as a group extension 0/ GLIK by G KIF so there is determined in a canonical way a 2-dimensional class (3 E H2(GK/F,GL/K), the. class of this extension. By means of the reciprocity law isomorphism, GLIK ::::: AK/NL/KAL (which is a G K / F isomorphism), fJ determines then a class {3' E H2(G K/ F ,AK/NL/KAL). This class fJ' is the image of the fundamental class OK/I" E H2(G KIF,AK) under the natural projection of AK onto AK/NL/KA L. PROOF. The proof is evident from the lattice dia.gram at the left. We identify the various galois groups with factor groups of subgroups of U, and we identify AK with UK/U'K by means of Ix. Then NL/KAL is identified with UL/U'K, and the reciprocity law i50WOrphisw become3 the identity map of Ux/Uf, by the preceding theorem. Hence fJ' is the class of the extension
AK/NLIKAL
-+
UF/UL
-+
GKIF
and is therefore obviously the image of OXIF because oX/F is the class of the extension
UCK
Ax ..... UFIU'K
-+
GX / F •
0
The theorem we have just proved shows that if K / F is a norma.l extension, and if B is a norm subgroup of Ax ·which is a. G KI ,,-submodule, and if L is the class field over K belonging to B (so that B = NLIKAL), then we can determine the structure of the galois group CLIF in terms of objects associated with the layer KIF. Indeed, GLIF is i50morphic to the group extension of AxlB by G K / F belonging to the image of the fundamental class. In the preceding paragraphs we have seen how a Weil group for a big normal Jayer Kd Fl contains information about all intermediate Jayers ElF, and in particular contains, as factor groups of subgroups, the Wei} groups for all intermediate normal layers Kj F. This suggests that we try to go to the limit and construct one universal group, a Wei! group for the whole formation so to speak, which will have all the Weil groups of all finite normal layers as factor groups of subgroups of itself. This is the next step on our program. In order to carry it through we must assume that our formation is a topological formation (ef. Definition, eh. XlV, §6) which satisfies a certain compactness condition, a condition which is satisfied in local and global class field theory. We shall also view the galois group, G, of our formation as a topological group, the neighborhoods of 1 in G being the subgroups GF. Thus 2This theorem is due to ShafareYich. He observed that it is a ~uenoe of a simple relation between the Akizuki-Witt map tI and the description of the Dorm residue correspondence via the Nal
xv. WElL GROUPS
.
17l!
all group!i and modules to be considered from DOW on are topological. As usuaJ. in the theory of topological groups we must distinguish between the notion of a representation and the Dotion of a homomorphillm. A representation is a continuous map of one group into another which is algebraically a. homomorphism. A homomorphism is a representation I: G - H such that I induces a. homeomorphism between G/Kernel! and Image j, in other words, such that f(U) is open in f(G) whenever U is open in G. These things being said, we can now define the type of object which we aim to construct: DEFINITION 3. Let (G, {GF}. A) be a topological class formation. A Wei! group (U, g, UF}) for the formation consists of the following objects: 1) A topological group U. 2) A representation, g, of U onto an everywhere dense subgroup of the gaJ.ois group G of the formation. Having 9 at our disposal, we can introduce, for each field F of our formation the subgroup UF = g-l{G F). The UF's will then be open subgroups of finite index in U whose lattice reflects exactly the lattice of the GF 's. We have UE C UF ¢;? FeE, and UE is normal in UF if and only if ElF is a normal layer. If this is the case, then 9 induces an isomorphism UF/UB R:: G E / F • (These things are true because y(U) is dense in G, and consequently, for any E, we have 9(U) . GE "" G, because
G E is open in G. In other words, every coset of G E contains an element of the form g{u).) The third ingredient of our Wei! group is 3) For each field F of our formation, an isomorphism (topoIogicaJ. and algebraic)
IF: AF ~ UF/Uj., where Uj. denotes here, and from now on, the closure of the commutator sub. group UF. In order to co~titute a Weil group for the formation, these objects U, g, and {IF} must have the following four properties: W 1) For each layer
E/F, the following diagram is commutative
where V is the transfer map. (It is eMY to verify, in case of a topological group G and an open subgroup H of finite index, that the transfer map VG,H carries GC, the closure of the commutator group of G into He, the closure of the commutator group of H, and eODSequently induces a map ofGICC into HIHc. It is this latter map which is meant by V here, and from now on.) W2) Let IL E U and let (f =g(u) E G. Then it is clear that U(UE)U- 1 = U(1)'')' Property W 2) states that the following diagram is coDUDutative for each field E:
IE R::
UEIU,/;;
!
coajugation by ..
I,;
UE"IU'j;.
]76
XV. WEIL GROUPS
W 3) For each uormallayer KIF, the clas5 of the group extension
(1) -- AK::::l UKIUk
-+
UF/Uk
-+
UF/UK ~ GKIF -+ (1)
is the fundamental class of the layer K / F. W 4) We finally require that
U .... ~u/UK be an isomorphism of topological groups. This concludes Definition 3. Suppose for the moment that U is a Weil group for the formation. It is obvious from the definition that for each normal layer K / F, UFlUK iB a Weil group for that layer. In particular, if k is the ground field of the formation, so that U = Uk, then the factor groups UIUk for variable K normal over k are Well groups for the varioua normal layers K/k. On the other hand, property W 4) above states that U is the projective (inverse) limit of these factor groups. This shows how we must go about com;tructing U; we must get it as the projective limit of Well groups of finite layers K/k. Let us carry out thiB program. From now on, unless specific mention is made to the contrary, all fields F, K, L, M, ... are understood to be normal over k. For each such field K, let (UK, gK, {If}) be a Weil group for the layer K/k. It will be convenient, to suppose that the isomorphism Iff: AK R:I Uff is the identity map, in other words to identify Ax with its isomorphic image for each K. This being said, we topologize UK by taking as fundamental system of neighborhoods of 1 in UK a fundamental system of neighborhoods of 1 in AK; in other words, we give UK the unique topology for which Ax is an open subgroup of UK, and for which the topology induced on AK iB the same as that which AK gets as a level of our topological formation.
uff,
LEMMA. ForeachfieldE, k c E c K, normai or not, the commtdatoraubgroup Df Uff i.!i clo.sed; in other words, (UI)C has the same meaning as before.
