Algebraic number fields: (L-functions and Galois properties) : proceedings of a symposium 9780122689604, 0122689607

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Algebraic Number Fields (L-functions and Galois properties)

Proceedings of a Symposium organised by the London Mathematical Society with the support of the Science Research Council and the Royal Society

Edited by

A. FROHLICH King's College, University of London

1977

ACADEMIC PRESS London: New York: San Francisco A Subsidiary of Harcourt Brace Jovanovich, Publishers

ACADEMIC PRESS INC. (LONDON) LTD. 24/28 Oval Road, London NW1 United States Edition published by ACADEMIC PRESS INC. 111 Fifth Avenue New York, New York 10003

Copyright ©1977 by ACADEMIC PRESS INC. (LONDON) LTD.

All Rights Reserved No part of this book may be reproduced in any form by photostat, microfilm, or any other means, without written permission from the publishers

Library of Congress Catalog Card Number: 76-016966

ISBN: 0-12-268960-7

Pnnted in Great Britain by Galliard (Printers) Ltd, Great Yarmouth, Norfolk

List of Contributors

C.J. Bushnell, Department of Mathematics, King’s College London, Strand, London WC2R 2LS W. Casselman, Department of Mathematics, University of British Columbia, 2075 Westbrook Place, Vancouver, B.C., Canada.

J. Coates, Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, 16 Mill Lane, Cambridge. J. Cougnard, Math emat i que s , University de Besanqon, Route de Gray - La Bouloie, 25030 Besancon Cedex, France. A. Frohlich, Department of Mathematics, King’s College London, Strand, London WC2R 2LS

H. Koch, ADW der DDR, ZI fur Mathematik und Mechanik, DDR 108 Berlin, Mohrenstr. 39.

V

vi

LIST OF CONTRIBUTORS

J.C. Lagarias, Bell Laboratories , Murray Hill, N.J. 0797^, U.S.A.

J. Martinet, Department de Mathematiques, Universite de Bordeaux, 351 Cours de la Liberation, 33^+05 Talence, France. J. Masley, Department of Mathematics, University of Illinois at Chicago Circle, Chicago, Ill. 60680, U.S.A.

L.R. McCulloh, Department of Mathematics, University of Illinois at Urbana, Urbana, Ill. 61801, U.S.A. 0. Neumann, DAW-Inst.-komplex Mathematik, IRM, DDR 1199 Berlin-Adlershof, Rudower Chaussee 5«

A.M. Odlyzko, Bell Laboratories , Murray Hill, N.J. 0797^, U.S.A. R. Odoni, Department University North Park Exeter EXU

of Mathematics, of Exeter, Road, i+QE

LIST OF CONTRIBUTORS

J-P. Serre, College de France, Paris,France.

H. Stark, Department of Mathematics, MIT, Cambridge, Mass. 02139, U.S.A.

J. Tate, Department of Mathematics, Harvard University, 1 Oxford Street, Cambridge, Mass. 02138, U.S.A.

M.J. Taylor, Department of Mathematics, King’s College London, Strand, London WC2R 2LS A.A. Terras, Department of Mathematics, University of California at San Diego, P.O. Box 109, La Jolla, Calif. 92037, U.S.A.

S.V. Ullom, Department of Mathematics, University of Illinois at Urbana, Urbana, Ill. 61801, U.S.A.

H.W. van der Waal 1, Mathematisch Institut, Katholieke Universiteit, Toernooiveld, Nijmegen, Holland.

vii

Preface

This volume is the outcome of a symposium on L-functions

and Galois properties of algebraic number fields, held from 2 to 12 September 19755

in the University of Durham.

organised by the London Mathematical Society,

It was

with the

generous financial support of the Science Research Council,

aided further by a grant from the Royal Society.

The smooth

running of the conference was made possible by the helpful attitude of the authorities of Durham University and the hard

work of the symposium secretary,

Dr. S.M.J. Wilson.

Almost all the lectures given at the symposium are

recorded here.

In many cases the presentation has been

expanded and new relevant material added.

My gratitude is

due to the lecturers for making publication of this volume possible by their willing cooperation,

as well as for their

original contribution to the success of the meeting itself. I also wish to express my thanks to Mrs. J. Bunn, who

typed the whole volume ready for publication, Smith,

who edited the manuscripts,

to Mrs. E.

to Dr. J.C. Bushnell

for help on all fronts and to Academic Press London for

continued cooperation.

A. Frohlich

IX

Contents Page

List of Contributors

v

1X

Preface J. Martinet,

Character theory and Artin

L-functions.1

J.T. Tate (prepared in collaboration with C.J. Bushnell and M. Taylor), Local constants.

89

A. Frohlich, Galois module structure.

133

J-P. Serre (prepared in collaboration with C.J. Bushnell), Modular forms of weight one and Galois representations.

193

J. Coates, p-adic L-functions and Iwasawa’s theory.269

H.M. Stark, Class fields for real quadratic fields and L-series at 1. A.M.

Odlyzko,

355

On conductors and discriminants.

377

J.C. Lagarias and A.M. Odlyzko, Effective versions of the Chebotarev density theorem.

U09

J. Masley, Odlyzko bounds and class number problems.

^65

A. Terras, A relation between SK(s) and ^(s-1) for algebraic number field K.

xi

^75

R. Odoni, Some global norm density results obtained from an extended Chebotarev density theorem.^85

S.V. Ullom, A survey of class groups of integral group rings. J.

^97

Martinet, H . o

J. Cougnard, Un contre-exemple a une conjecture de J. Martinet.

525

539

L.R. McCulloh, A Stickelberger condition on Galois module structure for Kummer extensions of prime degree. 561 A. Frohlich, Stickelberger without Gauss sums.

589

Fields of class two and Galois cohomology.

609

H. Koch,

0. Neumann, On p-closed number fields and an analogue of Riemann’s existence theorem.625

R.W. van der Waall, Holomorphy of quotients of zeta-functions. W.

Casselman, GL .

6^9

663

Character theory and Artin L-functions

J. Martinet

I. NON ABELIAN L-FUNCTIONS

The aim of this chapter is to describe the theory of Artin's non abelian L-functions,

theory of abelian L-functions.

taking for granted the This chapter owes much to a

talk by Serre (Fonctions L non abeliennes, Seminaire de Theorie des Nombres, Bordeaux, 10 avril 1973).

§1 . Frobenius

Two papers of Frobenius, both dating back to 1896, play a

key role in the theory we are going to describe.

The first

one is devoted to what is now called the "Frobenius sub­

stitution".

Let E/K be a finite normal extension of number

fields with Galois group G,

K.

and let p be a finite prime of

Assume E/K is unramified at p.

lying above p,

For every prime P of E

there is a unique element Op e G (the

1

2

MARTINET

Frobenius substitution) such that, for any integral x £ E, the congruence Qp(x) =

absolute norm of p.

mod p holds,

Moreover,

where N(p) is the

the conjugacy class of Cp

in G does not depend on the particular choice of P above p in E.

Frobenius stated in this paper a density theorem of

V the Cebotarev type,

and proved the following result:

every cyclic subgroup C of G,

there exist infinitely many

primes Psuch that Op is a generator of C.

questions of density,

for

Even disregarding

this is weaker than Cebotarev s *

theorem

which asserts that every generator of C is of the form Op for

infinitely many P. The second paper of Frobenius we are concerned with is devoted to the definition of the characters.

seen in a moment,

As will be

the theory of L-functions relies heavily

on the consideration of both the notion of a character and

of the Frobenius substitution. the connection,

But Frobenius did not see

and the sequel of his work deals mainly

with the theory of characters.

§2 . Weber For an ideal K prime to

of K,

and let P

let 1^ be the group of ideals of

be the subgroup of I, which consists

3

CHARACTER THEORY AND ARTIN L-FUNCTIONS

of ideals which can be generated by a totally positive ele­ ment a of K congruent to 1 mod 4 .

Let H be a subgroup of

T, containing Pz (we call such a subgroup a congruence

6

o

subgroup).

Weber called an abelian extension E of K ”a class field for H” if the prime ideals of K which decompose completely in E are precisely those which belong to H,

some sense the smallest possible ideal. the prime divisors of

and if

is in

In this situation,

are precisely the prime ideals of K

which are ramified in E.

* Now, for every character x • I^/H + (T s

there is an L-

function defined for Re(s) > 1 by:

l(s,x)

=

n --------i. l-x(p) N(p)

The question arises of comparing the zeta function £_(s) E with the product

Generally,

II L(s,x) when E is a class field for H.

they are not equal,

because of the possible

existence of prime ideals which are ramified in E/K but not in the subfield corresponding to the kernel of

I shall

write 5 (s) 'V

n

L(s,x)

£j

X to mean that the equality is true up to a finite number of factors.

4

MARTINET

To obtain an equality, replace

one must, for each character X,

by the conductor of X.

This was known to Weber

for those abelian extensions which were known to be class fi elds.

§3. Artin's first definition of L-functions

Artin's first definition of L-functions appeared in 1922 (on a new kind of L series).

In the meantime (1920) Takagi

had established in full generality the classical results of

class field theory,

namely the one-to-one correspondence

between abelian extensions of number fields and congruence

subgroups,

and also the isomorphism theorem,

which asserts

that the Galois group G of the extension is isomorphic to the quotient I^/H.

Using an isomorphism between

and G,

it would be

possible to define L-functions for degree one characters of

G.

But Takagi's theory does not give any canonical iso­

morphism between

and G.

