Abelian L-Adic Representations and Elliptic Curves (Advanced Book Classics)

Abelian l-Adic Representations and Elliptic Curves Jean-Pierre Serre College de France McGill University Lecture Note...

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Abelian l-Adic Representations and Elliptic Curves

Jean-Pierre Serre

College de France

McGill University Lecture Notes written with the collaboration of WILLEM K UYK and JOHN LABUIE

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"lIo ADDISON-WESLEY PUBLISHING COMPANY, INC. ��: THE ADVANCED BOOK PROGRAM +- PROC;

Redwood City, California. Menlo Park, California. Reading, M:

New York· Amsterdam· Don Mills, Ontario· Sydney. Madrid Singapore· Tokyo· San Juan· Wokingharn, United Kingdom

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Abelian l-A

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Originally published In

1968 b y

IIIPtlc Curves

.'

W.

A. Benjamin, Inc.

Library of Congress Cataloging-in-PubIlcation Data

Serre, Jean Pierre.

Abelian L-adic representation s and elliptic curves.

(Advanced book classics series)

On t.p. "I" in I-adic is transcribed in lower-case script.

Bibliography: p. Includes in dex.

1. Representations of groups. 2. Curves, Elliptic.

3. Fields, Algebraic. I. Title. QA171.S525 1988

ll. Series.

512'.22

88-19268

ISBN 0-201-09384-7

Copyr ight© 1989,1968 by Addison-Wesley Publishing Company All rights reserved. No part of this publication may be repro duc ed, stored in a retrieval system,

or transmitted, in any form or by any means, electronic, photocopying, recording, or otherwise,

withou t the pri or written permission of the publisher. Manufactured in the United States of America

Published simultaneously in Canada

Publisher's Foreword

"Advanced Book Classics" is a reprint series which has come into being as a direct result of public demand for the individual volumes in this program. That was our initial criterion for launching the series. Additional criteria for selection of a book's inclusion in the series include: •

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Its intrinsic value for the current scholarly buyer. It is not enough for the book to have some historic significance, but rather it must have a timeless quality attached to its content, as well. In a word, "uniqueness."

The book's global appeal A survey of our international markets revealed that readers of these volumes comprise a boundary less , worldwide audience. .

The copyright date and imprint status of the book. Titles in the program are frequently fifteen to twenty years old. Many have gone out o f prin t, some are about to go out of print. Our aim is to sustain the lifespan of these very special volumes.

We have devised an attractive design and trim-size for the "ABC" titles, giving the series a striking appearance, while len din g the individual titles unifying id entity as part of the "Advanced Book Classics" program. S ince "classic" books demand a long-lasting binding, we have made them available in hardcover at an affordable price. We envision them being p urchased by individuals for reference and research use, and for personal and public libraries. We also foresee their use as primary and recommended course materials for university level courses in the appropriate subject area. The "Advanced Book Classics" program is not static. Titles will continue to be

added to the series in ensuing years as works meet the criteria for inclusion which we've imposed. As the series grows, we naturally anticipate our book buying audience to grow with it. We welcome your support and your suggestions concerning future volumes in the program and invite you to communicate directly with us. vii

Vita

Jean-Pierre Serre

Professor of Algebm and Geometry at the College de France, Paris, was born in Bages, France, on September 15, 1926. He gmduated from Ecole Normale Superieure, Paris, in 1948, and obtained his Ph.D. from the Sorbonne in 1951. In 1954 he was awarded a Fields Medal for his work on topology (homotopy groups) and algebraic geometry (coherent sheaves). Since then, his main topics of interest have been number theory, group theory, and modular forms. Professor Serre has been a frequent visitor of the United States, especially at the Institute for Advanced Study, Princeton, and Harvard University. He is a foreign member of the National Academy of Sciences of the U.S.A.

viii

Special Preface

The present edition differs from the original one (published in 1968) by: •

the inclusion of short notes giving references to new results;

•

a supplementary bibliography.

Otherwise, the text has been left unchanged, except for the correction of a few misprints. The added bibliography does not claim to be complete. Its aim is just to help the reader get acquainted with some of the many developments of the past twenty years (for those prior to 1977, see also the survey [78]). Among these developments, one may especially mention the following:

l-adic representations associated to abelian varieties over number fields Deligne (cf. [52]) has proved that Hodge cohomology classes behave under the action of the Galois group as if they were algebraic, thus providing a very useful substitute for the still unproved Hodge conjecture. Faltings ([54], see also Szpiro [82] and Faltings-Wiistholz [56]), has proved Tate's conjecture that the map HomK (A,B) � Z, � Homa.J(T,(A), T ,(B» is an isomorphism (A and B being abelian varieties over a number field K), together with the semi-simplicity of the Galois module Q, � T/(A) and similar results for T /(A)/rr/(A) Ix

Preface

This book reproduces, with a few complements, a set of lectures given at McGill University, Montreal, from Sept.5 to Sepl18, 1967. It has been written in collaboration with John LABurn (Chap. I, IV) and Willem KUYK (Chap. II, III). To both of them, I want to express my heartiest thanks. Thanks also due to the secretarial staff of the Institute for Advanced Study for its careful typing of the manuscript. JEAN-PIERRE SERRE Princeton, Fall 1967

xi

Special Preface

x

when I is large enough. These results may be used to study the structure of the Galois group of the division points of A, cf. [80]. For instance, if dimA is odd and En
and l-adic representations

The existence of I-adic representa tio ns attached to modular forms, conjectured in the ftrst

edition of this book , has been proved by Deligne ([50], see also Langlands [65] and Carayol [49]). This has many applications for instance to the Ramanuj an conjec tu re (Deligne) and to congruence properties (Ribet [69], [71]; Swinnerton-Dyer [8 1 ]; [73], [77]). Some generalizations are known (e.g. Carayol [49]; Ohta [68]; Wiles [84 D, but one can hope for much more, in the setting of "Langlands' program": there should exist a diagram

motives . II-adlC I representaUons . raUona .

automorphic represen tat ions of reductive groups

where the vertical line is (essentially) bijective and the horizontal arrow injective with a precise description of its image (Deligne [51]; Langlands [66];[78]). Such a diagram would incorporate, among other things, the conjectures of Anin (on the holomorphy ofL functions) and Taniyama-Weil (on elliptic curves over Q). Chapters II and III of the present book, supplemented by the results ofDeligne ([53]) and Waldschmidt ([63] , [83 D, may be viewed as a partial realization of this ambitious program in the abelian case.

Local theory of l-adic representations Here the ground field K, instead of being a number field, is a local field of residue characteristic p. The most interesting case is charK = a and p = I, especially when a Hodge Tate decomposition exists: indeed this gives precious information on the image of the inertia group (Sen [72]; [79]; Wintenberger [85]). When the /-adic represen tat i on comes from a divisible group or an abelian variety, the existence of such a decomposition is well known (Tate [39]; see also Fontaine [60]); for representations coming from higher dimension cohomology, it has been proved recently by Fon tai ne -Me ssing (under some restrictions, cf. [62]) and Faltings ([55]). The results of Fonta ine- Mess ing are pans of a vast program by Fontaine, relating Galois representations and modules ofDieudonnc type (over some "Barsotti-Tate rings," cf. [58], [59], [61]).

I

I

I I

I

I I

I

I

I I

I

I I

I

I

I I

I

I I

I

I I

I

I

I

I I

I

I I

I

I

I I

I

I I

I

I

I I

I

I I

I

I I

I

I

I

I I

I

I I

I

I

I I

I

Contents

INfRODUCTION

xxi

NOTATIONS

Chapter I

xvii

I-adie Representations

§1

The notion of an I-adie representation

1-1

1.1

Definition

1.2

Examples

1-1 1-3

§2 I-adie representations of number fields

1-5

2.1

1-5

2.2 2.3 2.4 25

Preliminaries Cebotarev's density theorem Rational l-adic representations Representations with values in a linear algebraic group Ljunctions attached to rational representations

1-7

1-9 1-14

1-] 6

xIII

Contcnt�

xlv

Appendix

Equipartition and L-functions

A.1 A.2

Equipartition

A.3

Proof of theorem 1

The connection with Llunctions

Chapter II §1

1-18

1-18 1-2 I

1-26

The Groups Sm

Preliminaries

II-I

1 .1

The torus T

II-I

1.2

Cutting down T

11-2

1 .3

Enlarging groups

II-3

§2

Construction of Tm and Sm

11-6

2.1

Ideles and idele-classes

11-6

2.2

The groups T". and S".

II-8

2.3

The canonical l-adic representation with values in S".

2.4

Linear representations of S".

25

l-adic representations associated to a linear representation of S".