PROOF. Let for the moment (U:)C denote just the cOllllIlutator subgroup rather tha.n its closure. TheIl (Uf) n AK is closed in Ax because it iB the kernel of NX/B' and the norm map NK/E is continuous in a topological fonnation. Being closed in AK, it is closed in UK. On the other hand (Uf)C n AK is of finite index in (Uff)C because Ax is of finite index in UK. Therefore, being the union of c is closed. a finite number of closed oosets, 0
(Un
iff : As
LEMMA. The isomorphism only if NK/E i.!i an open mapping.
PROOF. The following diagram
::::l
U~ l(ujf)C is a homeomorphism if and
is commutative
{g U}f I(Ul)e .
/K;id. t::t
.t (map A
indUced)
by inclusion
t.
K
Since AK is open in UK, i. is an open map. And since IK is a homeomorphism, it follows that if IE is a homeomorphism, then NK/E must be an open map. Conversely, if NK/B is an open map. then NK/EAK is an open subgroup of As, and is
xv. WElL GROUPS
177
homeomorphic to AKINK~E(I). On the subgroup NK/EA K • the composition of two homeomorphisms, namely,
IE induces therefore
NK/EA K :;::: AK/Ni()E{l):;::: i.(AK). Since the left hand and right hand of these groups are open in AE and in respectively, we conclude that IE is a homeomorphism.
ut /(Ui)C 0
Now to proceed to the construction of our projective limit we must find a collection of "reasonable" homomorphisms C{JL/K: U L -+ UK, one for each pair (L, K) such that L :::l K, which are transitive in the sense that '{JMIK = 'PL/K'{JM/L whenever M :::l L ~ K. What is to be meant by "reasonable"? The answer is obvious once we recall that UI. /(U!
PROOF. In the following diagram
I
UI./(UL)c K
• UK
K restriction of tbe Weil
i.
~d
UI.I.
K
isomorphism lDduced by
...\.
0\ ~"
\.
'f,.\{/
~#
the left side square is commutative, and the top square is commutative, by the definition of Weil isomorphism. Factoring the diagonal arrow (restriction of 'P) into the product of i. and the Weil isomorphism, we see that the restriction of cp does induce NL/K on AL as contended. Now since CPLIK maps U I. onto UK, 'PL/K will be a topological homomorphism jf and only if it is an open map. Since the subgroups AL C U I. and AK C UK are open subgroups, it is clear that 'PL/K is an open map if and only if its restriction to AI. is open, i.e. if and only if NLIK is open. This concludes the proof of the lemma. 0 According to Theorem 2, a Weil map '{JI./K exists for each layer L/ K, but is not unique, bcing determined only up to an inner automorphism of UK by an element of AK = uff. This non-uniqueness will be the main obstacle we shall encounter in constructing our projective limit, since it forces us to make a selection. We must therefore discuss in some detail the inner automorphiBIllS by which the Wei! maps
XV. WElL GROUPS
178
can be changed. In order to have a notation for them, let us denote by aK the inner automorphism of UK effected by an element a E AK ; that is, we put aK(u) aua- 1 fur u E UK. When is aK the identity automorphism? Clearly, when, and only when, "commutes with every element of UK. Selecting representatives U~,q E GKlk, for the elements of UK I AK l'::j G K /k we see that the condition is that a commute with each ua, because, AK being abelian, it will then foHow that a commutes with each a for all element of UK. Since u.,.au cr 1 = a~, this condition means that a~ a E GKI"" in other words that a E A/c. Thus we have shown that the group of inner automorphisms of UK by elements of AK is isomorphic to AKIAk. For each layer L :J K, let Xl-,K denote the set of all Weil maps 'fIL/K: U L _ UK. If 'fI1/K is one fixed element of XL,K, then the other elements are of the form aK'fJ1IK' with 0 E A K . Since 'P~/K is an onto map we have aK'fI~/K = bK'fI1/K ¢} aK bK <=> ab- 1 E Ai; ~ aAk bA",. In other words, we have a one-one correspondence
=
=
=
=
aAk .... akIP1/K
between AKIA" and XL,K. This allows us to topologize XL,K by the topology which is induced on it by the topology of AKIAk, i.e. such that the above oneODe correspondence is a homeomorphism. This topologization of the set of Weil maps XL,I<. is independent of the choice of 'fI~/K because the topology of AKIAk is invariant under translation. LEMMA. The composition of Weil maps is continuous. More precisely, if M:J L :J K, then the composed map 'PLIK'PM/L E X M / K is a continuous function oj the tWQ variables 'PL/K E XL/K' flnd 'fIM/L E XM,L'
PROOF. Note first of all the rule
'PL/KOL
= (NL/Ka)KI.{)c,/K
for a E AL, 'PLJK E XL,K' Indeed, for u E U L we have 'PLIKaL(u)
='fIL/K(aua- 1) =I.{)L/K(a)'PLIK(U)('fIL/K(U)r 1 = =(Nc,IKa)'fIc,/K(U}(NLIKa)-l = (Nc,IKa)KIPc,IK(U),
because the restriction to Ac, of 'fIL/K is NL/K by the preceding lemma. For a E AI., bEAK we have therefore (bK'PL/K )(aL'PM/L)
= (bNc,/Ka)K'PL/Kf{JM/L.
Since bNL/Ka is a continuous fundloD of the variables a E AL and" contention follows.
e
AK, our
0
LEMMA. Suppose AK I Ak is compact for each K. Then there exists a transitille collection of Weil maps IPI./K: uL -+ UK,
PROOF. A random choice of Weil maps ma.y be thought of as an element 'P .,. ('PLIK)
of the cartesian product
X=
II XL,K.