Nevertheless,

Artin thought

that the L-series we defined previously with a congruence

class character could be identified with L-series defined for a degree one character ♦ of G by the formula: l(s,p(s) be a representation

Denote by x 'the character of p ,

defined by

6

MARTINET

x(s) = Tr(p(s)) for all s £ G.

For a prime p in K,

the

determinant det(l - N(p) S p(Op)) does not depend on the choice of P above p,

and takes the same value for two

isomorphic representations. l(s,x)

We can therefore define

------- i----------- —

n

=

P unramified

det(l-p(ap)N(p) S)

The series is convergent for Re(s) > 1.

It is then obvious that L is additive, i.e.

(a)

L(s,x +X9) = L(s,x )L(s,x ) 12 12

:

.

The following equalities, however, are true only up to

a finite number of Euler-factors (we use the notation Let H be a normal subgroup of G corresponding to an

Let p be a representation of G/H with

extension F/K .

character x and. let p’ be the lifting of p to G with chara-

acter x1-

(b)

Then we have the lifting formula

L(s,xf)

L(s,x)

Let H be a subgroup

. and let x be a character of

of G,

H which induces the character x

*

of G.

Then we have the

induction formula

(c)

L(s,x )

Moreover,

L(s,x)

Artin proved that L(s,l)

£ (s). K

7

CHARACTER THEORY AND ARTIN L-FUNCTIONS

Applying formula (c) to the unit character of a subgroup

H of G corresponding to an extension F/K, formula ^(s)

L(s,rG/H),

we obtain the

where rQ/H is the character of

the permutation representation of G on G/H. Let us take H = (1) in the above formula. the character r

G

is just the sum

of the regular representation of G,

which

£ x(l)x over all irreducible characters of

X Now applying formula (a),

G.

Then rG/H

?E(s)

we get

n

_L(s,x)x(1)

.

X irreducible Assuming the reciprocity law,

Artin gave a proof of the

theorem of density conjectured by Frobenius.

He stated the

existence of an analytic continuation for his L functions

(with perhaps ’’ramification” points) and of a functional equation relating L(s,x) and L(l-s, x) as had been proved in

191T by Hecke for abelian L-functions.

He also asked

whether his L functions are holomorphic in the whole-complex plane for a character which does not contain the unit chara­ cter.

We now call this statement "the Artin conjecture".

The general definition of non abelian L-functions Surprisingly,

Cebotarev proved in 1926 the density

theorem conjectured by Frobenius without using L-functions.

8

MARTINET

The main idea behind the proof is to reduce to the case of a cyclotomic extension.

In 1927,

using this device,

proved his general law of reciprocity.

In 1930,

Artin

he returned

to the problems of L-functions in his paper "on the theory of L series with general characters".

The two main problems

are:

(i)

To define local factors at ramified primes,

in such

a way as to put true equalities in the above formulae.

(ii)

To define local factors at infinity and an exponent­

ial factor in order to get an analytic continuation and a functional equation.

(i)

As always,

we consider a normal extension E/K of

number fields with Galois group G and a complex representation p : G -*■

G1(V) with character X.

choose a prime P above p. ly,

Now,

Let p be a prime of K;

Let Dp and Ip denote, respective­

the decomposition group and the inertia group of P.

the quotient group Dp/Ip is isomorphic to the Galois

group of the residue extension.

Hence,

we can define a

Frobenius substitution Op belonging to Dp/Ip.

The vector

space V is acted on by G via the formula a x = p^(x) Dor all

9

CHARACTER THEORY AND ARTIN L-FUNCTIONS

x £ V and all c e G.

Let

Ip V = {x e V |

a e Ip,

V

ox = x)

the subspace of elements of V fixed by Ip. determinant of the transformation (l-N(p)

Once more,

-s

Op) of V

not depend on the particular choice of P above p, the same for two isomorphic representations.

,

the does

and is

We can thus

define l(s,x)

---------- ------ —-------

n

=

P det (l-N(p) finite V P

ap)

for Re(s) > 1.

Now,

the induction formula and the lifting formula be­

come equalities.

We summarize the fundamental results

(notation as above):

Theorem

(a)

L(s,x +X„) = L(s,x ) L(s,x ) 1

(h)

L(s,x') = L(s,x)

(c)

* L(s,x ) = L(s,x).

Assume G is abelian.

of G,

1

2

Let x

2

a degree one character

and let ip be the corresponding congruence class

character.

Then, (d)

L(s,x) = L(s, 1. X

Then

A possesses a meromorphic continuation in the whole complex

plane,

and satisfies the functional equation A(l-s,x) =

W(x) A(s,x) for some constant W(x) of absolute value

1 (the

so-called "Artin root number").

In the theorem,

x is the complex conjugate of X.

If

15

CHARACTER THEORY AND ARTIN L-FUNCTIONS

X is the character of a representation p : G the character of the contragedient

representation

* * G1(V ) (V is the dual space of V) ,

p : G

=

x is

G1(V),

for all s e G,

defined by x e V,

* f e V .

Artin could not prove the existence of a meromorphic

continuation for the function A. 191+7 by Brauer.

The theorem was proved in

We now give the proof.

We must first establish properties (a), (b) , (c) for the

enlarged L-functions.

Properties (a) (additivity) and (b)

(lifting property) are easily verified for the functions L and y 9 X

as well as for the conductor ^(x). *

we can define y (s),

a virtual character of Q ).

^(x),

under induction.

For the Artin conductor,

the formula is a bit more complicated. of G with fixed field F,

' dp/k’ “f/k «< *»•

Let H be a subgroup

and let x be a character of H.

conductor of the character x

d(x‘)

A(x) and A(s,x) for

It is not difficult to show the

K

invariant of y

they are

and hence for the function A

true for the constant A(x),

(therefore,

Thus,

*

The

of G induced by x is given by:

where D

F/K

is the discriminant

of the extension F/K.

A simple calculation using the transitivity formula for discriminants gives the equality A(x ) = A(x),

and thus the

16

MARTINET

induction formula A(s,x ) = A(s,x) for the enlarged L-

function. there exist

We now apply Brauer’s induction theorem:

subgroups H. (1

n) of G,

i

degree one characters

X^ (1 $ i $ n) of H. and rational integers m

(1 £ i < n)

for some n such that the following equality holds: n. X-* •

IV

X =

1=1

We thus have, by properties (a) and (c): n.

n

A(s,x) =

For 1 $ i $ n,

n A(s,x«) 1. i=l

let F. be the fixed field of H., i ii

kernel of X- and F’ the fixed field of H’. 1 i i

H! the

The extensions

F’/F. are cyclic extensions with Galois group G. = H./H1. i 1 1 1 i

Writing x’ for the character of G^ defined by x^»

we then

have, by property (b):

A(s,Xi) = A(s,xp. We now use Hecke’s results. Artin map,

By composition with the

the characters X^ define congruence class chara­

cters (or idele class characters in modern language)

K,

and we know,

by property (d),

L(s,x|) is equal to L(s,^). function L(s,ip),

Now,

of

that the function given an abelian L-

Hecke defined an enlarged function Af(s,ip)

CHARACTER THEORY AND ARTIN L-FUNCTIONS

17

by the formula

A'(s,ip) = A’ (ip)S//2 Y^(s) L(s,ip)

,

(

Gal(Q /E)ab

p

„

Tv

K

inclusion --------------- E

p

„

Tv

35

CHARACTER THEORY AND ARTIN L-FUNCTIONS

In both diagrams,

the vertical maps are Artin maps.

In the left hand diagram,

E/K is a finite extension of

contained in a given al­

fields of finite degree over gebraic closure-of

In the right hand diagram, number fields,

E/K is a finite extension of

and !$., I£ are the corresponding idele

groups. We shall write Ver

E/K

for the transfers involved in these

2 diagrams.

Proof

This is a property of class formations (see e.g.

Artin-Tate, Class Field Theory,

chap. XIV, or Serre, Corps

Locaux, chap. XI).

b)

Induced representations.

Given a representation p

of a finite group G in a complex vector space V,

the de­

terminant of p depends only on the character of p. linearity,

By

we define the determinant of any virtual

character x of G.

(Notation : det ). X

Proposition 3.2.

Let G be a finite group and let H be a

subgroup of G.

Let x be a character of H,

and let X

be

36

MARTINET

the character of G induced by X.

For any element s £ G, let

eG/j/s) be the si6nature of the permutation of G/H defined by multiplication by s. det * (s)

Then:

= fcG/H(s)X^1^ det^(Ver^(s))

,

X or, more briefly: v ( 1)

det

Proof

* = £g/h °Ver a a W S,0

x) = det (x+n a W 0s’

x) = det (Ver(s)). A

let e.(l * i S X(D) be a basis of W.

6(a)e. (deG/H, 1 $ i $ X(l)) of V.

£g/h

Consider the basis For each i,

,

Corollary.

v per­

dety(v) =

As there are X(D indices i

£G/H (s)

Now

and the signature of the permutation is

mutes the 6(o)e.,

(s).

detw^ulwa)

first dety(u) =

Q.E.D.

If X is a character of trivial determinant and

of degree zero,

* so is the induced character x •

§U. Local Galois Gauss sums

Let p be a place of Q,

closure of

field,

(thus,

be an algebraic

and let

and

= ZR

= C).

we mean a finite extension of

By a local

which is contained P

in Q . P

Given a local field K,

we consider virtual char-

acters of Gal(Q /K) which are differences of two characters P

of representations of open kernel.