II-I8

2.6

Alternative construction

II-21

1I-1O 1I-l3

2.7

The real case

1I-23

2.8

An example: complex multiplication of abelian varieties

II-25

§3

Structure of Tm and applications

11-29

3.1

Structure ofX(T".J

II-29

3.3

Structure of T",

The morphism j* : G'" � T".

II-32

3.4

How to compute Frobeniuses

II-35

3.2

Appendix

Killing arithmetic groups in tori

II-31

JI-38

A.1

Arithmetic groups in tori

11-38

A.2

Killing arithmetic subgroups

II-40

xv

Contents

Chapter In

Locally Algebraic Abelian Representations

§1

The local case

III-I

1.1

Definitions

III-I

1.2

Alternative definition of "locally algebraic" via Hodge-Tate modules

III-5

§2

The global case

III-7

2.1

Definitions

III-7

2.2

Modulus of a locally algebraic abelian representation

III-9

2.3

Back to S'"

III-I2

A mild generalization The functionfield case

III-16

25

§3

The case of a composite of quadratic fields

IJI-20

3.1

Statement of the result A criterion for local algebraicity An auxiliary result on tori Proof of the theorem

III-20

2.4

3.2 3.3 3.4

Appendix Hodge-Tate decompositions and locally algebraic representations Al A.2 A.3 AA A5

A6 A7

III-I6

III-20 III-24 III-28

I1I-30

Invariance of Hodge-Tate decompositions

III-31

Admissible characters

III-34

A criterion for local triviality The character ;E

Characters associated with Hodge-Tate decompositions

III-38 III-40 I1I-42

Locally compact case

III-47

Tate's theorem

III-52

Contents

xvi

Ch a pter IV

l-adic Representations Attached to Elliptic Curves

§1

Preliminaries

1.1 1.2

Elliptic curves Good reduction

IV-2 IV-2

IV-3

1.3 1.4

Properties of V, related to good reduction Safarevic' s theorem

§2

The Galois modules attached to E

2.1 2.2 2.3

The irreducibility theorem Determination ofthe Lie algebra ofG, The isogeny th eor em

§3

Variation of G, and

3.1 3.2

Preliminaries The case of a non integral j Numerical example

3.3 3.4

Proof of the main lemma o f 3.1

Appendix A.1

C, with 1

Local results

The case v(j) < 0

A1.1 A1.2

The elliptic curves of Tate An exact sequence

A13 Determination of g, and i, A1.4 Application to isogenies A15 Existence of transvections in the inertia group

A2 The case v(j) � 0 A.2.1 The case I :t. p A2.2 Th e case I = p with go od reduction of height 2 A2.3 Auxiliary results on abelian var ieties A.2.4

The case I

BIBLIOORAPHY INDEX

=

p with good reduction of heig ht 1

IV-4

IV-7 IV-9 IV-9

IV-II IV-I 4

IV-I8 IV-I 8 IV-20

IV-21

IV-23

IV-29 IV-29 IV-29 IV-3I IV-33 IV-34 IV-36 IV-37 IV-37 IV-38 IV-41 IV-42 B-I

INTRODUCTION

The " L-ad ic representa t ions " consid ered in thi s book a re the algebra i c ana logue o f the locally constant sheaves ( o r " loca l c o e f fic i ents II ) o f To pology . A typical example n is given by the L _th d ivision po ints of abelian vari e t i es ( c f . cha p . I ,

1 . 2 ); the c o rresponding L-adic spac es , first

introduced by Weil [40]

are one o f our main too ls in the

s tudy o f these vari et i es . Even the

case o f d imension

presents non t rivia l problems; some o f them w i ll b e

1

stud i ed in cha p . IV . The general no t ion o f an L-adic representat ion was first d e f ined by Taniyama

[35] ( se e also the review o f

this paper given by Wei l i n Math . Rev . , 20,

1 959 , rev . 1 6 67 ) .

He showed how one can rela te L-adic represent a t ions re la t ive to di f ferent prime numbers

L

� the pro pert i es o f

the Frobenius e l ement s ( se e b elow ) . I n the same pa per , Taniyama a lso stud i ed some a b e l ian representa tions which a re c lo s ely rela ted to comp lex mult i plica t i on ( c f . We i l

[ 41 ] , [42 ] and Shimura-Taniya ma

[34)

. These abelian repre

sentations , to gether w i th s ome a pp li cat ions to ell i pt i c curves, a re t h e subjec t matter o f t h i s book .

There a re four Cha pters, who s e cont ents are as fo l lows :

xvii

I NTRODUCTI ON

xviii

Cha pt er

I

begins by giving the definition and s ome

examp l e s o f l-adic re pres ent a t ions ( §1 ) .

In §2 , the ground

fie ld is a s sumed to be ,a numb er field . Hence , Frob enius e l ements a re defined, a nd one ha s the no tion of a rationa l l-adic re pres enta t i o n

: one for which their chara c t e ri s

tic po lynomia l s hav e ra t io na l co efficien t s ( ins t ead of mere ly l-adic ones ) . Two repres enta t ions co rres ponding to different prime s a re compa t ib l e if t h e cha ra c t e ris tic po lynomia l s of their Frobenius e l ement s a re the s ame ( a t l ea s t a lmo s t everywhere )

; no t much i s known a bout this

no t ion in the non a b e l ian cas e ( c f . the l i s t o f o pen ques tions at the end of 2 . 3 ) . A la s t s ec t ion shows how one a t taches L- func t ions to ra t iona l l-ad ic repres enta t ions ; the wel l known c onnec tion b e tween equid i s tribution and ana lyt ic pro pert ies of L-func t ions i s dis cus s ed in the Appendix . Cha pter I I gives the cons truc t ion of s ome a b e l ia n l-adic repres enta t ions of a number fi e ld

K . As indi ca t ed

a bove , thi s cons t ruc t ion is es s entia l ly due to Sh imura , Taniyama and Weil . However,

I have found it conveni ent

to pres ent their results in a s l i ghtly d i fferent way , by de fining firs t s ome algebra ic groups over

Q ( the groups

Sm ) who s e repres enta tions - in the usual algebra ic s ens e co rres pond to the s ou ght for l-ad ic repre s entat ions o f

K.

The s ame groups had b e en cons id ered b e fore by Gro thend ieck in his s t i l l conjec tura l theo ry of " mo t ives " ( ind eed ,

mo t ives a re suppo s ed t o b e " l-adic cohomo logy without l " s o the connec tio n is not surpris ing ) . The construct ion o f a nd o f the l-adic repres entat ions a t ta Sm ched to them, i s given in §2 ( §1 conta ins some pre l i thes e groups

minary cons truc t ions o n a lgebraic groups , o f a ra ther

xix

I NTRODUCTI ON

elementary kind ) . I have a lso bri e fly ind ica t ed wha t

relations t h e s e groups have with complex mult i plica t ion

( c f . 2 . 8) . The la s t

� 's.

the

§

c o nt a ins s o me more pro pert i es o f

Cha pter I I I i s concerned with the following question : let fi e ld

p

be an a b elian l- ad ic representa t io n o f the number

K; can

The answer is

p

be o b t a ined by the method o f cha p . I I ?

: this i s so i f and only i f

a lgebra ic " in the sens e d e f ined in

§1 .

p

is " locally

In mos t applica

t i o ns , local algebra i c ity can be checked using a result o f Ta te s aying tha t it i s equivalent to th e existence of a " Hodge - Ta t e " d ec ompo s i t ion , at l east when the repre s entat ion is s emi-s imple . The proof o f this result of

Ta t e is ra ther long , a �d reli es heavily on his theorems on p-divi s ible groups

[39]; it is g iven in the Appendix .

One may als o a s k whether any abelian ra t ional semi-simp le l-ad ic repre s enta t i o n o f

K

is ipso facto locally alge

bra ic; this may well be so, but I can prove it o nly when K

is a compo s i t e o f quad ra t ic f ields; the proo f relies

on a t rans c end ency result o f S ie gel a nd Lang

( cf . §3).

Cha pter IV i s conc erned with the l-adic representation E . I t s a im is to d e t er d e f ined by an ellipt i c curve l mine, as prec i s ely as possible, the image of the Galo is

P

group by

Pl'

or at lea s t its Lie algebra . Here a ga i n

t h e ground f i eld is a s sumed to b e a numbe r f ield

( the

case of a func tion f i eld ha s been sett led by Igusa Most o f the results have b een sta t ed in a t bes t some sketches o f proo fs .

[10]).