L:JK
the product being taken over aU pairs (L,K) such that L :J K. Since we have assumed AKIAk is compact, we know each XI.,K is compact and consequently their
xv. WElL GROUPS
179
product X is compact in the product topology. Now for eadl triple M J L J K, let X (M, L, K) be the subset of X consisting of the element,s I{J such that C(JM/K
= C(JL/KC(JM/L'
Our task is to show that the subsets X (M, L, K) have a non-empty intersection, an element of their intersection being a choice of Weil maps which is transitive for each triple M :J L J K. Each of our subsets X(M,L,K) is closed, by the preceding lemma. And since X is compact, it will be enough if we show that the sets X(M, L, K) have the finite intersection property. To this effect, let (Mi,Li, K i ), llO; i lO; n, be a finite set of triples. Select afield P such that P :J Mi , for each i. We shall show that there exists an element cp E X such that C(JM/K = 'PL/KI{JM/L
for all triples M, L, K SIlch tha.t P J M J L :J K :J k, and therefore in particular for our given finite set of triples. To do this we select, for each field F between k and P, a Weil isomorphism 8F: uP I(uC)C ~ U F
and then we put, for each couple L J K between k and P: tpL/K
= 8i/'I/lL/K8L: U L -+ UK
where 'I/lL/K denotes the natural map of uP I(UfY onto uP I (UI::)c. These tpL/K'S are obviously Wei) maps, and their transitivity, in the levels between P and k, follows from the transitivity of the natural maps 'I/lLIK' Finally, whenever P 1> L, we choose a 'PLjK at random, obtaining then 8. tp = ('PL/K) BUch that I{J
E
n" X(M., L" Ki). i=1
This concludes the proof of the lemma.
0
THEOREM 7. Suppose (G, {GF}, A) is a topological cllw formation satisfying the following three conditions. WT 1) The norm map Nlf,/F= AE -+ AF is an open map, for each layer ElF. WT 2) The factor group AE/AF is compact for each layer E/ F. WT 3) The Galois group G is complete. Then there exists a Weil group (U, g, {iF}) for the /()'I"mO.tion, and it is unique tip to isomorphism. PROOF. For the details of the proofs ofthe basic facts about projective limits which we shall use in our proof we refer the reader to Weil [27, section 5]. Choose a transitive collection of Weil maps 'fILIK: U L -> UK (this is possible by hypothesis WT 2) and a lemma). Each 'PLIK is a (topological) homomorphism of U L onto UK (by hypothesis WT 1» and a lemma above. I contend that the kernel of 'PLIK is compact. Since AL is of finite index in U L , AL nKertpL/K is of finite index in KertpLIK so it suffices to prove that AL n Ker tpLIK is compact. Since the restriction of tpL/l< to ALIK is NL/K we are reduced to proving that Ker NL/K is compact. Now i(GLIK)AL = nO'EG L1K At-I) is of finite index in KerNL1K because the factor group is H- 1 (G L /K,Ad. It will therefore suffice to prove IGL1KAL is compact. For ellCb q E G L/ K • the map a -> aO'-l induces
180
XV. W£IL GROUPS
=
a representation of AdAK onto Ar:-l, because A'K- 1 1. Therefore, Ai:- 1 is compact, being a continuous image of AdAK which is compact by hypothesis WT 2}. Consequently IGL/KAL is compact as contended. We have now shown that the family of topological groups {UK} together with the family of homomorphisms {lPL/K} satisfies Weil's conditions LPI, LPII, and LPIII". It follows that we can build a projective limit U with all the desirable properties one could wish for. We form the direct product
11 UK K
of our Weil groups UK, /Wd in it we consider the subgroup U consisting of aJl elements UKEU K
U=(11,K),
such that UK = IPL/KUL for all pairs L :) K. We topologize U by giving it the topology which is induced by the product topology in the direct product. At first sight, this means that a neighborhood of 1 in U is given by a finite set of fields K i , together with a neighborhood Wi of 1 in UK for each i, the corresponding neighborhood in U consisting then of the elements u = (UK) such that UK, E W. for each i. However, taking into account the "coherence" of the components of 11" i.e. the fact that UK, = IPL/K.UL for a suitable L ::) K; aJl i, we see that it suffices to consider the neighborhoods of 1 in U which are given by a single field L together with a neighborhood WI. of 1 in U L , these constituting a fWldamental syBtem. For each L we have a map IPL: U -> UL defined by
!f1L(U)
= UL
for u = (UK) E U. Using the compactness of the kernels of the IDap5 'PM/L for fields M ::) L one shows easily that 'PL is onto, and consequently is a topological homomorphism of U onto UI.. We must now construct the representation g: U -0 G, and to do so we must use the hypothesis wr 3) which states that G is complete in the topology for which the subgroups GK are a fundamental system of neighborhoods of 1. This completeness assures us that G Is the projective limit of its factor groups GKI" = G/GK, 88 follows: Let 'l/;l./K! GLI" - 0 GK/r. be the natural map. Then, given any family of elements {D"K}, with D"K E GK/k, such that UK :0 'lJ!L/KD"L for each pair L :J K, there exists a unique element
=
y(u)
= (gK{UK))
E
G,
in other words, we define 9(1.£) to be the unique element of G such that g(U)GK
= gK(UK}
for each K. From another point of view, if we introduce the na.tural maps 1PK: G GK/k, then we see that our map g: U _ G is characterized by the property that ,pKg gKrpK for each K. Our 9 is obviously an algebraiC homomorphism. Its image, g(U), is everywhere dense in G. Indeed, for any neighborhood GK of 1 in G, we have g(U)G K = G,
=
XV. WElL GROUPS
181
because gK maps UK onto GKlk = G/GK' Finally, one checlq; inunediately that 9 is continuous. Having 9 at our disposal, we can introduce the sUbgrouP'UE = g-l(GE) of U, for each field E (not necessarily normal over k).
LEMMA. Let K :J E :::) k, with Klk nonnal. Then UE
Uk
= 'Pi1(Uff)
and
= 'Pi/ (Uff)c). PROOF.
the rule T/.IK9
The first statement is almost evident. The formal proof depends on = gKrpK and runs as follows: rpil(U~)
= 'Pi/(gK)-l{G K/ E )
=(gK'PK}-I(G K/ E) =(IPK9)-1(G K/ E ) =g-11/Ji1(G K / E ) =g-l(GE) =
Uf;.