We simply write GR for

the Galois group Gal(Q^/K). For a local field K and a (virtual) character 0 of G , beligne and Langlands defined a local root number W(0) (see

Tate’s lecture cf.

[1U]

).

The local root number is well

38

MARTINET

defined, by the following three properties:

(i)

W(O1 + e2) = w(ei) w(e2).

(ii)

Let 6 be a

irreducible character of degree one,

*

and let 0’ be the character of K

map.

defined by 6 via the Artin

Then, W(0) is the local root number W(0T) defined in

section 2.

Let E be a finite extension of K,

(iii)

let 6 be a

*

character of degree zero of G_ and let 0 h of G& induced by 6.

be the character

*

Then W(0 ) = W(0).

We are now able to define the local Galois Gauss sum.

Definition

Let K be a non archimedean local field,

let 0 be a character of Gal(Q /K). P

sum

t(0)

and

The local Galois Gauss

is defined by the formula:

-r(e) = w(e)

/n(6(0))

,

where ^(0) is the Artin conductor of 0 and the square root

is the positive square root.

Note that ^(0) = ^(0).

Hence

W(e)) The local Galois Gauss sum is well defined by the

39

CHARACTER THEORY AND ARTIN L-FUNCTIONS

following three properties which are obvious consequences of

the corresponding properties for local root numbers and

conductors: (i)

t(61

(ii)

Let 6 be an irreducible character of degree one,

and let 6*

be the character of K

Then,

map.

+ 02) = 1(6^) r(e2).

t(6)

*

defined by 6 via the Artin

the local Gauss sum defined in

= t(0,)5

section 2.

(iii)

Let E be a finite extension of K,

let 6 be a

* and let 6

character of degree 0 of G

be the character of

* G induced by 0. K

Notation.

Then

t(0

) = t(0).

Given a local field K,

an element x £ K

irreducible character of degree one 6 of G„, for the element 0(a)),

ab where u) e G„

*

and an

we write 0(x)

is the image of x under

the Artin map.

Proposition U.l.

Let K be a finite extension of

let 0 be a character of G . K

and

Then:

(i)

|t(S) | = /N(6(6))

(ii)

T(e) t(6) = N(^(e)) dete(-l).

The following corollary is an easy consequence of the

40

MARTINET

above proposition for an extension of Q , P for K = 1R

Corollary.

or K = d:

Let K be a local field.

(i)

Iw(e)| = i

(ii)

w(e) W(e) = det.(-l). U

Proof.

and is obvious

Then:

We have only to prove the proposition when 9 is an

irreducible character of degree 1,

and show that the 2

sides of the equalities are invariant under induction for characters of degree zero.

Now,

the case of an irreducible

character of degree 1 has already been dealt with in §2,

and both sides of the above equalities are invariant under induction when 0 is of degree 0

(for (ii),

just remark

_ * _* that (0) =0 ) .

Remark.

Using part (ii) of proposition U.I.,

immediately the formula

W(6)

t(6)

= det0(-l) 4(6(0)) •

one proves

41

CHARACTER THEORY AND ARTIN L-FUNCTIONS

Galois action on Galois Gauss sums and root numbers (local theory) Let K be local field, and let 6

be a character of G K

The values of 6 are algebraic numbers. we define 6

U)

For any to e Q

©^(s) = (0(s))W for every

by the formula:

We do not worry about left or right action of G

s e gk‘

as the results we are going to prove do not depend on the

choice we make. The aim of this section is to compute W(0W) in terms of W(0) and the theorem we shall prove is just a local version of a global theorem of Frohlich.

field, 0^ = 0,

For an archimedean local

and there is nothing to do.

We thus

restrict ourselves to finite extensions of Now,

is an algebraic number:

t(0)

of degree one,

p finite.

for a character

this is clear from the definition,

and the

general case is a consequence of the induction formula. Therefore, W(0) itself is an algebraic number. now compare

t(0w)

We shall

with t(0)W.

We first define a homomorphism u$ of

= Gal(Q/$) into

Sr°up of p-adic units.

definition

Given to e Q ,

u (to) is the unique p-adic F

42

MARTINET

^-1

= n p

unit such that n

,

for every pn-th root of unity

For any extension K of Q , P * morphism of ft into K .

n in Q.

Let K be a finite extension of ©

Theorem 5-1-------

Q

for some P

and let 6 be a character of G .

finite p, co e ft

we view u as a homoP

Then, for any

, -1

T(e“ )“ = T(e) det6fl (up («)).

Proof.

Step 1.

The proof is in 2 steps.

Let 0 be a character of degree 0,

subfield of K.

and let F be a

Assuming the formula is true for 0,

prove it for the character 0

*

of G

£

induced by 0.

we For the

* right hand side, observe that

t(0

) = t(0) and that

(u (u))) = det (u (w)) by propositions 3.1. and 3.2. . e p 6 p -1 v -1 . * x co co * that (0 ) = (0 ) , hence For the left hand side notice det

* w r = T(e

Step 2.

t((0w

r )*

=

t(0w

)\

We prove the formula for an irreducible character

of degree 1.

* Regarding 0 as a character on K ,

we write,

43

CHARACTER THEORY AND ARTIN L-FUNCTIONS

with the notation of §2,

t(0)

=

£

e(“)

) * ^(

•

= I

I

»(f) ♦( G1(V) where V is a real vector space of dimension n.

The group p(G) is contained in the ortho­

gonal group 0(V) of some positive definite bilinear form on

V.

By theorem 3.2,

p(G) is contained in the normalizer of

a maximal torus T of 0(V).

Let m = [^-] .

There exists a

subspace W of V of dimension 2m such that the matrix of T in

a suitable basis e, ..... 1 5

Let w.

2m

of W is of the form

(1 $ i $ m) be the subspace of W spanned by the

vectors e^^,

Now,

there are two possibilities:

65

CHARACTER THEORY AND ARTIN L-FUNCTIONS

Since W is invariant under the action of

a) n is odd.

V contains an invariant subspace W1 of dimension 1. 5

since p is irreducible,

W = (0) and V = W1 .

The character

x is then of type (i).

b) n is even,

bet H be the subgroup of those elements

s e G such that Pg(Wx) the normalizer of T, p is irreducible,

wi•

Since p(G) is contained in

p(G) permutes the subspaces W^.

this permutation is transitive.

Since This

means that p is induced by the representation p: H

deduced from p by restriction to H.

representation. group of 02(]R)

whether p

Therefore,

But p

G1(W1)

is a real

P (H) is isomorphic to a sub­

and x is of type (iii) or (ii) according to

is absolutely irreducible or not.

We shall now give a corollary of theorem 3.1. Deligne ([U]

;

due to

Deligne's paper also contains a purely group

theoretic proof of theorem 3.1).

We must first extend

slightly the definition of a dihedral character : we consider

that a character lifted from a character x’ of a quotient °f G isomorphic to D

is a dihedral character if X1 is the

sum of 2 distinct irreducible characters of degree 1.

66

MARTINET

Definition

Let G be a finite group.

Let x be a dihedral

character of G lifted from a character x1 of a dihedral quotient G’ of G.

x’ = Indus’),

Then,

is an irreducible

cyclic subgroup of G’ of index 2 and

character of degree 1.

where H’ is a

We call r^ the4 character of G

lifted from Ind^f(£’-1). has degree 0 and trivial determinant.

Note that r X

Theorem 3.3.

Let G be a finite group.

(Deligne)

Every

orthogonal character of G of degree 0 and trivial determinant Q. is a 2 -linear combination of characters of the form Ind^( $) where $ is either a character r

Proof.

with ^(1) = 0.

or a sum

Let x be a character of G of degree 0 and trivial

determinant.

By Brauer-Witt’s theorem,

the unit character

of G can be written as a sum 1=1 n__Ind2($ ) n

ranges over the r

_1K

ri

ri

where H

n

-elementary subgroups of G and $ n

orthogonal character of H.

I nH Ind^(Res$(x) •$). H trivial determinant, assume that G is a f

Wow,

is an

X = X.l =

Since Res (x) has degree 0 and G

so does Res (x).$. G

We may'therefore

-elementary group.

Let A be the subgroup of R° generated by the characters G

67

CHARACTER THEORY AND ARTIN L-FUNCTIONS

of the form of theorem 3.3. _ 9

With the notation of theorem

let B (resp. C, D) be the subgroup of R° generated by Cr

characters of type (i) (resp. (ii), (iii)).

If G is

Lemma 3-H.

Proof of lemma 3A

-elementary,

then R° = A+B.

It is enough to prove that every

irreducible orthogonal character X belongs to A+B. x(l) = 1,

there is nothing to prove.

prove the lemma by induction on x(l).

X = Ind^(ip +

with ip(l) = 1,

We can therefore If X e C3

say

write

X = IndJ [( ® I7L + Q/ZZ PPP

2iri

HR /2Z

->

S1,

where S1 denotes the unit circle in C.

If k is an algebraic number field,

and v is a place

of k:

= the completion of k at v;

k

o

°k ; k

= the idele group of k;

ck = the idele class group of k.

§1. Root Numbers in the Abelian Case

Let Q be an algebraic closure of the rational field ($,

and k c Q an algebraic number field (of finite degree over

Q).

Let ft

= Gal(flj/k),

and let

K.