[25] , [311 but with

I have g iven here comple

te proo fs , granted s ome ba sic facts on elliptic curves , which are collec ted in

§1 . The method fol lowed is more

INTRODUCTION

xx

" global " than the one indica t ed in

[ 2 5] . One starts

from the fact, noticed by Cassels and others, that the numb�r of isomorphism classes of elliptic curves isoge nous to

E

S afarevic 's

is finite; this is an easy consequence of theorem

( cf.1.4 )

on the finiteness of the

number of elliptic curves having good reduction outside

a given finite set of places. From this, one gets an

( cf.2.1 ) . Im ( pz. ) then

irreducibility theorem

The determination of

the Lie algebra of

follows, using the

properties of abelian representations given in chap. II, Ill; one has to know that

Pz.

'

if abelian, is locally

algebraic, but this is a consequence of the result of

Tate given in chap. III. The variation of 1 m ( P L) with L is dealt with in § 3. Similar results for the local case are given in the Appendix.

NOTATIONS

Gene r al notations Pos itive means Z (r e s p . Q ,

> O.

R, C)

i s the r ing (r e s p . the field) of inte g e r s (r e s p . o f r ational numb e r s , o f r e al numbe r s , o f c omplex numb e r s ) . If

is a pr ime numb e r , F denote s the pr ime field Z / p Z P and Z p (r e s p . Q p ) the r ing of p - adic inte g e r s (re s p. the field of p - adic r ational numbe r s ) . One has : p

Q

p

=

Z [!..] p p

P r im e numbe r s They are denote d by 1

for

"1

1, 1',

p , . . . ; we m o s tly us e the letter

- adic r e pr e s entations" and the letter p for the r e s idue

characte r i stic of s ome valuation .

Fields If K is a fie l d , we denote by K an alg ebr aic closur e of K, and by K

the s e parable c l o s ur e of K in K ; most of the fields we s c ons ide r are perfe c t , in which c a s e K = K . s If I,. / K i s a (po s s ibl y infinite ) Galois extens ion , we denote its Galo is g r oup by Gal (L / K} ; it is a pr oj e c tive limit of finite gr oups . xxi

NOTATIONS

xxii

Algebraic groups If

G i s an alg e b r aic g r oup ove r a fie ld K, and if K '

c ommutative K - algeb r a , we de note b y G (K') the g r oup of K' - p o ints of G (the "K ' -r ational" points of G) . When K'

is a is a

field, we denote by G the K ' - algeb r aic g r oup G X K K ' ob I K' taine d from G by extending the gr ound fie ld from K to K ' . Le t Y b e a finite dim ens ional K -vec tor spac e . W e denote by

Aut (V ) , or Aut (V ) , the gr oup of its K - line ar automorphisms , and K by GLy the c o r r e s ponding K - algebr aic gr oup (ef. chap. I, 2.4). F or any c ommutative K - alg ebr a K ' , the gr oup GL y (K ' ) of

K' -points of GL y is Aut , (V S K') ; for ins tanc e , K K GL y (K) = Aut (V ) .

•

Abelian l-Adic Representations and Elliptic Curves

CHAPTER I £-ADIC REPRESENTATIONS

§l. 1. 1.

THE NOTION OF AN 1 -ADIC RE PRESENTATION

Definition Let K

be

a

field , and let

K s be a s epa rable alg ebraic clo

Gal(K / K) be the Galo i s g roup of the exten s ion s K / K . The g ro up G, with the Krull top ology, is c ompa ct and totally s dis c onne c ted . Let 1 b e a pr ime numb er, and let V be a finite sur e of K . Let G

=

dimens ional v e c to r spac e ove r the field 01 o f 1-a dic numbe r s. The

full linear g r oup Aut( V) is an 1 - adic Lie g r oup , its top ology being induced by the natural topology of End(V ) ; Aut(V ) ::: GL(n, °1), DEFINITION -

An 1 -adic r ep r e s entation of

if

n

=

dim(V),

G ( o r,

we have

by abus e of

langua ge , of K ) i s a continuous hom o m o r p hism p: G

--?>

Aut(V ) .

Remarks 1)

A lattic e of V i s a sub - Z .t - mo dul e T which i s f r e e of

finite rank, and genera te

with T �

Z

1.

s table under

V

ov e r

01 ,

s o that V can be identified

01' Notice that th e r e exis t s a lattic e of V which is G.

T hi s follow s

from the fac t that

G

i s c ompact .

1-2

ABELIAN l-ADIC RE PRESENTATIONS

Indeed, l e t L be any lattic e of V , and l e t H be the s e t of elements g e: G such tha t p(g)L = L . T hi s is an open subgr oup of G , and G/ H is finite . The lattic e T g e ne rated by the latti c e s p( g ) L , g

i s s table under G .

E:

G/ H,

Notic e that L may b e ide ntified with th e p r oj e ctive l im it of m m the fr e e ( Z/ l Z ) -modul e s T / l T , on which G ac t s ; the vector space V may be r ec on s tructed from T by V = T � Z 01' 1 2 ) If p i s an l-adic r ep r e s entation of G, the g roup

p = Im( p) i s a c l o s e d s ubgr oup of Aut(V ) , and henc e , by the l-adic a nalogue of Cartan1s the or em ( cf . [28 ] , LG, p. 5 - 4 2 ) G i s it s elf an p l-adic Lie g r oup . Its Lie alg ebra .[ = Lie ( G ) i s a subalg ebra of p p End(V ) = Lie ( Aut(V». The Lie alg ebra g i s eas ily s e e n to b e in -p var iant unde r extens ions of finit e type of the gr ound field K G

( c f . [ 2 4 ], 1 . 2 ) . Exe r c i s e s 1 ) L e t V be a v e c tor spa c e of dimension 2 ove r a field k and l e t H be a subg roup of Aut(V ) . A s sume that det(l -h)

=

0 for

all he H . Show the exi stenc e of a ba s is of V with re spect to which H is c ontained eithe r in the s ubg roup

(� �)

(�

:)

or in the subg roup

of Aut( V ) . �

Aut(V ) be an l-adic r epre s e ntation of G, 1 whe r e V is a 0 -vector spa c e of dimen s ion 2. As s ume 1 l det( l - p ( s »:: 0 mod . 1 for all s E: G . Let T be a lattic e of V s table 1 by G. Show the exi s tence of a lattice T' of V with the following 1 two prop e r tie s . 2 ) Let p :

G

a ) T ' i s s tabl e by

G

b) Either T ' i s a s ublattic e of index 1 of T and trivially on

T/ T'

or

T

i s a sublattice of index

1

of

T'

G

acts

and G

1.

1-3

-ADIC REPRESENTATIONS

acts tr ivially on

T'/T.

(Apply exe r c ise 1 ) above to k = F and V = TILT.) 1. 3 ) Let p be a semi-s imple 1 - adi c r ep r e s enta tion of G and U

be an inv a r iant subg r oup of G . A s s ume that , for all x€" U , pix ) i s unipotent ( all its e ig envalue s a r e e qua l to 1 ). Sh ow that pix ) = 1 for aH x e: U . (Show that the r e s tri ction of p to U is

let

semi - s imple and use Kolchin' s the orem to b r ing it to triangular form. ) 4 ) Let

p:

and T a lattic e of

G

�

V

following p r op e rtie s :

1

Aut ( V ) be an 1 - adic rep re s entation of G, 1 s table under G . Show the equivalenc e of the

a ) The rep re s entation of G in the F - vec tor s pace T ILT i s i i r r e ducibl e . n b ) The only lattic e s of VL s table unde r G are the L T , with n e. Z . 1 . 2 . Example s 1 . Roots of unity . Let 1

#

cha r ( K) . The g roup m G = Gal(K I K) ac ts on the group IJ. of L -th roots of unity , and s m hence al s o on T V-t) = ?-l im . IJ. The Q . -ve ctor spac e m i .r; Q is of dimen s ion I , and the homomo rphi sm V (IJ. ) = T V-t) � Z 1. 1 L 1 X : G � Aut(V ) = Q/ define d by the action of G on V is a 1 i L I - dime nsional 1. -adic rep r e s entation of G . The character Xi take s •

its value s in the group of unit s U g(z)

=

z

x

l

(g}

i

of Z i by definition

if g

e.

G,

z

1

m

=

1 .

2 . Elliptic curve s . Let 1 # char(K} . Let E be an elliptic cur ve define d ove r K with a g iven rational p oint O. One know s that

AB E LIAN I. -ADIC REPR E S ENT A T IONS :=:'cre is a un iq u e s t r ucture of g r oup v a ri e ty on c:ement.

Let E

a:ld let

The T ate

a.:ts

.l:

(d.

m

be

the kernel of multiplication

module T (E) is I

[12],

c ha p .