To prove the second statement, let U~ denote the commutator group of UE so that Uk is, by definition, the closure of UE in U. Let L be any field containing
K such that L/k is normal. Then from what we already have proved, namely UE = 'PLl(U~), and from the fact that 'PL is an onto mapping, we can conclude that
UHKer'Pd = 'PLl«U~)C). (Reca.ll that (Ui)C = (Ui)' by a lemroa.) Now from the theory of Weil groups of finite layers and from the definition ofWeil map we know that (Uk)C = 'PLjK«U~)C). And since 'PK = 'PL/K'PL, we conclude U~(Ker'PL) = 'Pi/(Uff)C),
a formula in which the left side contains an arbitrary L :::) K, and the right side is independent of L. From the definition of the topology in U, we can make Ker'PL an arbitrarily small subgroup by selecting L sufficiently large. Hence U~
= UE: :) 'PKl«U~)c).
The opposite inclusion is trivial since 'P'i/«Uff)C) contains U~ and is closed. This concludes the proof of the lemma and we now continue with our construction of the Weil group U. 0 Since 'PK: U ..... UK is onto, we conclude from the lelWl1& that 'PK induces an isomorphism 'P~: UE/U~ ~ l(Uff)C, and this isomorphism is topological as well as algebraic because 'PK is a topological homomorphism. Thus there p.xista a unique isomorphism IE such that 'PK 0 IE Iff, i.e. such that the diagram
uff
=
AE .
~
,. UE/U'i!.
~ /.~) Uf/(Uff)C
182
XV. WElL GROUPS
is commutative. Moreover this isomorphism IE is topological as well as algebraic because the same is true of f~, by hypothesis WTl and the second of the above lemmas. The propertie:> WI-W4 for the Weil group (U,g, {ld) of the forma.tion DOW follow readily from the corresponding properties for the Weil groups of finite layel'll K/E, and our existence proof is complete. 0 Unicity was not discussed in the earlier editions. In these last several page:>, added in 2008, we fix that gap and go on to discuss a few more things about Well UOUps, including an account of Well's original proof of their existence. Concerning unicity, it is proved in !25, Proposition (1.3.1)] for the classical situations, and the proof there works in the present a.bstract context. If U and U' lIIe two Weil groups for the same formation one shows, by the same type of compactness argument used to prove the lemma just before Theorem 4 of this chapter, that there exists a compatible family of isomorphisms (JK : U/UK -. V'IU'/i commuting with IK and Ik and with the maps induced by 9 and 9', for normal extensions K of the ground field k. Then (J = ~9K: U -. V' is an isomorphism orWell groups. 0 THEOREM 8. (a) Let (U.g,{fF}) be a Weil group Jor II (G,GF, A). For rockfield F, the composed map
f
.b
10f'frUltion
cat
=~ where g~ is the map induced by 9, is the reciprocity mapping. (b) If for every normal layer K/k there is a cyclic layer Llk of the same d.egree, then in the definition of Weil group for a class formation (Definition 3), one can substitute statement (a) 01 this theorem for the condition W 3) mvolving the fundamental classes.
(1)
Ap 2.., UFb ~
daBS
PR.ooF. For (a) it suffices to prove that for every abelian normalla)W KIF, that the composed map AF -. Vp/UK
-I
Gp/GK :::; GKIF .'
where the arrows are induced by IF and Gab, is the reciprocity map for that layer, and that is true by Theorem 5. The fundamental class of a layer of degree n was defined ill Chapter XIV as the class with invariant lin, so to prove (b) we have only to show that in a class formation satisfying the hypothesis of (b) the reciprocity maps determine the invariant functions. That hypothesis implies that every two-dimensional class comes by inflation from a cyclic extension. For a cyclic exteIIBion KIF cut out by a character X every class is of the form aOx and h
If (U,g,{jF}) is a Weil group for a class formation (G, {Gp},A) in whic.hG is complete, it is clear from the definitions that we can recover the formation up to isomorphism from knowledge of only the topological group U and the collection of subgroups {Up}, for we have isomorphisms G = ~(GIGK) = !!.!!!-(V/UK),
GF
""
lli!!{GF/GK):::; ll!!!-(UFIUK),
the projective limit being taken over all small V K normal in U and the maps IF give an isomorphism
xv. WElL
GROUPS
183
the injective limit on the right being taken over all UF'S rel(l.tive to the transfer homomorphisms VE / P ' This suggests we define a new type of mathematical objects which we will call W-group. DEFINITION 4. A W-group (U, {UF}) is a topological group U, together with a family of subgroups {Up} such that WG 1) The family {Up} consists of open subgroups of finite index in U, is closed Wlder finite intersection and conjugation, and contains all subgroups containing any one of its members. WG 2) For every pair UK
It is clear that the Weil group of a class formation satisfying the conditions WT 1,2,3 of Theorem 7, gives us a W-group. In fact, the converse is also true. THEOREM 9. Up to isomorphism, every W-group cornea from a class formation lJaUsfyi,ng the three conditions of Theorem 7.
PROOF. Let (U, {UF }) be a W-group. As explained in a paragraph preceding Definition 4, we can construct from U a formation (G, {G F }, A) by putting
G=~(G/GK)=~([[/UK)
and
A=~(uF/UH
and taking for G F the inverse image of UF/UK for all sufficiently small UK normal in U. The level AF, which by definition is AGp- can be identified with the isomorphic image of UF/U'} in A because the injective limit is made with the transfer maps, and for all small UK normal in U we have VK/F(UFIU'j.) = (UK/U'KF,,/UK (UK/Uk)GF/GK. If 9: U -+ G is the canonical map, and IF : AF - UF/U'F is our identification map, then (U,g, {IF}) is a Weil group for the forma.tion. This formation does satisfy the three conditions of Theorem 1. The norm map NE/pAs -+ AF is reflected in the canonical map Us/UE--+ UF/Up which is open; the inclusion map AF ~ AE is reflected in the transfer map which has compact cokernel by the definition of W-group; and G is profinite, hence complete. The only question is whether our formation is a class formation. To see that it is, let K / F be a normal layer and consider the exact sequence
=
0--> AK ~ UK/Uk ..... Up/Uk ..... UF/UK ~ GFIGK - 0 which to simplify notation we will denote by
0--+ AK
--+
uff -+ GKIF --> O.