X: \ + ®* X be a continuous 1-dimensional linear representation of ft . K ab • Then x factors through ft , the Galois group of the maximal k.

abelian extension of k, group of S1.

ab C, -> ftn , k k

and the image of X is a finite sub­

Composing with the Artin reciprocity map

• and the canonical quotient J

k

C , k

we obtain

characters of finite order of the locally compact abelian

93

LOCAL CONSTANTS

groups C

and- Jlk 3

which we denote also by x«

X: C^SL X: Jk-SL To the representation X of

,

we can attach A(s,x),

the Artin L-function with factors corresponding to the

Archimedean primes of K ([Dur.M]).

This function is the

same as the "abelian’' L-function attached to the idele-class

character X

( [T] or EW1 ) >

811(1 111 satisfies the functional

equation:

A(l-s,x) = W(x)A(s,x) ,

where W(x) £

is a constant.

determined locally in this case. of J , k

(cf.

let X = xlk , v 1 v

The root number W(x)

be

Viewing x as a character

for each place v of k.

Then

[T] or [W] ) ,

w(x) = n w(x ), V where the constants W(x^) depend only on

and x^,

and are

given by explicit formulas which we now recall.

Let K be a local field of characteristic 0,

character of KX of finite order.

Then the root number W(a)

is a complex number of absolute value 1,

value is: (i)

K = (T,

W(a) = 1.

and a a

and its precise

94

TATE

(ii)

K = JR, I if a is trivial,

J

-i otherwise, i.e. if

a(x) = sgn(x).

(iii)

K non-Archimedean: Let 4(a) he the conductor of

a, P the absolute different of K, P(a) = 4(a)P , h. K such that do

= P(a). iv iv of K+ defined above.

W(a) = Nf{(a) 1/2

Let ip

and d £ K

he the canonical character

Then:

1

a(d Lujd h). K

x xEOk

mod

Here,

4(a)

N denotes the absolute norm,

and the sum is taken

over a set of representatives of the cosets of 1 + 4(a) in X

X

(with the convention 1 + 4(a) =

0

IV

iv

if 4(a) = d ). K

In the

notation of [Dur.M]:

W(a) = N4(a) ^^(a) •

Notice that if a is non-ramified, W(a)

Proposition 1

a( V K K

Let K be a non-Archimedean local field of X

characteristic 0,

and let a be a character of K

,

.

of finite

95

LOCAL CONSTANTS

Let a be an ideal of 0 such that a2 | /j(a) , K

order-

and let

Then there exists c e K such that:

a-1^(a).

(i)

cdK = P(a),

(ii)

a(i+y) = *K(C !y) for a11 y e b-

and

Further, for any such c: (iii)

W(a) = N(ba-1)-1/2

E

a(c-1* x)

(c * x) .

xe(l+a) X L mod o

Proof

If a =

then b = 4(a),

and the assertion (iii)

iy.

if c is any element of

is just the formula above for W(a), Further,

K satisfying (i).

in this case.

satisfies (ii), Suppose d

any c satisfying (i) also

0 .

Then p |&|b|^(a),

and if y, y’ e b,

then yy1 e ^(a) so that: a(l+y)a(l+y») = a(l+y+y»). That is, y i—* a(l+y) is a character of the additive group b. This character extends to a character of K+ and, by local

additive duality,

there is some c e K such that

a(1ty) = ^(c^y) for all y e b.

y P

The character

y) of K+ is trivial on 6(a),

6(a) c b.

The character

but not on

is trivial on D \ K K.

but not

96

TATE -ln-l

on p

K

u

K

.

Therefore cO

= P(a).

K

Now: v

a(c

Z

—1

—1

x)d(c

x

x)

K

X£°K X mod 4(a)

I X

I

zeo

a(c Td+y))^ (c Td+y))

yd IY

x mod b

=

mod ^j(a)

h“(c 1z)d(c h) £ ip (e Xy(z-1))] A.

r,

IY

z

y

However,

by the construction of c.

the inner sum is zero

unless yi—y(z-l)) is the trivial character of the IY

group.b/^(a);

that is,

unless z = 1 (mod a).

So this

double sum reduces to:

Na

a(c Xz)ip (c Tz) , ze(l+a) K

modX 0T

and the assertion (iii) follows.

(Lamprecht, Dwork)

Corollary 1 ramified,

or

| £(cx)

(i.e.

If either a is non­

a is ’’truly wildly ramified”),

97

LOCAL CONSTANTS

then W(«) is a root of unity.

pr00f

If t$(a) = cl2

then CL — b

for some ideal cl C- 0 9 K

and

W(a) = a(c t'l' (c b ,

which is clearly a root of unity.

Now assume that ^(a) = CL2p

for some proper ideal cl of K

0 . Let p be the residual characteristic of K. K Proposition, we have b = ap^., and

W(a) = Np

In the

a(c 1x)ip (c \). Xe(l-Hl) K

Y

K X

mod

CLp

Since CL is a proper ideal of 0

is a p-group and so a(x), root of unity.

of unity.

Also,

the group (1+&)/(l+t$( ot))

for x £ (1 + cl) ,

the values of ip

are p-power roots

Hence the quantity C, = (a(c ^WCa))2

N the field E of p -th roots of unity,

show that C is a root of unity. °nly one place above p, II £ II

K

is a p-power

for some N.

lies in We must

Since the field E has

this will follow if we show that

= 1 for each place v of E which is not above p.

^■his is a consequence of:

98

TATE

Lemma

Let E be a subfield of X containing W(a).

Then

l|w(ot) || v = 1 for each place v of E not dividing p,

the

residual characteristic of K.

Proof

Suppose that v is non-Archimedean (and does not

divide p).

The explicit formula for W(a) shows that W(a)

is a local integer at v.

One also knows ( [Dur.M,2.2])

that: ) * (

W(a)w(a) = a(-l),

so that W(a) is a local unit at v, i.e. Suppose that v is Archimedean.

||w(a) ||

= 1.

The ratio W(a)Q/W(a°)

is a root of unity for every automorphism o of 0 (cf. [Dur.M,5.1]) •

Choosing a so that

we have

absolute value),

||w(oc) ||

||x ||^ = |x°|

(ordinary

= |w(a°)| and we know

|w(a°)| = 1.

X Corollary 2

Let 3 be a character of K

such that t$(B)|a.

of finite order

Then: W(S.a) = 3(c)W(a).

In particular,

Proof

if 6 is non-ramified,

W(S.a) = B(P(a))w(a).

The hypothesis implies that either p^(3)|a) or K

99

LOCAL CONSTANTS

else both a and 3 are non-ramified.

Hence

from the Proposition applied to

qo

= o(a).

instead of a,

we

have: W(e.a) = N(^a

lyZ2

3ot(c Tx)^ (c Xx).

£ xe(l+a)

mod For any x = 1 (mod a),

K

X L 0

we have 3a(c

x) = 3(c)a(c

x) ,

and

the Corollary follows.

When we have an extension L/K and an ideal a of 0

ao

L

is an ideal of 0

which we shall again denote by d when L

there is no fear of confusion.

Corollary 3

different P

Let L/K be a finite extension with relative

Suppose the ideal d satisfies:

(a)

(b)

X Let 3 be a character of L

I of finite order such that

denote the character x

and let a

L

x * a(NT /T^(x)) of L . L/K

if c e K is as in the Proposition, W(3.a ) = g(c)W(a ). L L

Then

100

TATE

This Corollary will follow from Corollary 2, applied

Proof

to the field L and the characters

and 3 of L ,

once we

show that in Proposition 1 we can replace K, a, and d by

L, a , and (LO , L

and still keep the same c. Then we have only to verify that

Suppose first CL = 0^.

cO

= P(a) implies that cO = D(a ), i.e. that D(a ) = 'P(ct) K L 1j L '

Since L/K is non-ramified in this case, there is no problem. (We have D = D and, since N L K L/K.

,

maps 1 +

onto 1 + K

L

also 6(“ ) = ($(«)-) Li

Assume now P^|d. Tr

Let y £ CL

l^ien

=

, (y) £ b, and the product of any two conjugates of y L/K

over K is divisible by CL

—2 —2

2

and hence by ^(ot).

It

follows that ) * (

ajl+y) = a(NL/K(l+y)) = a( l+Tr^Cy)) = ^(c ^^(y))

for all y £ a

= ’PL(c b),

Using Pl - Dl//k,

andpja,

and cO

)

implies Li

) * (

it is easy to see that ( ) *

L

= P(a ). L

That being so,

shows that c satisfies (ii) of Proposition 1 for the

field L,

the character a ,

to be shown.

L

and the ideal CLO , Li

as was

101

LOCAL CONSTANTS

Existence of Local Constants Throughout this section, we consider only local and

If K is such a field

lobal fields of characteristic zero.

S and K is an algebraic closure of K, L cK,

extension L/K,

we write

If G is a profinite group,

then for any finite

= Gal(K/L). a virtual representation of

G is an element of the free abelian group on the set of isomorphism classes of irreducible continuous finite-dimen ­ sional complex linear representations of G.

If K is a local

or global field,

let R(K) denote the set of pairs (L,p),

where K C- LcK,

L/K is finite,

and P is a virtual repre­

sentation of to . ii If E/K is a finite Galois extension contained in K/K,

R(E/K) denotes the set of pairs (L,p),

where Kc L c E,

and p is a virtual representation of Gal(E/L) . natural way,

In a

we may regard R(E/K) as a subset of R.(K) ,

and then:

U R(E/K) E/K as E ranges over all finite Galois extensions of K in K. R(K) =

Let R^K) denote the set of pairs (L,x),

mite extension of K in K, °rder of L

where L is a

and X is a character of finite

(if K is local) or C

(if K is global). L

Via

102

TATE

we may view R^K) as a subset of R(r)

class field theory,

We also write R}(E/K) = R (K) H R(e/K). Suppose we have a function F defined on R^K) taking

values in some abelian group A.