E

VII).

a

wi t

by

h 0 I.

m

as n eutral

i n E(K ), s

fr e e 2 -module on which G = Gal(K / K) s I.

The corr e s p ond i ng homomo r p h i s m

G ---;. Aut(V (E)) is an l-adic r epresentation of G. The group I Gl = Im(1T ) is a closed subgroup of Aut(T (E)), a 4-dimensional I I. Lie group isomorphic to GL(2, 2 ). (In chapter IV, we will determine 1. :he L i e a lge bra of GI, und e r t h e a s s um p t ion t hat K. i s a number

::eld. )

Since we can identify E with its dual (in the sense of the duality of abelian varieties) the symbol (x, y) (d. [ 1 2], loco cit. ) defines canonical isomorphisms

Hence det{1T ) is the character X defined in example 1 . I I 3 . Abelian varieties. Let A be a n abelian variety over K

we define T,t< A), V {A) in the I same way as in example 2 . The group T,t< A) is a free 2 -module 1 of rank 2d (d. [12], loco cit. ) on which G = Gal(K /K) acts. s 4. Cohomology r epresentations. Let X be an algebraic of

dimension d.

If I I- char(K),

variety defined over the field K,

and let

Xs

=

X XK K s

be the

corresponding variety over K . Let I I- char{K), and let i be an s integer. Using the etale cohomology of Artin-Grothendieck [ 3 ] we let

l-ADI C REPRESENT ATIONS

1- 5

�

Q1 on which G = Gal(K s / K) s a c t s (via the ac tion of G on X ) It i s finite dimens ional, at lea st if s c ha r{K ) = 0 or if X i s pr ope r . W e thus g e t a n 1- adic r ep re s enta tion of G a s soc iated to H i (X ) ; by taking dual s we al s o get homology 1 s l -adic rep re s e nta tion s . Example s 1 , 2 , 3 a r e particular c a s e s of homol ogy 1-adic r ep r e s entations whe r e i = 1 and X i s r e sp e c tively The g roup H (X ) i s a ve c tor spac e ove r .

the multiplic ative group G , the ellip tic cu rv e E , and the abelian m variety A. Exe r c i s e

( a ) Show that the r e is a n elliptic c urve E , defined ov e r K = Q(T ) , with j - inva riant e qual to T . a

( b ) Show that for s uch a c urve , ov e r K = C ( T ) , one ha s GL = SL(T (E» (d. 19u s a [ 10] fo r an algebraic p r oof) . L ( c ) U sing (b) , s how tha t, ove r K ' we hav e G = GL( T /E » . o L

( d ) Show that fo r any c lo s ed subgr oup H of GL( 2 , Z L ) the re i s an elliptic curve ( defined ove r s ome field) for which G L = H. §2 . L -AD1C REPRESENTATIONS OF NUMB ER FIELDS 2 . 1 . Prelimina r ie s ( For the b a s i c notions conc e r ning numbe r field s , s e e for i n s tanc e Ca s s el s-F rtlhlich [ 6], Lang [13] or W e il [44] . ) Let K b e a numbe r f ield ( i . e . a finite extens ion of Q ) . Denote by L

the s e t K o f a l l finite pla c e s of K , i . e . , the s e t o f all normalized dis c r ete valuations of K ( o r, alte r natively, the s et of p r ime ideals in tbe r ing A K of integ e r s of K) . The r e s idue fie l d k of a pla c e v deg ( v ) v Eo L elements, whe r e is a finite field with Nv = p v K

A B E LIAN .l-ADIC R E P RESE N T ATIONS

1 -6

The ralTIiover F = char (k ) and deg{v) I S the degree of k v p v v v fication index e of v is v{p ). v v Let L/K be a finite Galois extension with Galois group G, p

of G consisting of those g e: G � . The subgroup D L w for which gw = w is the decolTIposition group of w. The restriction and let w

e:

�

� i by abuse K of language, we also say that v is the restriction of w to K, and

of w to K is an integral lTIultiple of an elelTIent v we wr ite wi v ("w divides v"). pletion of L (resp.

Let L (resp . K ) be the COITIv w K) with respect to w (resp. v). We have

w :: Gal(Lw /Kv ). The gr oup D w is lTIapped hOlTIolTIorphically onto the Galois group Gal(l /k ) of the corresponding residue extension w v .l /k . The kernel of G ---> Gal(.t /k ) is the inertia group I of w v w w v w. The quotient group D /I is a finite cyclic group generated by w w Nv the Frobenius elelTIent F ; we have F(x') = X, for all X, � 1 . w w The valuation w (resp. v) is called unralTIified if I = {l}. AllTIost w all places of K ar e unralTIified. D

If L is an arbitrary algebraic extension of Q , �

L

to be the pr ojective lilTIit of the sets �

over the finite sub-extensions of L/ Q .

L

'

a

one defines

where L

a

ranges

Then, if L/K is an arbi-

trary Galois extension of the nUlTIber field K,

and

w

E

� ' one deL If v is an unralTIified place of K,

fines D , I , F as before. w w w and w is a place of L extending v, we denote by F class of F

w

in G = Gal(L/K).

DEFINITION - Let p: Gal(K/K) tion of K, and let v

e

�

v

the conjugacy

Aut(V) be an .l-adic representa

� . We say that p is unralTIified at v if K p(I ) = {l} for any valuation w of K extending v. w If the representation

p

is unralTIified at v,

then the

1.

1-7

-ADIC REPRESENTA TIONS

fac t or s th r ou g h D /1 for a ny w j v; h e nc e w w w p ( F ) E; Au t ( V) i s d e f in e d; w e c a l l p( F ) the Frob e niu s of w in th e w w r epre s entation p, an d we denot e it by F T h e c onju gacy c l a s s w, p If in Au t( V ) d e p e n d s only on v; it is deno t e d by F of F w,p v, p L/ K i s the e x t e n s ion of K c orr e s p on d in g t o H Ker(p) , th en p r e s tric ti on of p

to

D

=

is unraITlifi e d a t v if a n d only if v i s unraITlified in L/ K.

2.2.

C e b otare v ' s d e n s i ty the or eITl

P

Let

L: . For each integer n, let a ( P ) K n P such that N v < n. If a is a real nUITlber,

be a subset of

b e the nUITlber o f o ne s ay s that P

v E

h a s density a if

a ( P) n a (L: ) n K

liITl.

=

a

when

n-->oo .

Note that a (L: ) -- n/log(n), by the priITle nUITlber theoreITl n K (d. Appen dix or [13 ], chap. Vm), so that the above relation ITlay be ,

rewritten:

a ( P) n

=

Ex aITlple s

a. n/log(n) + o(n/log(n» .

A finite set has density

O.

of degree 1 K The set of ordinary

The set of ve: L:

( i . e. such that Nv is priITle) has density

1.

priITle nUITlbers whose first digit (in the deciITlal systeITl, say) is 1 has no density. We can now state THEOREM K,

-

C ebotarev's density

theorem:

Let L be a finite Galois extension of the number field

with Gal ois group

G.

Let X

b e a s ub s e t of

G.

stabl e by

A B E LIAN £-A DI C REPR E S E N T ATIONS

1 -8

c onjugati o n .

Let P

X

b e the s et of place s

s uch that the F r obenius cla s s F

v den s ity equal to Card(X)/ Card(G).

Eo

v

�K'

i s cont aine d in

X.

unr amifie d in T he n P

x

L,

ha s

F o r the proof, see (7], [ 1 ], or the Ap p e n dix.

COROLLARY

1

-

For every g

ramified places w

e:

�

L

€.

G,

there exist infinitely many un

such that F

w

= g.

For infinite extensions, we have: COROLLARY 2

-

Let L be a Galois extension of K,

which is un

ramified outside a finite set S . a) The Frobenius elements of the unramified places of L are dense in Gal(L/ K). b) Let

X

be

a

subset of Gal(L/ K),

Assume that the boundary of Haar measure is 1.

J.l

of

X,

has measure zero with respect to the

and normalize

Then the set of places

equal to

X v

stable by conjugation.

J.l

such that its total mass

¢ S such that F

v

eX

has a density

J.l(X).

Assertion (b) follows from the theorem, by writing L as an increasing union of finite Galois extensions and passing to the limit (one may also use Prop. 1 of the Appendix).

Assertion (a) follows

from (b) applied to a suitable neighborhood of a given class of Gal(L/K). Exercise Let G be an .l-adic Lie group and let

X

be an analytic sub

set of G (i. e. a set defined by the vanishing of a family of analytic functions on G).

Show that the boundary of

X

has measure zero

1-9

/.-ADIC REPRESENTATIONS

wi th r e s p ect to th e Ha a r lTI e a su r e of

G.