Let Q E 'H,z(GKIF,AK) be the clll$ of this group extension. Let {U~y be the algebraic commutator subgroup of Uff. LEMMA.
(Uff)' iB compact, hence (UIf)'
= (UIf)e = U'j./U'k, and U;.. is com-
pact. We postpone the proof of the lemma. Assume the lemma is true. Then the --+ A~Ktr is bijective, by condition WG 2) of Definition 4,
transfer map ulf I (Ulfl'
XV. WElL GROUPS
and it follows from (Chapter XIII, Section 2, Theorem 4) and its corollary, (in both of which UC denotes U'), that (La is surjective and a-2 is bijective, where an : 'Hn(G K / F , Z) --> 'H.n+2(GKIP, AK) is the map given by cup product with a. ALso, a-I Winjective because 'H.-l(G K/ F , Z) = O. Hence, by the general cohomological theorem alluded to in the proof of Theorem 1 of Chapter XIV, §4, an is bijective for all n E Z. For n = -1 this implies our formation is a field formation, and for n = 0 it implies that 1-{,2(GK / F ,Z) is cyclic of order lGK/FI == [K : Fj, generated by a = ao(l). We denote this class byaK/F when different layers are involved. For K C L normal over F, (XIII, §3, Th.6) shows that inflL/K(aK/F) = [L : K]a:L/F, because aK/F = veaL/F), by definition of v and the a's. Also for FeE c K we have aK{E rf:1!,E/FaK/F. These facts allow us to define "invariant maps" invF : 'H2(Gp. A) ..... Q/Z which satisfy the class formation axiom, Axiom II of Chapter XIV, Section 3, by putting invp(OI.K/F) =-= !K~FI'
=
PROOF OF THE LEMMA. Consider
IK/pA K :=
the inclusions
IT Aka c (Uff)' n AK c (U:)'. tleG K / F
The index of the first inclusion is finite, because, by Theorem 3 in Chapter XIII, Section 2, the quotient is a homomorphic image of 'H.- 3 (G K / P ,Z). The index of u is the second inclusion is finite because AK is of finite index in Uff. Each compact, as continuous image of AK/AF. Thus IK/pAK is compact and so also is (Uff)', as finite union of compacts. Hence (U}f)' is closed, i.e, equal to its closure (Uff)C = Uf;./U'k,. Finally, UF is compact as projective limit of the compacts U'j-/U'k. This proves the lemma and the theorem. 0
Ak-
THEOREM 10. The existence theorem holtL; for the class formation associated to a W-graup (U, {UF}) if and only if the family {UF} consist.5 of all open subgroups oj finite index in U.
PROOF. The existence theorem for a "field" F means that every open subgroup
of finite index in UF which contains Ui is of the form UK for some ''field'' K, which is then the "class field over F to the subgroup UK/U'j;. of Up/UP. = Ap". This is certainly true if every open subgroup of finite index in U is a UK, because finite index in UF implies finite index in U. Conversely, suppose the existence theorem holds. Let W be open of finite index in U = ~K(U/Uk)' A neighborhood of 1 in the projective limit topology contains the inverse image in U of a neighborhood of 1 in U/U'k for some K. Thus, W::> Uk for some K. By the existence theorem for K, there exists a "c1assfield Lover K belonging to the subgroup (W n UK )/U'K of UK/U'k", that is, a field L such that UL UK. and since W contains UL. it is of the form Up for some F. 0
= wn
It follows that a topological class formation satisfying the existence theorem and the three conditions of TheoreUi 7 is mathematically equivalent to a special type of topological group U, one which satisfies the two conditions WG 2) and WG 3) of Definition 4, if we take the collection of all open subgroups of finite index as the family denoted there by {UF}' In class fonnations in which the reciprocity maps AF -+ GpIG~ are injective, the map U ..... G is injective, and one can "find" 11. Weil group U for ~he formation
XV. WElL GROUPS
185
inside the Galois group G. This is the case for nonarchimedean local fields and global function fields. We discuss the latter case as an example. Let k be a global function field, p its characteristic, k.ep 'a separable algebraic closure, ko the (finite) const8J1t field, and ko the algebraic closure of ko in ks..p • Let G = Gal(ksep/k) and let U be the subgroup of G consisting of the element8 which act on leo as a positive or negative power of the F'robenius automorphism x ...... x p • Let VI = Gl = GaI(k.ey/kok) be the subgroup of U consisting of the elemenu. of G which act trivially on k o. Topologize V by declaring UI to be open in V and giving VI the profinite Krull topology of G I . ThU5 U/VI = Z with the discrete topology, whereas G/G I = Z. Let 9 : U --+ G be the inclusion map. For each finite extellSion F of k in k.ep, let GF' = Gal(ksep/F), and VF' == V n GF "" g-I(GF')' Let TF : GF ..... GF/G"F be the reciprocity map (denoted by w in Chapter VIII). The image of rF is UF/Gp.. We have G'j, = V~, because UF is dense in GF and ~ c G) = Ui' Let iF : CF -+ UF/U'} be the bijection induced by rF. Our choice of the topology of U is such that the iF'S are lliomorphisms of topological groups (cf. Ch. VIII, §3). Thus (U,9,{jF}) is a Wei! group for the formation (G,{GF}'C.....,I,). This should be clear from the preceding discussion, except perhaps for the fact that for each normal layer K/ F the class of the group extension
is the fundamental class. But tha.t is true by part (b) of Theorem 8, because part (a) holds, by our construction of U: It was Well who first focused on the fact that the Takagi-Artin class field theory, 88 expressed for infinite extensions by Cheva.lley with ideJ.es, could be interpreted, in the case of function fields, in terms of the group U we have jU5t describt;d. His belief in the deep analogy between function fields and number fields and his hope to find a non-abelian analog of Heeke's L-functions with Grossencharacters led him to expect that a similar group V might exist in the case of number fields, a group in which the quotients UF/U<j. are isomorphic to the idele class groups CF, so that one would have a Galois-like interpretation of the fun group GF, 88 a refinement of Chevalley's interpretation of the group of connected components Cp/DF 88 Gp/G'j,. Wei! believed that to lind a theory in which such a group U appears as naturally for number fields as It does for function fields is a very important problem, perhaps holding the key to the Riemann hypothesis. He was at least able to show that this idea was not a pipe dream, by proving in (28) by an artificial construction that such a group U, with all the properties one would expect, does indeed exist for number fields, and is characterized up to isomorphism by those properties. To find it occurring naturally is still, 60 years later, an open problem. How did Wei! conatruct the group U nsing only the classical Takagi-Artin theory, without the theory of global fundamental classes which was the basis of our method? Although our way is more general, and perhaps more natural, once one has the theory of fundamental classes, Weil's is a natural direct attack on the specific problem of global number fields, and is certainly of intrinsic and historic interest. We finish this chapter by describing WeB's method in an abstract situation briefly, in a series of exercises.