We say F is extendible if

F can be extended to an A-valued function on R(K) satisfying, (a) F(L,Pi + P2) = F(L,P1) ,F(L,p^) for all (L,Pi) e R(K), and

(b) if (L,p) e R(K) with dim(p) = 0, and L □ L1

K,

then:

F(L,p) = F(L’, IndL/L,(p)),

where Ind^^, (p) is the virtual representation of

induced

from p . If E/K is finite Galois,

we say F is extendible in E/K

if F can be extended to a function on R(E/K) satisfying (a)

and (b) with (L,p^) and (L,p) in R(E/K).

Remarks

1)

If F is extendible (or extendible in E/K) ,

there is a unique extension of F to R(K) (or R(E/K)) satisfying (a) and (b).

extensions F^ and F^.

For,

suppose we have two such

Then, if (L,p) £ R(E/K):

F.(L,p) = F.(L,P - dim(p)[1 ]).F(L,[1 ])dlm(p), for i = 1, 2, 11 1j 1j where [1 ] denotes the unit representation of Gal(E/L). Li

By

103

LOCAL CONSTANTS

grauer induction ( [s,p.9$ Ex.2]): p - dim(p)[lLl = I niIndL^L(xi ~

■

i

1

1

for some rational integers n£ and some (L^X^ £ R^E/L).

Consequently: p (L,P - dim(p)[lT]) = n F(L. ,X-) ^F(L.,[1 1

])

11

= F (L,P - dim(p)[1 ]), Z jj

and therefore Fx = F£. By the uniqueness just proved,

2)

it is clear th

is extendible if and only if it is extendible in E/K for

all E. Suppose F is extendible and let F denote its

3)

extension.

In the situation of (b),

hypothesis dim(p) =0,

but without the

we have:

F(L' .Ind^Jp)) = \/L, (F)dlm(p)F(L,p) ,

where:

F(L', XL/L'(F) FtL.ll.]) Li ls a constant depending only on F,

L/L’.

Indeed,

and on the extension

this formula follows immediately on writing

p = PQ + dim(p)[lL],

where dim(pQ) = 0,

and applying (a)

104

TATE

and (b).

= 1 for all L/L’,

If

without the hypothesis dim(p) = 0,

if (b) holds

i.e.,

then we shall call F

strongly extendible.

Examples

(l)

If K is global,

(L,x)

-> Ms,x) is strongly

The extension (L,p) 1—► A(s,p) is given by

extendible.

ArtinTs theory of non-abelian L-series. (II)

(L,X)

If K is global or local non-Archimedean,

Nl/k(^(x)) is extendible.

where ^(p) is the Artin conductor of

(L,P) —>Nl/k(/j(p)),

P.

In the sense of Remark 3),

\/L'

we have in this case

where d. denotes the discriminant.

NL'/K^L/L'^’

(ill)

The extension is

If c £ CT^ (K global) or c £ K K

* x(c) is extendible by (L,p)

then (L,x)

(K local),

detp(c).

Here

x

we view c £ C x

x

or K H L .

L

or L

via the canonical inclusions C

K

*

C

L

In this case we haVe \/L' = T/L'(C) = ±ls

where £L/L' is the character corresponding by class field

theory to the extension L’^dJ/L’,

where d is the discrimin­

ant of L/L’ . (IV)

Suppose that F(L,x) depends only on

L, F(L,x) = a(L) , say. F(L,») - »(L)ai"(p).

Then F is extendible by

105

LOCAL CONSTANTS

(V) If K is global,

ible by (L,p)

(L,x) '—>W(x) is strongly extend­

W(p) = A(l-s,p)A(s,p)

.

Notice also that a product of extendible functions with values in the same group A is extendible.

Theorem 1

(Langlands)

istic zero ( ), *

If K is a local field of character­

then (L,x) l~~* W(x) is extendible.

This result was proved, up to sign, by Dwork [Dw] ;

see

Corollary 2 below. The proof we give of Theorem 1 is a modified version of In the terminology of [D] ,

that of Deligne [D] .

our local

£(p,i ,dx,i) = £(pah ,x) 1—> W(x) is extendible in E/K. (*)

The restriction to characteristic 0 is just to fix

ideas;

the result is true,

and can be proved in essentially

fhe same way, in any characteristic.

106

TATE

There exists a finite Galois extension e/k of global

Lemma

fields and a place v of k such that: o

(i)

there is a unique place u of e lying over v and o o

the extension e1

is isomorphic to our given local

/k o

o

extension E/K; (ii)

k is totally complex (i.e.

k has no real

Archimedean place).

Proof

Let e’ be a global field which is dense in E and

which contains some imaginary quadratic subfield of K.

Let

e be the compositum of the fields (e1)0 for a e Gal(E/K),

and let k = e O K.

Then Gal(E/K) acts on e,

fixed field for this action,

and k is the

so e/k is Galois and we may

identify Gal(E/K) with Gal(e/k). Since k contains an / imaginary quadratic field, it is totally complex. Let v$ be the place of k induced by the inclusion k c K,

u

o

be the place of e induced by e c. E.

invariant under Gal(e/k),

v .

The completion e

Then u o

and let

is

so is the only place of e above is E,

since e was chosen dense in

o

E.

The completion k

is obviously contained in K,

and

o must be all of K by comparison of degrees.

Let k, e, vq, and u$ be as in the lemma.

Identifying

107

LOCAL CONSTANTS

/k^ we have an isomorphism Gal(E/K) - Gal(e/k) o o end hence a bijection (£,p) *—> (£ ,p ) between R(e/k) and o o r(E/K) 9 where wq is the unique place of £ above vq, for

E/K with

k,

„ o £

C

is the restriction of p to

and where p O

Gal(E/^ )• course, this bijection commutes with "W o addition and induction. Our problem is therefore to prove that the function: (£,x)

W(x ) w o

(the local root number)

is extendible in e/k.

If e o

k and v is a place of k,

for primes of e and £ such that u|w|v.

Archimedean v / v , o

we write u and w For each non-

let a be an ideal of 0 vv

6(3) |a for each (F,3) e R (e /k ), v 1 U V

such that

and such that d V V

= 0

if v is non-ramified in e (in which case each 2 is non­

ramified) .

Let a be a character of finite order of C, such k

=1 and such that d2P2 . |6(a ) for each nonv v e /k u v o U V Archimedean v / v . (if v is non-ramified in e, this last that a

condition is no condition at all. a is that it be 1 at one place,

Thus the requirement on and highly ramified at a

finite set of the remaining places. 9X1 a

“

The existence of such

indeed of an a having preassigned local components

at a finite set of places - is guaranteed by the Grunwald-

108

TATE

Wang theorem,

[AT,p.103,th.5].)

e.g.

cf.

Let c = (c^) be an idele of k constructed as follows:

c c

v

= 1, if v is Archimedean or if v = v ; o

v

= the element of k v

and & v

as in

for non-Archimedean v / v . o

Proposition 1, Let ( ,x) *

associated to a v

e R^e/k),

and let

Then for each

= a o

place w of £ we have (ct^)^ = % °

/k ’ w' v

and:

X (c )W((a ) ) if w is non-Archimedean and W V w w # wo;

{

W(Xw ) if w = wo; o 1 if w is Archimedean.

The first case follows from Corollary 3 of Proposition 1,

the second from the fact that a

= 1 and the third from v o the fact that k is totally complex so ot^ and X^ are 1 for

Archimedean v. Expressing the global root numbers as a product of local ones,

we find: W(xa„) = n W(x .(%) ) w

= w(xW ). o

n_L

w^w

o w non-Arch.

= w(xw )x(c)a(£), o

xW (c v )w((a x ) w )

109

LOCAL CONSTANTS

where

a(£) =

Bv example III, (

n / WfW

o

x) »-> x(c) is extendible.

(£ y)

•—> a( &) is extendible.

(£,X)

1(ii) —* W(x.

because

W((a ) ) w

By example IV,

By example V,

is extendible by (£,p) I—> W( p 0 a^) ,

corresponds to the restriction of a to

Ind(P 0 res (a)) = (Ind p) 0 a.

(^,x) H-> W(x w

is extendible,

Corollary 1

o

Hence:

) = W(x.a )x(c)"1a(Jl)"1 X/

as was to be shown.

Let K be a local field of characteristic 0,

and let (L,p) e R(K).

Then:

(i)

|w(p)| = 1;

(ii)

W(p)W(p) = detp(-l);

(iii)

if P = P,

(i)

and

then W(p) is a fourth root of unity.

If (L,x) £ R/K),

then |w(x) I = 1.

Clearly

(R»X) 1 * |w(x) | = 1 is extendible by (L,p) ‘—> |w(p)|. Hence, by uniqueness of extension,

(ii) cf-

If (L,x) e R^K),

[Dur.M,2.2].

|w(p)| =1.

then W(x)W(x) = x(“l),

110

TATE

Now, (L,x)

W(x)W(x) is clearly extendible by

(L,p) •—> W(p)W(p),

so by uniqueness of extension and exampie

we have W(p)W(p) = det (-1) .

III,

) is now immediate,

(iii

Remark

since det (-1) = ±1.