2. 3 . Rati onal /.-a dic r ep r e s e nt a tion s L e t p b e a n l-a dic r ep r e s enta t i o n of t h e nUlTIb e r fi eld K. E:

2: , and if v i s u n r alTIifi e d wi th r e s p ec t to p , K P (T) de not e th e p olynolTIi a l d e t ( l - F T). v, p v, p

If v

we l e t

DEFINITION - The /.-adic r e p re s e ntati on p i s said to ·b e rational (resp. integral) if there exists a finite subset S of 2: such that K (a) Any elelTIent of 2: - S is unralTIified with respect to p. K (b) If v ¢ S , t h e coefficients of P (T) belong to Q v, p (resp. t o Z ) . RelTIark Let K ' /K be a finite extension.

An

l-adic representation p

of K defines (by restriction) an 1-adic representation p / K' of K ' . p

is rational (resp. integral) , then the salTIe is true for p / K ; ' this follows frolTI t h e fact that the Frobenius elelTIents relative to K ' If

are powers of those relative to K . ExalTIples The 1-adic representations of K given in exalTIples 1, 2, 3 of section 1. 2 are rational (even integral) representations.

1, one can take for S the set S

l

of elelTIents v of 2:

the corresponding Frobenius is Nv,

K

In exalTIple

with p

v

= 1;

viewed as an elelTIent of UJ."

In e xalTI pl e s 2 , 3 , one can take for S the union of S

and the set S 1 A where A has "bad reduction"; the fact that the corresponding Frobenius has an integral characteristic polynolTIial (which is inde pendent of 1) is a consequence of Weil's results on endolTIorphisms of abelian varieties (d. [4 0 ] and [1 2 ] , chap. VII) .

The rationality of

ABELIAN 1-ADIC REPRESENTATIONS

1-10

the cohomology representations is a well-known open question.

K,

and assume that

p'

be a prime.

DEFINITION - Let l'

p, p'

�

l' -adic representation of

ar e rational. Then

p, p'

are said to be

compatible if the re exists a finite subset S of � p'

K such that are un ramified outs ide of S -and P , ( T) -for ( T) = P v,p v ,p

p

and

(In othe r wor d s , the c haract e ri s tic polynomial s of the Frobenius element s are the s ame for p and p ' , at least for almost all v ' s . ) If p : Gal(K/ K)

�

Au t ( V ) i s a rational 1-adic representation

of K, then V ha s a c ompo s ition s e r ie s V = V :JV :J . . . :JV = 0 l o q

of p - inva riant sub s pac e s with V./V.

1+

1

1

(0 < i< q - l ) siznpl e

(i. e . i r r e ducibl e ) . The 1 - adic r ep r e s entation p ' of K defined by q -l V' = � V./V. 1 is semi - s iznpl e , rational, and c oznpatible with p ; . 1 1+ 1= 0

it i s the " s ezni - siznplification " of V. THEOREM - Let p be a rational 1 -adic r epre s entation of K, and let l ' be a prizne . Then the re exi s t s at znos t one (up to i s omorphi szn) l'

-adic rational rep r e s enta tion

c ompatible with

p.

p'

of K which i s s ezni - s iznple and

(Henc e the r e exi s t s a unique (up to is ozno rphis m) rational , s emi - siznple 1 - adic repre s entation c oznpatible with p . ) Pr oof . Let p � , P z be s emi - s iznple

l'

- adic rep r e s entations of K

l-ADIC REPRESENTATIONS

I -ll

which are rational and compatible with p .

H

We first prove that Tr(pi(g)) = Tr(pz,(g)) for all g E G. Let = G/(Ker(p p n Ker(p z )); the representations pi, Pz may be re

garded a s representations of H, Tr(p i (h)) = Tr(p Z (h)) for all h

H.

e:

a nd it suffices to show that

H. Let Me K be the fixed field of

Then by the compatibility of p i , P z the r e i s a fini te subset S of such th at for all v € :E S, WE: :EM' w I v, we have

:E K T r (p i (F )) w

=

K

-

T r(p Z (F ))· But, by cor. 2 to Cebotarev's theorem w

2 . 2 ) the F w are dense in H. Hence Tr(p i (h)) = Tr(p Z (h)) fo r all h £ H s inc e T r . p i , Tr .. Pz a re continuous. The theorem now follow s from the f oll owin g result applied to the group r ing /\ = QiH]. (d.

LEMMA - Let k be a field of characte ri s tic ze ro, let

1\

be a

k - algebra, and let P ' P be two finite -dirnens ional linear rep re I 2 s entations of 1\. g PI' P2 a r e semi - s impl e and have the same trace (T r .. P I = T r c P 2 ) , the n they are i s omo rphic . . F o r the proof s e e B ourbaki, Alg . , ch. 8, §1 2 , no I, prop. 3. DEFINITION - For each prim e 1 let P1 be a rational l - adic rep re s entation of K. The sy s te m ( Pl ) i s said t o be c ompatible if P l , Pl ' are compatible for any two prime s 1 , 1 ' . The sys tem ( p ) i s said 1 to be strictly c ompatible if the r e exists a finite subset S of L K

such that:

= {vi Pv = d. Th e n , for every viaS u Sl' Pl is (T) ha s rational coeffic ients . unramifie d at v and P v, Pl (b) P ( T ) = Pv l (T ) g v , S u Sl u 51' . v, P , P ' 1 (a) Let

S 1

1-12

ABELIAN l -ADIC RE PRESENTATIONS

Whe n a s y s te ITl ( P ) IS s t r i c tly cOITlpatible , the r e i s a s mall I e st finite s et S having p rope rti e s ( a ) and (b) above . We call it the exc eptional s et of the systeITl. ExaITlple s The s y s teITl s of i -adic repr e s entations g iven in exaITlple s 1, 2, 3 of s e ction 1. 2 are each s t ri c tly cOITlpatible . The exc eptional s e t o f the fir s t one is eITlpty . The exc eptional s e t of exaITlple 2 ( r e sp . 3 ) i s the s e t of plac e s whe r e the elliptic c urve ( re sp . the abelian variety ) ha s "bad r educ tion ", cf . [3 2]. Que stions 1. Let P

P

be a rational i-adic rep r e s entation. Is it true that

v, p ha s rational c o efficients f o r all v such that at v?

P

i s unraITlified

A sOITlewhat s iITlila r que stion i s :

I s any c OITlpatibl e sy steITl strictly c OITlpatible? 2. Can any rational i-adic r ep re s entation be obtained (by

ten sor p r oduc t s , dir ect SUITlS, etc . ) froITl one s c OITling froITl i-adic c ohoITlology ? 3 . Given a rationa l i-adic r ep r e s entati on p o f K,

and a

p riITle i' , doe s the r e exist a rational i' -adic r ep r e s e ntation p ' of K

c OITlpatible with p? .... [ n o : easy co u nter-exam ples]. 4 . Let p, p ' be rational i, i' -adic r ep re s entation s of K

which are c OITlpatible and s eITli - s iITlpl e . ( i ) If p is abelian ( i . e . ,

if

IITl( p ) i s abelian) , i s it true th_a t

p ' is abelian? ( W e shall s ee in c hapte r III that thi s is true at lea st if

p is "loc ally algebraic " . )

....

[yes: th is follows fro m [63].]

( ii ) I s it true that IITl( p ) and IITl( p ' ) are Lie g r oup s of the

1-13

J. - ADIC REPRESENT ATIONS

More optimistically, is it true that there exists a

same dimension?

Li e alg eb r a Li e

(I m (p l » 5.

K,

�

� over

Q such that

Lie (Im ( p »

?

= � 8 Q ' Q J.

= .& �

Q

Q ' J.

Let X be a non - s ingula r p r o je c tive va riety defined over

and let i be an inte ge r . Is the i -th c ohomology rep r e sentation

H (X ) semi-simple? Does its Lie algebra contain the homotheties s if i > l? (When i = 1, an affi rmative an swe r to eithe r one of the s e

que stions woul d imply a p o s itive solution f o r the "c ong ruenc e sub g roup pr oblem" on abelian va rietie s ,

d.

[24], §3 . )

-.

[yes fo r i=l:

see [48] an d also [75].] Remark The concept of a n l-adic representation can be generalized by replacing the prime

1

by a place }.. of

a

E.

number field

}..-adic representation is then a continuous homomorphism Gal(K /K} s

�

A

where V is a finite-dimensional vector

Aut(V},

spa ce over the local field E}...

The concepts of ra tional k-adic

representation, comp atible representations, etc., can be defined in a way similar to the 1-adic case.