XV. WElL GROUPS
186
To begin, we must define what Well constructed. For that we have only to modify oW' Definition 1 at the beginning of this chapter as follows: (i) Replace "class formation" by "field formation with reciprocity maps AF ~ GFIG'];. satisfying (Chapter XIV, Section 5, Theorem 6)" (ti) Have 9 he a map of U into GpIGl<. rather than into GKIF = GFIGK . (iii) Replace condition W 3) involving the funda.mental class by: W 3') For each intermediate field FeE map induced by g. Then the composed map
c K,
let g~b :
ut
-I
G~ denote the
g~b rtG~ A EJ-Bvnab E - ue
is the reciprocity map rEo Call a triple (U, g, {IE}) satisfying this modified definition a. Weil group in Weil's sense for the layer KIF. Note that it has a key feature, the map gin (li) above, which is missing from our definition of Wei! group of a layer. We recover the existence of 9 only after passing to the inverse limit to construct a Well group for the whole formation. In [28J, Weil showed for the formation of idele classes of global number fields that for normal KIF a Weil group in Well's sense exists and is unique up to isomorphism. The key special properties of number fields which he used are: (1) For each intermediate field FeE c K the reciprocity map rE : CE GBIG"E is surjective. (2) The GKlrmodule DK := Ker(rK) is isomorphic to a direct sum of modules induced from subgroups of order 1 or 2 of G KI F. (3) For each intermediate E, H.l(GK/E,DK) = 0 and NK/BDK = DE' THEOREM 11. Let (G, {G F }, A) be a field formation with reciprocity laws satisfying Theorem 6 in Chapter XIV, §5, properties (1) and (3) just above and, instead of (2), the slightly more liberal condition (2') The G K/ F -module D K := K er{ r K) is isomorphic to a direct sum 0/ modules induced from cyclic subgroups O/C K / F • Assume also th4t G i.s complete so that we have reciprocity maps rF : Ap CW. Th.en a Wei! group in Weil's sense exist.s Jor a normal layer KIF of the formation, and is unique up to isomorphism. PROOF. (Sketch, in a series of exercises. The method is essentially Weil's, except for the additional technical difficulty posed by assuming (2') illlltead of (2).)
1. Define a W-diagf'affl for K / F to be an exoct commutative diagram
o --
AK
!
.!..
rK
UF
!9
1...
GKIF
--
0
! id.
o -- GK/G"K ~ GFIG"r<, ~ GK/ F - + O. Using the long exact cohomology sequence associated with the short exact sequence of G KI rrnodules 0 - DK
.....
AK ~ GKIG"K =
G"J --+ 0,
show that a W-diagram exists.
2. The group 1i2 (GK/P, DK) acts simply and transitively on the set of isomorphism classes of W-diagrams for KIF. To see this, fix a section €I i-+ iu of the
xv. WElL GROUPS
1111
canonical map GFIGk -+ G K1F which occurs in the bottom row of W-diagrams for KIF. Let a~~ = 'Ya'YT'Y;! E Z2(GK/F,GKIG'k) denote the corresponding 2-cocycle, which represents the class of the Galois group extension in the bottom row of each such diagram. The set of lifts of this cocycle to AK if; a coset of Z2(GK / F ,DK) in Z2(GKIP,AK), where Z denotes cocycle. Barh lift acr,T defines a W-diagram (U,g, ... ) with elements Ua E U such that -1
u"u.,..u.....
="",,,,
Show that the product action of Z2(GKIF, D K) on the coset of lifts induces a simply transitive action of?{J ( GK/ p, D K) on the set of isomorphism classes of W -diagrwns for KIF. Call this action "twblting". 3. Suppose (Up,gp, i,j) is a. W-diagram for KIF. For each intermediate field E, FeE c K, let Us = r1(GK/s) = g-l(GdG'kJ and note that by replacing F's by E's in the diagram in (1) we obtain a W-diagram for KI E as "subdiagram" of the one there for KIF. The transfer map U;;b - u;t = AK factors through the inclusion AE '-I AK. Let VK / E : U1;b -+ AE denote the map it induces. Show that a Weil group (U,9,{!E}) for KIF in Weil's senae is the "same" as a W-dlagram (U, g, ... ) in which, for each intermediate E, the map VKIE = fi/ is an isomorphism and the composed map rEo VK / E is equal to the map 9Eb : -+ Ge,b induced by g. Our first goal is to show that there exists a W-diagram in which rB 0 VK / E = ~b for all E, and that it is Wlique up to isomorphism. Then we shall prove that for such a diagram the transfer maps VK/E are isomorphisms.
ut
4. Suppose (UF' g1' •.•• ) is a W-diagram for KIF. Show that the maps rp 0 VK/F and9~b coincide on the imageofAK = U'Kb in UFb and that their compositions with Vfc7F : G}" -+ Gft are equal. (Use the ''translation'' and "transfer" theore~, i.e., parts (b) and (a) of Theorem 6 in Chapter XIV, Section 5).
5. Note that the quotient of UFb by the image of AK is ca;/F = 'H.- 2(GK / F ,Z). Also, show that rF maps 'H.°(GK1F , DK) = D~KIF /NK/FDK isomorphicaIly onto Ker(Vfc1F)' 6. By 4 and 5, the ratio rFo~~/'" 9,..