Using the lemma which is stated before Corollary 2

of Proposition 1,

we can obviously generalise (i) as

follows:

If E is any subfield of C containing W(p),

|| W(p) ||

= 1 for every place v of E which does not lie above

p,

then

the residual characteristic of K.

Corollary 2 (Dwork,

x(“1)W(x)2

[Dw] ) The function (L,x)

is strongly extendible on R(K).

Indeed,

Corolla,ry 1,

the extension is (L,p)

this is the same as W(p - p),

dim(p - p) = 0,

Corollary 3 (K,p) e R(K).

det^ (-1)W(P ) 2.

and since

it is a "strong” extension.

Let K be an algebraic number field,

For each place v of K,

let P^ be the

restriction of p to a decomposition group of v. (K ,p ) e R(K ) and: v v v

and

Then

By

111

LOCAL CONSTANTS

w(p) = n V

w(pv).

where the product is taken over all places v of K.

proof

It follows from the group-theoretic properties of

induction and restriction ([S,Prop.22]) that if (L,0) £ R(K) ,

and if v is a place of K,

then:

IndL/K(0)v=

1 I w| V

W

V

where the sum is taken over all places w of L above v.

This

H w(0 ) is an extension of w (L,X) n w(x ), (L,x) e R (K). Since W(x) = H W(X ) for W 1 w w w (L,x) e R1(K), the result follows from uniqueness of

implies that (L,0)

extension.

Now let K be a non-Archimedean local field,

Galois extension,

E/K a finite

and P:Gal(E/K) + Aut (V) a representation

of Gal(E/K) on a complex vector space V.

Let P(E/K) denote

fhe first ("wild") ramification group of E/K,

and let

the subspace of all elements of V fixed by P(P(E/K)).

p induces a representation: PP : Gal(E/K)/P(E/K) ->Aut (VP).

Notice that for representations P

and P^ we have

be Then

112

TATE

(Pl + P2)

PPP = Pi + p^3

so that P

P

is defined even when P is

a virtual representation.

Let K be a non-Archimedean local field,

Corollary U

let (K,P) e R(K).

and

Then:

P W(p)/W(p ) is a root of unity.

Remark

Since

P Hp

is additive,

the Corollary for irreducible P.

either P else P

P

unity.

P

If p is irreducible,

. . ... m which case the result is trivial,

= P,

=0,

it is enough to prove then

or

in which case it states that W(p) is a root of

That statement is Dwork’s [Dw, Th.6(b)];

the version

above is Deligne's [D, Appendix].

Proof

a

For two non-zero complex numbers a and b,

b if ab

Lemma 1

-1

.

is a root of unity.

Let (L,0) e R(K) ,

w(e)

Proof

we write

and suppose L

LT

K.

Then:

wdnc^ /L,(e)).

w(e) .WUnd^, (e)) 1 = W( Ind^, [1L])

which

113

LOCAL CONSTANTS

(Alternatively,

a root of unity by Corollary 1 (iii).

is

is obvious from Corollary 2 that W(6)/W(Ind(6)) is a fourth root of unity.)

Let (K,a) e R^K),

Lemma 2

and let L/K be a totally wildly

the maximal tamely ramified exten-

ramified extension (i.e.

sion of K in L is K itself). W(a)

Proof

If « is tamely ramified, then: W(a ). L

Recall the notation a

ra.mi fi ed,

then ot

L

and the assertion is immediate.

is also

Then 6(«T ) = PT •

So assume that f$(a) = p K

W(a) -v Npl/2

X

X

= 0

K

We can view

1j

X

a as a character of (0 /p ) K

If a is non-

= a o n

K

mod

p . K

Then:

a(x)A(x)

y

K x£Ok

X mod

p„ K

for any non-trivial character X of the additive group

^°K^PK^

*

As x runs through a set of representatives of

r,x x K mo—^y^’^

-U

Li IK

is an automorphism of the field (lip, K.

K

So:

114

TATE

W(aL)

NPl1/2

aL(x)X(x[L:K])

[

* xto

K x mod p K

= Np"1/2 £ a(x[L:K])A(x[L:K]) x W(ct).

it is enough to prove the Corollary

By Brauer induction,

when P is a representation of Gal(E/K) of the form IndL/K(x)>

for some (L,X) £ RjCE/K).

Either:

X

P

=0,

or

x

P

= X.

In the first case, it follows from [S,Prop.22] that the

restriction of Ind^tx) to P(E/K) does not contain the unit p

representation.

Therefore Ind^^x)

= 0,

and the result

follows from Lemma 1 and Corollary 1 to Proposition 1. p

So assume that x Gal(E/L) Ci P(E/K),

= X.

Then x is trivial on

and we may extend x to a representation

of Gal(E/L) .P(E/K) by giving it the value 1 on P(E/K). Call this representation x * 5

W(x) % W(x!).

IndE'/K^X'j ’

and then by Lemma 2 we have

The representation Ind^^X^ contains

where Gal(E/E') = Gal(E/L).P(E/K).

Further,

115

LOCAL CONSTANTS

it follows from Frobenius Reciprocity ([S,Th.l3]) and the

properties of restriction that the unit representation occurs ith multiplicity exactly [E*:K] in the restriction of p

I»«1/K(X) ‘° P(E/K>'

ame degree,

w(p)

Hence:

p W(lndEt/K(x’)) = w(p ).

W(x’)

(i)

(K,p) e R(K),

and IndE' /K^') have the

and are therefore equal.

W(x)

Corollary 5

So Ind^/^x)

If K is local non-Archimedean,

and if x is a non-ramified character of K

X

of

finite order, then: W(x 0

p)

= X(t5(p)) .w(x)dim(p)w(p) = x(P(p))W(p),

where P(p) = (ii)

.

If K is an algebraic number field,

representations of

and p and a are

with relatively prime conductors, then

W(p 8 a) = (-l)adetp(^a))deto(^(p)).W(p)dim(a)W(a)dim(p),

where a is the number of Archimedean primes of K at which ^etp and det^ are both non-trivial.

^S^rk

The symbol detp(^(o)) is to be

sense detp(^(o)) = detp(f),

where f is

hat fv - 1 if v fs Archimedean or if p

understood in the an idele of K such

is ramified at v,

1 16

TATE

and f 0 = ^(o ) otherwise, v v v

(i)

Proof

Corollary 2,

Likewise for det (4(p)).

If (L,a) c R (K) ,

then by Proposition 1

we have W(XT-a) = XT(^(a))W(a). L

But:

L

XTL(P(“)) = X(NLiT /n/K(^(a))).XLiT(PLTi ) = X(NL/K(^(a))).W(XL).

So by Theorem 1 and examples II and IV,

(L,ot)

h->>W(XT L

0 a)

is extendible by:

x(NT/KU(e))).w(xT)dim(e)w(e). L/iL L

(L,e)

Hence by uniqueness of extension:

W(x (ii)

0

= x(^(p)).w(x)dim(p)w(p).

Suppose that v is non-Archimedean,

non-ramified.

sentations, W(p

where det

p)

Then o

v

is a sum of one-dimensional repre-

all of which are non-ramified,

® o ) = det v v c,v

o ,v

is

and that

= (det ) . O V

(^(p )).W(p )^m^ v v

and by (i):

v

At any other non-Archimedean

place v: W(pv ® Cy) = d-etp y( d( Gv))-w( pv)

•

Notice that these expressions are symmetric in P and o if

neither is ramified at v.

117

LOCAL CONSTANTS

are both sums of one-

If v is Archimedean, P^ and

^imensional representations and one verifies directly that: ufp 0 a ) = (-1) n \ rv V uhere a

.W(P ) V

W(a ) V

= 0 unless both det^

and detp

are non-trivial,

Taking the product over all v:

in which case a^ = 1.

W(p 0 a) = (-l)a.detp(($(o))de-to(^P)).W(p) ||x|| $ of K . &

(Here,

||x|| __ is the K

normalised absolute value function on our local field K.)

Furthermore,

we

have p 0 to$ = p1 0 (Dg| if and only if

P 0 w

9 in which case to is necessarily of b b S“S finite order, i.e. s - s’ is a rational multiple of

P

27ri/log(NpK).

* = P 0 oo ;

That being so,

we define for irreducible

118

TATE

w(4>) = w(p0 U)s) = (NP(p))

(*)

sw(p),

as suggested by Corollary 5 (i) above.

By that Corollarv V

it is obvious that this definition is valid, dent of the decomposition $ = p 0 to$. for irreducible

,

indepen­

Having defined W(

for some representation 6:

In both cases,

Here we have w(e) = w(e),

W(p) = 1.

so w(p) = w(e)w(e) = i.

(iii) P is dihedral:

is cyclic,

L/K is quadratic, E/L

Gal(E/K) is dihedral,

That is,

and P = Ind^Cx) for some 1-dimensional repre­

sentation x of Gal(E/L);

the situation is the

in short,

global analogue of that in Theorem 2.

as idele

By the global transfer theorem, [Dur.M,3.1], class character,

we have x|c = 1. IY.

Let L = K( cl(^2(p)) satisfies

"these three conditions and is therefore equal to c(p). details, see [DO].

orm ) ** (

above,

We note that condition (b) ,

corresponds to the rule:

in the

For

130

TATE

S(PL + %) =

+ MM + \

where G = Gal(E/K) ,

H2(Gv,2Z/2 2Z) ,

and G^ is a decomposition group of v.