Exe rc i s e s 1) Let

p

and p I be two rational, s emi - s imple , c ompatible

r ep re s entations . Show that, if Im( p ) is finite , the same is true for Im( p l ) and that Ke r ( p)

=

K e r ( p ' ) . (Apply exe r . 3 of 1 . 1 to p ' and

to U = Ke r ( p) . ) Gene ralize this to },, - adic rep r e s e ntations (with r e spect to a numbe r field E). 2) Let p ( re sp . pI) be a rational J. -adic ( r e s p . representation of K, patible.

If

s

E.

G

=

of degree n.

Assume

p

and

pI

l'

-adic )

a re c om -

Gal(K/K), let a.(s) (resp. a !(s» be the 1

1

1-14

ABELIAN J.-ADIC REPRESENTATIONS

i-th coefficient of the characteristic polynomial of p(s) (resp. of p'(s». Let P(Xo , ... , Xn ) be a polynomial with rational coefficients, and let Xp (resp. Xp) be the set of s e: G such that P(a o (s ) , . . ,an (s» = 0 (resp. P(a'o (s), . .. ,a'n (5» = 0). a) Show that the boundaries of Xp and Xp have measure zero for the Haar measure J.l of G (use Exer. of 2. 2). b) Assume that J.l is normalized, i. e. J.l(G) = 1. Let TP be the set of v e 1:K at which p is un ramified, and for which the coefficients ao, .. . , an of the characteristic polynomial of Fv, p satisfy the equation P (ao, , an) = O. Show that Tp has density equal to J.l(Xp), c) Show that J.l(Xp) = J.l(Xp). .

•

.

•

Representations with values in a linear algebraic group Let H be a linear algebraic group defined over a field k. If k ' is a commutative k-algebra, let H(k') denote the group of points of H with values in k'. Let A denote the coordinate ring (or "affine ring") of H. An element f A is said to be central if f(xy) = f(yx) for any x, y E: H(k') and any commutative k-algebra k' . If x e: H(k '), we say that the conjugacy class of x in H is rational over k if f(x) e: k for any central element f of A . 2 . 4.

4i:.

DEFINITION - Let H be a linear algebraic group over Q, and let K be a field. A continuous homomorphism p: Gal(Ksl K) --:;:. H(Ql) is called an J.-adic representation of K with values in H. (Note that H(QJ.) is, in a natural way, a topological group and even an J.-adic Lie group. ) If K is a number field, one defines in an obvious way what it

1-15

I.-A DIC R E P R E S E NT A T IONS

m eans for p

VE

t o b e unra mifi e d a t a plac e

fin es th e Fr obe nius e l em en t F

E:

F

is rati o na l

v,p

We say, as b e for e,

(a )

i s a finite s e t S

th ere

ou tside S,

(b ) if

v

w, p that p

¢

H(Q)

of �

I.

K

�

K

;

if wlv , one de

an d it s conjugacy c la s s if

such that

i s unramified

i s rational ove r Q. Tw o r a ti onal r e p r e s en t a t ions p, p ' (f or p r ime s 1 , 1' ) a r e said to be compatible if there exists a finite subset S of � such that p S,

the c onjugacy cla s s F

p

v,p

K

a re unramifie d out s ide S and such that fo r any c entral ele ment f E A and any v E � - S we have f( F ) = f(F v , p ,) . One K v,p define s in the same way the notions of c ompatibl e and stric tl y

an d p '

c ompatible s ys tem s o f rational rep r e s e ntation s . Re ma r k s

1 . If t he algebraic g roup H is abelian, then condition (b) above means that F (whi c h i s now an element of H(O » is l v,p rational ove r 0, i. e . b elong s to H ( O ) . 2.

L et

V

o b e a finite -dimens ional vec tor spa ce ov e r 0, and

be the linear algebraic group ove r Q who s e g roup of o point s in any c ommutative Q - alg eb ra k is Aut( V 0 3Q k); in parti c ula r , if V1 = V 0 � 01 ' the n GL ( 01 ) = Aut( V )· If O V 1 o 1/>: H�GL V i s a homomo rphi sm of linea r algebraic g r oup s ov er let GL

0,

V

call

1/>1.

o

th e induc ed homomorphi sm o f H(O ) into l

(01) = Aut(V )· If p is an 1-adic representation of Gal(K/K) 1 o into H(Q ) ' one g et s by c ompo sition a linea r 1 - adic r ep r e s entation L q, c p : Gal(K /K) � Aut( V 1 ) . U sing the fa ct that the coeffi cient s of s I. th e c h a r acte r isti c p olynomial are c e nt ral func tion s , one s e e s that GL

V

AB E LIAN L -ADIC REPRESENT A TIONS

1-16 4>L

is rational if p rational (K a number field). Of course, compatible representations in H give compatible linear representa tions. We will use this method of constructing compatible repre sentations in the case where H is abelian (see ch. II, 2 . 5 ) . 0

P

1S

L-functions attached to rational representations Let K be a number field and let P = (P ) be a strictly com L patible system of rational l-adic representations, with exceptional set S. If v ¢ S, denote by Pv, ( T ) the rational . polynomial P det(l - Fv, P T ) , for any L � P ; by assumption, this polynomial v 1 does not depend on the choice of L. Let s be a complex number. One has: 2. 5.

P

-s v, p (Nv) ) = det(l = IT ( 1 i

>--.

1,

v

I (Nv)

s),

where the >--. V I S are the eigenvalues of Fv, (note that the >--i, v I S P are algebraic numbers and hence may be identified with complex numbers). Put: 1,

1 L (s) = n P v ,. s Pv, p ((Nv)-s) co

� an Ins , with coefficients in Q . n=l In all known cases, there exists a constant k such that I >--. v I -< (Nv) k, and this implies that L is convergent in some half plane R(s) > C ; P one conjectures it extends to a meromorphic function in the whole This is a formal Dirichlet series

1,

1 -1 7

l -ADIC REPRESENTATIONS

pla n e .

When

P

C OITl e s f r o m l -a d i c c ohoITl o l og y , the r e a r e s o me

fu r the r c o nj e c tu r e s o n th e z e r o s and p o l e s o f the s e , a s indic a t e d by Tat e , ITlay

be appl i e d

p r o p e r ti e s of th e F r ob e n iu s e l e ITl ent s ,

d.

to

L , P

d. T a te

[36 ] ;

g e t equidi st r ibut ion

Appendix .

R e ITla r k s

1 ) One can al s o a s soc iate L -func tions to E - rational sy s te ITl s

o f A - a dic r ep re s entations ( 2. 3 , Rema r k) , whe r e E i s a numbe r field, onc e a n eITlb e dding o f E into C ha s been cho s en . 2) W e have g iven a definition of th e local fa cto r s of L

only

P

at the pla c e s v ¢ S . One c an giv e a ITl o r e s ophi sticated definition in which local factor s a r e defined fo r all plac e s , even (with suitable hypoth e s e s ) for p r ime s at infinity ( gaITlma fa c tor s ) ; thi s i s nec e s s a ry when one wants to study func tional e quation s . W e don ' t g o into thi s he r e .

�

[ se e [ 5 1 ] . [ 7 4 ] . ]

3 ) Let cP ( s ) =

� a n / n s b e a Di richlet s e rie s . . U s ing the

theo rem in 2 . 3 , one s e e s that the re is (up to is omorphism) at mo s t one s e mi - s imple sy stem P = ( PI ) ove r Q such that Lp = cP o Whethe r the r e doe s exi s t one ( fo r a given t/l ) i s often a quite in t e r e sting qu e stion . F or in stanc e , i s it so for RaITlanujan ' s 00 s cP ( s ) = � T ( n) / n , whe r e T ( n) i s define d by the identity n= 1 00 n n 4 00 x n ( 1 - x / = � T ( n) x ? n= l n= l The re is cons ide rable nume rical evidence fo r thi s , ba s e d on the c on g ruenc e p r op e r tie s of

T

(Swinne rton - Dye r , unpublished) ; of c our s e ,

such a P would b e o f dime ns ion 2, and i t s exc eptional s e t J be empty . � [ p r o v e d b y D elign e : s e e [ 4 9 ] . [ 5 0 ] . [ 6 5 ] •

S

woul d

. . .

M o r e g e ne rally, the r e s e e m s to b e a cl o s e c onne c tion between

AB ELIAN

1 -18

mo dula r fo r m s , . s

u c

h as

L:

7 ( n}

x

n

,

£

- A DI C R E P R E S E N T A TIONS

and r a t i onal ( o r a l g eb r a ic )

£ - a dic r e p r e s e nta t i on s ; s e e fo r in s ta nc e Shimu r a [ 3 3 ] and W e il [4 5 ] . �

[ see also [ 4 9 ) , [ 5 1 ) , [ 6 5 ) , [ 6 6 ) , [ 6 8 ) , [ 8 4 ) . )

Example s Ii

is an Artin ( non abelian) L - s e rie s , at lea st up to a finite number of fa c to r s 1.