:
= (DK n AF)IDF
ut -> GV can be factored as follows
up -> G'fl/F ='H.- 2(GK/p, Z) .!!... 'IfJ(G K/ P , DK) = (DK n AF}I DF !.!... G1<-b. The same comideration applies to each subdiagram (UE, gE, .. ). In a this way, a W-diagram determines a collection of homomorphisms 1/JE : 'H.- 2(GK/E,Z) -+ 1{°(GK1E ,DK), one for each intennediate field E. Our first goal is to show the existence and uniqueness up to isomorphism of a W-diagram for which WE = 0, i.e., rE 0 VK / S = 91;b, for all E. 7. As notation, for a group G, a G-module M and Q E ?{J(G, M), let eQ : 'H.- 2 (G, '1.) -+ 1{°(G, M) denote the Nakayama map ( ...... ('01 given by cup product with Q. With notation as in 2., show that VK/P(u..) = npEGK I F Gp.cr and 9~b(uO'} = • 'YO'. Conclude that twisting a W-diagram for KIF by a class 0 E 1i?(GK / F ,DK) multiplies its 1/JE map by the map eresE/FD , for each intermediate field E. ','
:...
-,:.
xv.
188
WElL GROUPS
8. For finite cyclic G and G-module M, show tha.t 0: 1---+ eo: is an isomorphi5Ill from rl 2 {G, M) to Hom("H- 2(G, Z). "H°(G, M». Thus a W-diagram defines, for each intermediate field E such that KlEis cyclic, a unique class aE E 1,.2(G K/ E ,DK) such that 1{;s = .as. 9. Let (U,9, ... ) be a W-diagranl for KIF. let E c E' be intermediate fields corresponding to subgroups H:J H' of G K/ F • Let E 11.2(H, DK). Check that the following diagram is commutative with the middle horizontal arrows being either 1{;E and 1{;E" as in the diagram, or being eo and ereso (V denotes the transfer map and id the map induced by the identity on DK):
a
nab
=
.!!..
1fJ(H,DK)
Lres
LV H,al>
11- 2 (H, Z)
=
==
DIJINflD.K
tid
ires
'H- 2 (H' , Z) -.p~- '1fJ(H',DK) = DIJ'/Nn,DK.
Prove that if S is the set of cyclic subgroups of G KIF, then the family (OE)GKIBES defined in 8 is coherent in the sense of the following
Let G be ajinite group and MaG-module. Let S be a set of subis in S and each conjugate H" = rIHu- 1 is in S. Call a/amity of elements as = (aH)HEs E nHES 'W(H, M) coherent if resaH = Qn' for all pairs H' c HE 5, and O(H') = rIaH for all pairs PROPOSITION.
groups
0/ G such that if H E 5, then each subgroup of H
H E 5 and (J E G. Suppose that M is a finite direct sum of modules induced from subgroups H E 5. Then the map "Hr(G, M) -+ flHES "W(H, M) defined. by Q -+ (reS~Q)HES i.s an isomorphism oflt.r(G, M) onto the set of coherent families
(QH)HES' 10. To prove the above proposition, suppose H and G are in S. Let
G=
II HrGD and
BrGD =
-rET
II
Then H ;: IID"ER.. u(HnrGo.,.-l) and MasH-module is a direct sum, M ;::: }:: M.. , where M .. L"ER. u.,.Mo is /Ill H-module induced from the (Hn-rGo.,.-l)-module
=
.,.Mo· Let
'W(H, M) be the H-component of a coherent family as. Let Let o~,.,. = pr(resD:H:r) E "Hr(Hn TGOT- 1, T Mo) be the class corresponding to aH,.. by the semilocal theory. Show that Q~, .. ;::: r(resr:-lHT)nGoQGo,Il. Conclude that all res~Q, where Q E 'H.r(H,M) is the unique class such that OGo,l E 1f"(Go,Mo) is the class corresponding to a: by the semilocal theory. Since H was arbitrary in S, this proves the
an
OH E
= }::"ETQH,-r with QH, .. E rtr(H,M.,.).
=
theorem.
=
GK / F , show that 11. Taking S to be the set of cyclic subgroups of G 8, 9, and 10 imply that any W-dia.gram for KIF can be twisted by a unique 6 E 1-{.2(G KIP,DK) to a W-diagram for KIF in which WE = 0 for all intermediate E. 12. 'Ib finish we must show that in a W-diagram (U,9,i,j) for KIF in which = g'!D for all intermediate E, the tra.nsfer maps VK / E are isomorphisms. It suffices to do this for E F. We write U = UF, 9 gP, and V = VK/F' f's
0 VK / E
=
=
XV. WElL GROUPS
189
Surjectivity: Let a E AF. Since gaJ> is surjective, there ,exists x E
that gab(x)
= rF(a).
uab such
Then
Vex)
- - E Ker(TF);::: DF a
= NK/FDK. =
Hence Vex) =a.NK/Fd = aV(i(d» for some dE DK, and a V(~). Injectivit'll: Let u E U such that V(uUC ) = O. Then gab(uUC) TF(V(UUC» = O. Hence there is a d E DK such that U E i(d)Uc. because the k.eroel of 9 iu a W-diagram is i(DK). We must show i(d) E UC. We know th~t 1 = V(uUC} = V(i(d)UC) = NK/Fd. By our hypotheses 011 DK, 'H,i(GK/F,D K } 0 and the cohomology of DK has period 2. Thus, 'H,-l(GK/F,DK) 0 and consequently, Ii = rIv d~·-l for some finite set of pajrs (d". all) E DK x GK / F . Let i(dv ) = II" and j(u,,} = q". Then i{d) = rIll UIIII"u;ly;! E UC as was to be shown. 0
=
=
=
That finishes our sketch of the existence and uniqueness of a Wei! groups in Weil's sense for layers in the special type of field formation we are considering, following Weil's proof for the fonnation of idele class groups of number fields. If, as in tbe case of number fields, the formation is a class forma.tion with cyclic layers over the ground field of arbitrary degree, then the class of the group extension of GKI F by AK given by the Wei! group, is the fundamental class. This was proved for number fields by Nakayama in a paper in the same Takagi memorial volume of the Journal of the Japanese Mathematical Society as [28J, by the same method we used to prove Theorem 8(b). The fundamental class was discovered at almost the same time by Nakayama and Weil, in completely different ways, Wei! as a byproduct of his discovery of the Wei! group ap.d Nakayama by a systematic study of the Galois cohomology of class field theory, partly in collaboration with G. Hochschild, leading to most of the cohomological results we have presented in Chapter XlV (d. [11]).