Thus the elements cl^^p )) = c(p^) are the local invariants

of an element (of order dividing 2) in the global Brauer

group.

Therefore c(p),

the product of these invariants,

is indeed 1.

REFERENCES

This volume:

[Dur.M]

J. Martinet, Character theory and Artin L-functions.

LOCAL CONSTANTS

131

E. Artin & J. Tate, Class Field Theory (Benjamin, New York, 196?). J.W.S. Cassels & A. Frohlich, Algebraic Number Theory (Academic Press, London, 1967).

J-P.Serre,

Corps Locaux (Hermann, Paris, 1962).

p. Deligne, Les constantes des equations fonctionnelles des fonctions L (Springer Lecture Notes 3^9 (197^) PP- 501-597)• P. Deligne, Les constantes locales de 1'equation fonctionnelle de la fonction L d'Artin d'une representation orthogonale (to appear in Inv. Math.) B. Dwork, On the Artin root number (Amer. J. Math. 78, 1956, iUU-1+72).

A. Frohlich & J. Queyrut, On the functional equation of the Artin L-function for characters of real representations (inv. Math. 1H, 19715 173-183). R.P. Langlands, On Artin's L-functions, in Complex Analysis (Rice University Studies 56, 1970, 23-28). J-P. Serre, Representations Lineaires des Groupes Finis (Hermann, Paris, 1971 (2nd. ed.)).

J.T. Tate, Fourier analysis in number fields and Hecke's zeta-functions (thesis, Princeton University 1950; published in [CF], 3O5~3^7)« A. Weil, Basic Number Theory (Springer-Verlag, Berlin, 197^ (2nd. ed.)).

Galois module structure

A. Frohlich

The central topic of these notes is the global structure

of the ring of algebraic integers in a normal extension N/K of number fields,

as a module over the Galois group.

pared to the original notes,

distributed in Durham,

presentation here has been expanded and recast,

covering essentially the same material.

results have been incorporated,

of an adelic resolvent.

Com­

the

although

Some new ideas and

in particular the notion

We also give a new treatment of

properties on change of base field or of group;

the conn­

ection between norms of homomorphisms and restriction of

scalars has now been clarified. The notes have been subdivided into three main parts.

e

Part is concerned principally with the theorems

°n Galois module structure, esolvent theory,

the second with the underlying

and the third part - which may be read

dependently of part II and which may be viewed as a

133

134

FROHLICH

supplement to Martinet’s lectures (cf.

root numbers and Galois Gauss sums. brief outlines of proofs.

[M3]) - deals with

We mostly give only

In the nature of things these are

mainly contained in Parts II and III.

There is an appendix

which deals with some of the subject matter of Part II in a

more formal manner.

Notations.

As usual,

7L,

(ft, ]R, C are the ring of integers

and the field of rational, of real, and of complex numbers,

respectively.

* noted by S .

The multiplicative group of a ring S is deThe Galois group of a normal field extension

E/F is Gal(E/F) .

The algebraic closure of (ft in (T is de­

noted by ft and a ’’number field” K is always a subfield of ft

of finite degree over (ft.

symbol 0 , or just 0, integers in K.

We then write Gal(ft/K) = & . K

The

stands for the ring of algebraic

Completions at prime divisors of K are

indicated by appropriate subscripts,

with the convention

that if p is an infinite prime divisor and M an 0-lattice then M is actually the completion of MK. P

We shall often

consider completions with respect to prime divisors of sub­ fields (semi-local completions), denoted in the same way.

135

GALOIS MODULE STRUCTURE

Part I. Theorems on Galois module structure

Background Throughout F is a finite group and K a number field.

Assume we are given a surjective homomorphism tt:

(1.1)

Q

K

with open kernel and we denote by N the fixed field of Ker

tt.

Then we have an isomorphism

which we use to make N,

§1U,

f = Gal(N/K) ,

tt:

(1.2)

0^ etc. into f-modules.

(Except in

iris really fixed and need not be referred to).

wish specifically to study

= 0

We

(notation used throughout)

as a module over the group ring d(F Hom(2 5C ) is the transfer. K K,

equations, in conjunction with [M3]

2, Theorem 7-2,

These

now

imply the first part of Theorem 1. We also note for later reference that

A.. (a |x) = (a|x) Det (X)

(7-3)

for X e K(T)

*

.

X

One can now proceed either idele theoretically, ideal - and module theoretically,

or

corresponding to the two

descriptions (2.1) and (2.2) of Cl(o(F)).

lay the emphasis on the first approach,

Although we shall

we shall indicate

briefly also the second one - each offers its own advantages.

We shall have to extend the notion of resolvent even further.

It is clear in principle that it can be defined in

the context of commutative rings.

Here we are specifically

concerned with local (or semilocal) completions.

p be a prime divisor of a subfield k of K. =ND(=N®kkD)’ r K p

i-6'

Let then

If a K (?)

a is a free generator of N

GALOIS MODULE STRUCTURE

over &p(r)>

157

then we can define the resolvent (a|x),

similarly for k c F c K also NK/F(a|x)5

same way as in §3.

EP (EP = E0kV-

For E big enough,

and

in essentially the

these will be in

We shall specifically be interested in

the case when a 0 (T) - 0^9 P

i.e.,

when a generates a

local normal integral basis at p.

More generally we shall look at products of completions, with p running over some set S.

comment is required. we shall need,

in OJ).

no further

We consider the only other case which

namely S the set of all prime divisors (say

We are now considering adele rings Ad(K), Ad(N) For an adele a of N,

etc.

For S finite,

i.e.,

for a e Ad(N),

we again

define (a|x) = det I a1' T(y)

\

WK/k(a|x) = n

(a|x°

,

a just as in (3.1), (3.2),

provided of course that a generates

a normal basis of Ad(N)/Ad(K),

ap Kp(r) = Np ’

a

=

(a|x) e J(E).

obvious manner.

i.e.,

that

for a11 10

’

for aijnos't all P

Formulae (7.1) - (7.3) extend in the

Moreover via the embedding N C Ad(N) ,

original resolvents may be viewed as adele resolvents.

the

158

FROHLICH

Finally for local components we have

(a|x). = (a |x).

(7.5)

IJ

The ideal

theoretical

counterpart to adele resolvents

are the resolvent-modules (0:x).

Note that, for given x,

the resolvents (a|x)5 a e N span a one-dimensional K(x)~ subspace of

(or of E).

define (0:x) to be the 0

This follows from (7.1).

generated by the (a

K(x) -module

This is then rank one,

with a e 0.

Analogously we let

K/k

conclusions!). adele ct of N,

x)

finitely generated.

(0:X) be the

by the ^K/k(alx) with a e 0

We

-module generated

(use (7-2) and draw the same

then there exists an

If now N/K is tame,

= °n9

%

so that for all p ,

and the

connection with resolvent modules is given by

(“|X)P

(

(7.6)

°K(x),p"(l,:X)P

'

j

i (W . (ct|x) )$ / \ = N , (d:x) < v K/k K k(x),P K/k Kip The next theorem describes the class of 0 in terms of

resolvents.

Theorem 8. (i)

a

p

Suppose N/K to be tame. Let a £ N,

a K(F) = N.

0 (T) = 0 , for all p. p p * ------- ‘

Write -----

Let a s Ad(N),

159

GALOIS MODULE STRUCTURE

f(x) = (a|x) (alx)

1

whence_

WK/^

* NK/(x) = (^x) (a|x)

foiL(0)o(r),

.

Then

X) J is a family of invariants

b(x)l one for (0) g(r)

and

We indicate the proof of (i),

version of the theorem.

We now have M = 0,

Then the adele a of the theorem is of

v = a.

form a = a3,

the idele theoretic

Go back to the description of

(M) e Cl(o(r)) given after (2.2). V = N,

Write

3 an idele of K(F).

represented by f,

Thus (°)0(r) is

with f(x) = Det (3). X

But by (7-3),

or

rather its extension to adeles we see that Det (3) = X Nx) (a|x) . This yields the stated description of

p(r)’

For (0)Z(r) apply (2,8)‘

Now we come to the connection with Galois Gauss sums. This,

as we shall see,

looks neater in terms of resolvent

160

FROHLICH

modules - possibly because we do not yet have the full story

Theorem 9*

(i)

u(x) =

Suppose that N/K is tame. Let a e Ad(N), a 0 (r) = 0^ for all p. r P

t(x)

Nk/^

(a|x) \

Let

Then

u e Homo (R 5 U(E)). $

(ii) T(x) %(x) It is now clear that Theorem 1. is a consequence of the

We shall only give an outline of the

last two theorems.

strategy of proof for Theorem 9,

using most of the theory

presented in the next few sections (see §10).

Remark

Theorem 9,

to the wild case.

say in its adelic form, One can only demand that

at the tame primes,

u £ Hom

(R , U’(E)),

can be extended

o^(r) =

and one can then only assert that where U’(E) consists of those ideles

which are units at the tame prime divisors.

§8.

Change of field or of group This section contains a down to earth,

explicit

161

GALOIS MODULE STRUCTURE

account of the topic in the title.

stage bringing in new concepts,

We shall avoid at this

in accordance with our aim

to give a quick and accessible introduction to the essential rather than a formally presented theory.

results,

The

appropriate formal background for the proper theoretical setting of these results will be briefly outlined in the appendix (which the reader can omit!).

In the sequel let

be one of the three following

functors of number fields L.