( d . [1]) .

G acts thr ough a finite g roup, L

Al l

c our s e one

p

A rt i n L - s e ries are g otten i n this way, provided of

use s E - rational repre s e ntati o n s ( d . Remark 1 ) a nd

not

m erely ratio n al o n es . If

p

i s the system a s s o c iated with an elliptic curve E ( d . 1 . 2 ) . the c o r r e spond in g L -func tion give s the non -trivial part of the zeta function of E . The symmetric powe r s of p g ive the zeta functions of the products E X X E, d . Tate [ 3 6 ] . 2.

•

.

.

A PPENDIX

Equipa rtition and L -func tions A. 1.

Equipa rtition Let

X b e a c ompa c t t o po l o g i c a l s p ac e a nd

C(X)

th e B ana c h

space of c ontinuou s , compl ex -valued, functions on X, with its u sual norm II f ll = Sup I f(x) l . For each x X let 0 x be the Dirac X E: X mea sure as s oc iated to x; if f € C ( X ) , we have 15 x (f) = f(x) . L e t (x ) n n> l b e a s e q e nc e of p o int s of X . F o r n > I , l e t e:

u

J-l n = ( 15

Xl

+

•

•

•

+ 0

xn l / n

I -19

l-ADI C REPRESENTATIONS a nd l e t iJ. on

C(X),

b e a R a d o n m ea s u r e o n X

d. Bourbak i, Int. , chap.

( i . e . a c o ntinuo u s l in e a r f o r m

III,

The

§l).

s eq ue nce

(x ) n

is

to be iJ. - e quidistributed, or iJ. - unif o rm l y distributed, if iJ. n � iJ. weak ly a s n -..;> i . e . if iJ. (f) --..;;. iJ. (f) a s n � co fo r any n f E: C ( X ) . Note that this implies that iJ. is positiv e and of total mas s said

co ,

1.

N o t e al s o that iJ. (f) -..;> iJ. ( f )

n

iJ. (f) =

m e a n s that

n 1 lim 1:: f(x . ) n-»co n i= 1 1

LEMMA 1 - Let (c/J ) be a family of continuous functions on X with the property that their linear c ombinati ons a r e dense in C ( X ) . Sup pas e that, for all the s equenc e (p n (c/J » n>l ha s a limit . Then the sequenc e (xn ) is equidi stributed with re spect to some measure iJ. ; it is the unique measure such that iJ. (c/J ) lim iJ. (c/J ) for all n-»co n --

a

a,

a

a

If

f

e

=

a

a.

a n argument us ing equic ontinuity shows that the sequence (iJ. n (f » has a limit iJ. (f ) , which is continuous and linear in f ; hence the lemma . C (X ) ,

PROPOSITION 1 - Suppo se that (xn ) is iJ. -equidistributed. Let U be a subs et of X who se boundary ha s iJ. -mea sure ze ro, and, for all n, let n be the numbe r of m < n such that ·x e U . Then U lim (nU l n) iJ. ( U ) . n-»co Let U O be the inte r ior of U . W e have iJ. ( U o ) = iJ. ( U) . Let o E > O . B y the definition o f iJ. (U ) the r e i s a continuous function IjJ E C (X) , 0 � 1jJ � l, wi th c/J = 0 on X - U O and iJ. ( c/J ) � y ( U) - E . < n I n we have Sinc e iJ. n ( cf» U =

m

A B E LIAN I. -ADI C R E PRESENT ATIONS

I - 20 lim

inf n / n :::' I b n J.l. n ( cP) n�co U n�co

=

J.I. ( cP ) :::' J.I. ( U ) -

£ ,

f r om which we obtain lim inf nU l n :::' J.l (U) . The same a r gument applied to X - U shows that lim inf ( n - nU ) I n :::, J.I. (X

-

U) .

Henc e lim sup n U l n < J.I. (U ) < lim inf n I n, which implies the propo U s ition . Example s 1.

Let X [ 0 . 1 ] . and let J.l be the Lebe sgue mea sur e . A s equenc e (xn ) of point s of X is J.I. -equidi stributed if and only if for each inte rval [a, b], of length d > ° in [ 0 , 1] the numbe r of m < n such that xm £ [a , b] is equivalent to dn as n ----> co . 2 . Let G be a c ompact g roup and le t X b e the spac e of c onjugacy c las s e s of G (i. e . the quotient spac e of G by the equi valenc e relation induc ed by inner automorphi sm s of G ) . Let J.I. be a mea sur e on G; its imag e of G ----> X i s a measur e on X , which we al s o denote by J.I. . W e the n have =

PROPOSITION 2 - The s equenc e (x ) of elements of X is n

J.I. -equidi s tribute d if and only if for any irr educ ible characte r X of G we have 1

n

lim r; X (x . ) n�co n i= l 1

=

J.I. (X ) .

T he map C(X) � C( G ) i s an i s omo rphi sm of C(X) onto the space of c entral func tions on G; by the Pete r -Weyl theorem, the

1 -21

.l - ADIC R E PR E S E NT A T IONS

i r r e ducible cha ra c te r s X of G generate a den s e sub spac e of Henc e the p r opo sition follows f r ozn leznzna 1.

C (X) .

- Let I-' be the Haa r zn ea sur e of G � I-' ( G) = l . Then a s e quenc e ( xn ) o f eleznent s o f X is I-' - e quidi stributed if and only if for any i r r e duc ible c ha racte r X of G , X f:. 1, we have COROLLAR Y 1

n 1 lizn !: X (x. ) n�CX) n i= l 1

=

0

This follow s frozn P rop . 2 and the following fact s : I-' ( X ) = 0 if X is i r r e duc ible f:. 1 I-' ( 1) = 1 . COROLLAR Y 2 - (H. Weyl [46 ]) Let G = R/ Z , and let I-' be the norznalize d Haar znea sure on G. -Then (xn ) i s I-' -equidistributed if and only if for any integ e r zn f:. 0 we have !:

n
e

2 7Tznixn

= o(N)

(N --->

(0)

•

For the proof, it suffic e s to rezna rk that the irreduc ibl e . s x ........-> e 27Tznix (zn e; Z ) . characte r s of R/ Z are the znappmg A. 2 . The c onnec tion with L -functions Let G and X be as in Exaznple 2 abov e : G a c oznpac t group and X the spa c e of its c onjugacy clas s e s . L et x v , v Ei !: , b e a family of elements of X , indexed by a denume rable s et !: , and let v .,......:;> Nv be a function on !: with values in the set of intege r s � 2 .

1-22

AB ELIA N £ - ADI C R E PRES ENTATIONS

W e make the following hyp o the s e s : ( 1) T he infinite p r oduc t

1

n

c onv e r g e s fo r every -s v E. � 1 - ( Nv ) and ext e nd s to a m e r omo rphic func tion on S EO C with R( s ) > I , R ( s ) :::' l having ne ithe r z e r o no r pole exc ept for a s imple p ole at s = 1.

Let p be an i r r e duc ibl e repr e sentati on of G, with cha ra c te r X , and put (2)

L( s , p) =

n

1

v € � de t(l - p ( xv ) ( Nv)

Then thi s pr oduct conve rge s fo r R( s )

> I,

-s

)

and extend s to a me ro

mo rphic func tion on R( s ) :::' l having ne ither z e r o nor pole exc ept p o s s ibly fo r s = 1 .

The o rder of L( s , p ) a t s = 1 will b e denoted by -c Hence, X if L( s , p ) ha s a pole ( r e sp . a z e r o ) of orde r m at s = I , one ha s •

c

X

= rn. ( re sp .

c

X

= - m) .

U nde r the s e a s surn.ptions , we have :

THEOREM 1 - (a) The numbe r of v to n/log n (� n � (0 ) .

e:

� with Nv < n i s e quival ent

( b ) F o r any i r r e duc ibl e c haracter X of G, w e have

�

Nv
X

(x ) = c n/l og n + o ( n/log n) v X

(n

�

The the orem r e s ult s , by a standar d argume nt, f r om the the o r e m of Wiene r -Ikehara,

d.

A. 3 below .

Supp o s e now that the func tion v p r ope r ty :

�

Nv ha s the following

(0).

1- 2 3

I. - ADIC REPRESENTATIONS ( 3 ) The re exi s t s a c onstant C such that, for eve ry n the numbe r o f v e L with Nv = n i s < C .