Bibliography III
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Y. Akizuki, Bme ~ Z~ tier Elemente deo- galoil&chct Grvppe zu deft EIe-
mente einer Untc,¥",JI1IC dcr NormJJ.as.ei&~ Math. Annalen, 112 (1935-1936), 560-571. (2) E. Artin. Algeb"". Number. and Algebraic FunctiOM. AMS Chelsea, Providence, RI, 1967. [3) _ _ , GoJou T~, Notre Dame Mathenu>-tice.l Lectures, no. 2, Notre Dame, IN, 1942; Reprinted in E:tp(mti0n4 by Emil Artin: A Selection, M. Rosen (ed.). History of Math., vol. 30" Amer. Math. Sox., Providence, RI, 2007, pp. 61-107. [4) N. Bourbaki. ACyebrE, Ch4p.tn: 5: Curp& oommv/;atijs. Hermann, Paris, 1959. [5) J. W. S. C_ls and A. Froblkh (ed&.), Algebraic Number Theory, Academic Preas, LondoD, 1976.
[6) C. Chevalley, La t.IIhwie4Io - - . . . dlwe3, Ann. of Math. (2) 41 (1940), no. 2, 391-418. (7] B. Dwork, Nann ruiAe ~ in loaM number jield&, Abh. Math. Sem. Hamburg, 22 (1958). 180-190. (8) I. V. FesenkD aDd S. V. ~. Local Ficld& and Their Em:nsioM: Seam" EditiDn, Amer. Math. Soc., Provid~. RI. 2002. (9) E. S. Golod and 1. R. Sbafarevicb. On c/a&, number toWenl, Amer. Math. Soc. '!'raDII!', Ser. 2, vol. 48, Amer. Math. Soc.. Pro\-ienC1!. RI, 1964, pp. 91-102. [10) W. Grullwald. Ein allgrnltlnc.s E.n.den%thcorcm fUr algebrauche ZohlkOrper, J, Reine Angew, Math. 169 (1933). 10l-107. . (11) G. Hochschild aud T. !'iaa,,-, CohomoIon in clcu~ field tIieor1J. Ann. Math. (2) 66 (1952), 348-366. (12] K. Iwasawa, Local CLu, PidtI 'lTacotv, Oxford Univ. Press, 1986. (13) _ _ . An eqAc;t /~ fin' the norm reMue symboL, J. Math. Soc. Japan 20 (1968), 151-165. (14) S. lyanaga. Z.", Bc.a. W H4up~atze.5, Ahh. Math. Sem. Hamburg, 10 (1934), 34~ 3Q'T. lUi) S. Lang, On quG3t alge6twc c:t.o.ure. Ann. of Math. (2) 55 (1952), 373-390, [16) _ _ , Algebnuc Jtluntkr TfIeot-y. Second edition. Springer-Verlag, New York, 1994. [17) J. Lubin 8.Dd J. Tau:. F.",... complu multiplication in 100000jielda, Ann. Math. (2) 80 (1964), 464-484.
(lB] K. Miyake (cd.). CI&u FIdI 77Ieory: Ita Centenary and Pro~pect Ad~ Studles Pure Math., vol. 30. Math. Sox. Japan, Tokyo, 2001. (19) T. Nakayama. l'ber die lkaehungen ztDdchen den Faktoren Slistemen und der Normclcusengruppe elnel galo,u~ Erv.'tIterungs Korpers, Math. Annalen, 112 (1935-1936), 85-91. (20] J. NeuIdrch. A. Schmidt, aDd K. Wingberg, Cohoo.oklgy of number fields, Springer-Verlag, Heidelberg. 2000. (21) J-P. Serre. Looal Fie.Ido. Springer-Verlag, New York-Berlin, 1979. [22) _ _ , GaloU C~. Springer-Verlag, Berlin, 1997. (23) _ _ , CohomGlogie d .n&IImttic, 8eminaire Bourbaki, Volume 2, Expose 77, Soc. Math. France. Paris. 1995. lIP 263-269. [24) I. R. Shafare-;ch. On 1M Galou grtIUIM of p-adic Jield8, Dok!. Akad. Nauk SSSR 53 (1946), no. 1, 15-11;: see aIIO C. R. Acad. Sci, Paris 53 (1946), 1:>-16 and Collected Mathematical Papers of Sbafare-'ich. Springer-Verlag, Heidelberg, 1989, pp. 5~. [25) J. Tate, Number tJu:on:hc background, Automorphic Forms, R.epreooentations and L functiona, A. Borel and W. C~ (edl;.), Proc. Syrup. Pure Math., vol. 33, Part 2, Amer. Matll. Soc., PtO\·i
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/27) A. Weil, L'integration lioN les groupcs tDrlflogiquu et applicatiofl$, Hermann, Paris, 1938. !28) _ _ , Sur 10 thiarie du CSJ-rps de classe.... J. Math. Soc. Japan 3 (1951), 1··35; see also A. Weil, Collected POp"rs, Volume 1, 11951b}, Springer-Verlag, New York-Heidelberg, 1979, pp. 487-581. (29) _ _ , Basic NumbeT Theory. Springer-Verlag, Berlin-Heidelberg-New York, 1967. I30J G. Whaples, Now-analytic class field theo1ll and Grunwald'~ theorem, Duke Math. J. D (1942).
455-473. 131) E. Witt, Zwei Re9e1n tiber ve ...chriinkte Protiukte, J. Jreine Angew. Math. 1'73 (1935), 191192.
ISBN 978-0-8218-44Z6-1
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