1 f(i) (8.1)

Z

(ii)

= L‘

\ = T

divisors P of L)

j ((iii)

XJ

0 , u, y

where p is a prime divisor

and in this case L is restricted to

of a number field k, extension

0^ p (product over all prime

n P

fields of k.

Let N, K, f be as before,

assume N/K to be tame,

with k c K.

In case (ii)

in case (iii) to be tame above p.

We consider the maps (8-2)

X i-»(a|x)

(X e Rf)

where a A_r(I) = A h N

(i.e.

5

a generates a ’’normal basis of \T”)«

It is

162

FROHLICH

important for the interpretation of the theorems in this that the maps (8.2),

section,

for varying a,

coincide

exactly with the maps

X»—> (ajx) Det (X)

(8.3)

a$ ^(T) = A^ and with X varying over

with a$ fixed, A$(r)

K(r)

(i.e.,

in case (i),

U(o(F)) in case (ii),

*

0 (F) r

in case (iii)).

Let △ be a normal subgroup of F,

field,

£: R

and write E = F/A.

its fixed

F = N

Thus 2 - Gal(F/K).

Let

be the lifting of characters.

-> R 1

Zj

Theorem 10.

Assume in case (8.1) (ii) that N/K is tame.

in the case (8.1) (iii) tame above p. tL/F(a) ^(E) = A^,

If a A$(F) =

where t^^ is the trace.

Also,

then

for

X £ re

= ^L/F^^F/K

Remark on notation.

*

We indicate resolvents with

respect to N/K by subscripts as above - strictly speaking

they depend on

Proof.

tt

.

Obvious.

163

GALOIS MODULE STRUCTURE

Next let F be any number field containing K,

) = A c F.

let

Then A = Gal(NF/F).

F

Theorem 11.

In’case (8.1) (ii) assume that N/K is tame and

that furthermore each prime divisor P of K is non-ramified

In case (8.1) (iii) make the corresp­

either in N or in F.

onding assumption .just above, P. b e

Then given a e A^,

sq that aAK(r)=AK-

'

*

3 X e A„(r) r

with

(a|x)N/K De\(A) = (b|(x|A))NF/F

’

for all x e Rp. Here (x|A) is the restriction of x to △.

Outline of proof.

We consider the induced T-module

Map^T, A^p) of the A-module A^p. pointwise operation on maps e.g.

This is an A^-algebra by f1 f2(y) = f^(y) f^(y).

Moreover f acts by algebra automorphisms. On the other hand consider the A^-algebra A^ ® with F acting via the tensor factor A^.

homomorphism

A^,

K Then we have a

164

FROHLICH

(8.U)

Y where 9(x 0 y) (y) = x y.

of algebras and r-modules,

view of the hypotheses made,

In

this is an isomorphism.

With b as given in the theorem let f, e Map^T,^) be b defined by

y e △

ftY,
E be the map g(x 0 y) = xy.

also by g the induced maps A^ 0^ K

A$ -> A$.

Denote

Then, recalling

GALOIS MODULE STRUCTURE

the definition of

it follows that

,

f;/1)) = N(xl»»„F/F

e(oetx(J

I

165

ye i

while trivially g(Detx( Ha 8 1)\ b) = (a|x)N/K,

j

g(Det (1 0 A)) = Det (A). X X The theorem is now seen to follow from (8.6) on applying g.

We note a Corollary,

Corollary

K,

above p,

to be used subsequently.

In case (iii) (so that each prime divisor P of

is non-ramified either in N or in F),

we have

\((a|x)N/K) = \((bl(x|A))NF/F),

!

£or all x e Rp 9

and all prime divisors q of E,

(a, b as in the theorem).

above p.

In particular if (x|a) =0 then

' °-

Here

* E 7L

is the standard valuation.

Next let △ be again a subgroup of f,

Then again △ - Gal(N/F).

For $ e R

and let now F = n\

denote by

the

induced character of I.

Theorem 12.

Assume in case (8.1) (ii) that N/K is tame,

in

166

FROHLICH

Let

case (8.1) (iii) that N/K is tame above p.

W4’"4!,

aAK(r)=\'

and {o} a right

and let {cJ be a free basis of transversal of

r

in

K.

(or of A in T). '—----

Then for some

A £ * AUA) IY

(al ** )N/K DeMA) = WF/K

Outline of proof

A^.(A) and {o}

c.°)degU) .

Let {a^} be a free basis of

a right transversal of

in

over If, say.

T: A -> GL^(E) is a representation, we consider the block matrix with row index o,

column index i whose a, i entry

is the matrix T( £ a.° 6 1). 6eA 1 depend on the character

the choice of {a.}, i

corresponding to T,

and on the choice of {c}.

denote it by Det, ({a.}). 1 of m,

correspond biuniquely to

under a map

ip i—> d

where ip is

the character of a representation lifted from a faithful

representation of the quotient group H n (d = d ) of H 4d ip 4m We shall then write W(N,ip) = W(N) when d = d . d ip

We thus

have to show: given a map f from the set of divisors d > 1 of

m to

± 1,

there exist infinitely many N,

tame

(1H.1) and normal with Gal (N/(ft) = F,

and N

F^

so

that W (N) = f(d) for all d. d

2)

Let p be the quadratic idele class character of

GALOIS MODULE STRUCTURE

p corresponding to its extension F .

179

Let be an idele

class character of F of order m, not ramified at the prime

divisors of m9with 4>((x,^) (x,ww') = (x,“)

(16.3)

‘Hx0

-1

,

(x,“'),

0

= (x,w),

»“)

a e ft , X, X' e R?.

for a>, W eft,

We denote by X of groups

Homn ,$(Rr’x) the set of homomorphisms f : Rp F with -i

f(xW

(16.U) for all

r

)W = f(x) (x,^) ,

X e Rr1

If

(f>

are two such pairings

then multiplication in Hom(Rp,X) yields a pairing

Hom,. , * Horn. , -> Hom , . %’S %’% %’S %

In particular,

is either empty or else is a coset of HomQ

Hom

F’

in Hom.

T

Restriction of base fields.

(i)

Let k be a subfield of the number field K.

We have

already considered the norm map MK/k(ef (2.6)) on the groups

Hom^ (Rr,X). X=J(E),

for

One can show that the map

maps Det(u(o(T))) into Det(U(o (I))) (similarly K.

for Det(K(F) ) etc).

Hence we get,

via (2.1),

a homo­

morphism

K.-. Ci(o(r)) -> ci(o (r)). A/K.

-K-

Comparing with restriction of scalars,

one gets,

for a

186

FROHLICH

locally free o(F)-module M of rank r,

(16.5)

(M)

the formula

(r) . W))’ (r) • ■K.

(W0(r)).

K

When k = ® the first factor on the right vanishes and we

get (2.8). We define also a map

N ■. Hom„ (Rr,X) -> Homo ,(Rr»X) K/k (j) r ^>v(j) r

by choosing a right transversal {a} of Q

in Q K

and setting k &

f(x° V .

(WK/k f) (X) =

(16.6)

\/K *(X’

Here v(x.

under the transfer map.

a different map N’

X i—>

) is the image of (x, ) e Hom(fi ,X) K

A different choice of {o} leads to but only to within a factor

Cl(Op(r)).

One then has

hc/F »(r)> '

(iii)

(H9 °F>oF(r)’

Transition to quotient groups.

187

GALOIS MODULE STRUCTURE

Let 2 = T/A,

I-

Lifting

of characters yields homomorphisms

-> R

r

A a normal subgroup of f.

L>

r E

Homfi (Rr,X) K

which takes e.g.

Hom K

(R£,X)

Det(U(o(F))) into Det(U(0(E))).

induces a homomorphism X

on class groups,

It thus

and we have

Zj

r

(16.8)

Analogous when a (iv)

a

^o(E)

is present.

Restriction to subgroups.

Here A is a subgroup of F. X' i—>

w

Induction of characters

is a homomorphism R.Al -> Rr

Pr/A : Homfi (Rr,x) K

which yields maps

Homn (ra’x)’ K

and analogously when a is present.

that e.g.

for X = J(E),

into Det(u(o(A))),

the map Pr/A takes Det(U(o(T)))

hence yields a homomorphism

Pp: Cl(o(f)) -> Cl(o(A)).

Pr/A ^M)o(rp

(v)

Again one can show

Now we have

(m)o(a)

’

Induction.

With A and F as under (iv),

characters yields maps

restriction R^

R^ of

188

FROHLICH

A/r: HomQ (Ra,X) -> HomQ (Rf,X) K K (and analogously when a pairing is present).

We get an

induced homomorphism i △/r: C1(O(A)) -> C1(O(F)), (16.10)

i-A/r^(m)o(a)(M*)fl(r)

where M *

is the f-module induced by M.

and have

5

§17• Factorial behaviour of resolvent classes and module classes Now we assume again as given a surjection

tt:

ft

K —Ke r "ii with N = Q .If first I = f/A (as in 16. (iii)) then by composition we get a surjection 7ir: ft -*■ E, and tt K -■^erE A Q = N = F, say. If N/K is tame we have from (16.8)

= (»F>0(£)-

(1T.1)

Next let A be any subgroup of I,

If N/K is tame,

let again F =

n\

(16.9)

then by (16.5)5

(A))

(17.2)

Pr/A

F Next let F be some number field containing K and let

) = A. k F'

tt(Q

Suppose that N/K is tame and that each prime

divisor of K is non-ramified either in N or in F.

Then

189

GALOIS MODULE STRUCTURE

(17.3)

1A/r^°NF^O (Ap r

1K/F^