One may th en ar range the element s o f

1:

£

Z,

a s a s e quence

(v i ) > l so that i :: j implie s Nv i :: NVj ( in g e ne ral, thi s is po s s ibl e i in ma ny way s ) . It the n make s s e ns e to speak about the equidi s tribu tion of the s e quenc e of xv ' s ; u sing ( 3 ) , one show s easily that this doe s not depend on the cho s en ordering of 1: . Applying the o r e m 1

and propo s ition 2 , we obtain

THEOREM 2 - T he element s xv (v &: L ) are equidistributed in X with r e spect to a mea sure I-' such that for any irreduc ible c harac te r X of G we have I-' ( X ) = c X

COROLLAR Y - The elements xv (v &: L) a r e equidis tribute d for the normalize d Haa r mea sur e of G if and only if c = 0 fo r every X i r r e duc ible c haracte r X -F. 1 of G, i . e . , if and only if the L - functions relativ e to the non trivial i r r e du c ible character s of G are holomo rphic and non z e r o at s = 1 . Example s

1 . Let G be the Galoi s g r oup of a finite Galoi s extens ion L/ K of the numbe r field K, let of K, let x

L

be the s et of unramified plac e s

be the F r obeniu s c onjugacy clas s defined by v v and let Nv be the no rm of v , cf . 2 . 1 .

�

L,

( 2 ) , ( 3 ) ar e sati sfie d with c = 0 for all X irr e ducible X -F. 1 . This i s trivial for ( 3 ) . F o r (1), one r emarks that Propertie s

(I ) ,

L ( s , l ) i s the zeta function of K (up to a finite numbe r of te rm s ) , henc e ha s a s impl e pole at s = 1 and i s h ol omo rphic on the re st of

1 -24

ABELIAN l -ADIC REPRESENTATI ONS

the line R( s ) = 1, cf . fo r ins tanc e Lang [ 1 3 ], chap . V II; fo r a pr oof of ( 2 ) , cf. Artin [ 1 ] , p . 121. Henc e theorem 2 give s the equidistribu . tion of the Frobenius element s , 1. . e . the C ebota r ev dens 1ty theor em, cf. 2 . 2 . 2 . Let C be the idMe c las s g roup of a numb e r field K , and let p be a c ontinuous homomo rphi sm of C into a c ompact abelian Lie gr oup G. An easy a rgument (cf. ch. III, 2 . 2 ) shows that p is almos t eve rywhe re unramifie d ( i . e . , if U v denot e s the g r oup of units at v , then p(U v ) = 1 fo r almost all v ) . Choo s e 1f v € K with v(1f v ) = 1. If p is unramified at v , then p ( 1f v ) depends only on v , and we s e t xv = A 1fv ) . W e make the following a s s umption: O ( * ) The homomo rphism p map s the g roup C of id�les of volume 1 onto G. (Recall that the volume of an id�le a = (av ) is defined as the produc t of the normaliz e d absolute values of its c omponent s a v , cf. Lang [13J o r Wei! [44 J. ) Then, the elements x v are uniformly di stributed in G with r e spec t to the normaliz ed Haar mea su r e . This follow s from theorem 1 and the fact that the L -func tion s r elative to the irreduc ible characte r s X of G a r e Hecke L -func tions with Grtf s sencharacte r s ; the se L -functions are holomo rphic and non -zero for R( s ) :: 1 if X #. 1 , s e e [13 J, c hap . VII . '"

Remark This example ( e s s entially due to Hecke ) is given in Lang (loc . c it. , ch. VIII, §5 ) exc ept that Lang ha s replaced the c ondition ( * ) by the c ondition "p is surj e c tiv e " , which is insuffic ient. This led him to affirm that, for example , the s equenc e (log p) (and also the s e que nc e

( l o g n» i s unif o r m.1y distr ibute d m o dul o 1 ; how ev e r ,

1 -2 5

l -ADIC REPRESENTATIONS

one knows that thi s sequenc e i s not uniformly di stributed for any mea sure on R/ Z (d. P olya -Sze gtl [ 2 2 ] , p . 1 7 9 -18 0 ) . 3 . ( Conj e c tural example ) . Let E b e an elliptic curve defined ove r a numbe r field K and let L be the s e t of finite plac e s v of K suc h that E ha s g ood reduction at v, d . 1 . 2 and chap . I V . Let v € L , let 1 F. pv and let Fv be the Frobenius conjugacy cla s s of v in Aut( T1 (E» . The eigenvalue s of Fv are algebraic numbe r s ; when embedded into C they give c onjugate c omplex numbe r s / 7rv ;v with 1 7r v I = Nvl 2 . We may write then ,

7r = ( Nv ) l / 2 e V

-i4J v

with

0

< 7r - 4J v -

<

On the othe r hand, let G = SU( 2 ) be the Lie group of 2 X 2 unitary matric e s with dete rminant 1 . Any element of the spac e X of c onjugacy clas s e s of G contains a unique matrix of the form i4J 0 (e "IP," 0 � 4J � The ima g e in X of the Haar measure of G � 0 e -l� 2 s in 2 4J d4J. The irreduc ible r ep r e s entations of G i s known to be 7r .

7r

a r e the m -th symmetric powe r s P m of the natural repre sentation P I of deg ree 2 . T ake now for xv the element of X corre sponding to the angle 4J = q, v defined abov e . The corre sponding L function, rela tive to P m is: ,

L If we put:

a= m 1 (s) = n n Pm v a= 0 1 - e i(m - 2a)q, v (Nv) - s

ABELIAN l -ADIC REPRESENTATIONS

1 -2 6

a= m = n n v a= O

1

w e hav e = L 1m ( s - m/ 2 ) 1 ha s b e en conside r ed by Tate ( 3 6 1 . He c onj e c ture s The func tion L m 1 for m > 1 . i s ho1omorphic and non z e r o for R( s ) '::' 1 + m / 2. that L m p rovided that E has no complex multiplication. Granting this conjectur e . the c o rollary to theor em 2 would yield the uniform dis tribu tion of the xvI s , or, e quivalently, that the angle s q, v of the Frobenius el em ents a r e uniformly distributed in [ 0 . 11' 1 with re sp e ct to the measure � s in 2 q, dq, ( l lc onje ctur e of Sato - Tate ") . One can expect analogous re sults to be true for othe r l -adic rep re s e ntations . •

11'

A. 3 . Proof of the o r e m 1 The logarithmic de rivative of L i s m )log(Nv) X (x v L ' / L = - 2:: v , m>l whe r e xm v i s the c onjugacy c la s s c onsisting of the m -th pow e r s of eleme nts in the cla s s xv One see s this by wr iting L a s the pr oduct •

n

i, v 1

_

1 ( i) Av (Nv) - s

J. -ADI C REPRE S E N TA T IONS

1 -2 7

(i) a r e th e e ig e nv al u e s o f v Now th e s e r i e s

wh e r e th e

:\

L:

v , m> 2 c onv e r g e s f o r

R( s ) >

1/ 2 .

I

v

in th e g i v e n r e p r e s e nt a t i o n .

lo g ( Nv ) ms (Nv) I

i t s uffi c e s t o show that

Ind e e d,

� ... ..., --'o:...:.--=-

l o g ( Nv )

v ( Nv)

if

x

a

< 00

a > 1; but thi s s e r i e s i s majo r i z e d by ( C onstant) X

1

1;

(£ > 0)

a+ £

--

v (Nv)

•

On the othe r hand, the c onve r g enc e for a > 1 of the pr oduct

n v

l

____

1 - ( Nv ) -

a

shows that

L: v

1

a <

---

( Nv )

00

f o r a > 1 ; henc e o u r a s s ertion. One c an the refo re w r ite

L ' / L = - L: v

x (x v )log ( Nv ) s

-----

( Nv)

+ 9>( s ) ,

1 whe r e 9> ( s ) is ho1 omorphic fo r R ( s ) > 2 . Mor e ov e r , by hypothe si s ,

AB E L IAN P. -ADI C R E PR E S E N T A TIONS

1-28

L'/

L

can be extended to a me r OITlO rphic func tion on R ( s) :::. 1 which is holomo rphic exc ept p o s s ibly for a s impl e pole at s = 1 with r e s i due - c . One may th en apply the Wiene r -Ikeha ra the o r em (d.

X

[ 1 3 ], p . 1 2 3 ):

S be a Di richl et s e rie s with c omplex n + s c oeffic ie nts . Supp o s e the r e exi s t s a Dirichl e t s er i e s F + ( s ) = � a n / n

T HEOREM - -Let F(s)

=

�a /n

with p o s itiv e r eal c o effic ients suc h that + ( a) I a I n < a n for all n; (b) T he s e rie s F + c onv e rg e s for R ( s ) > 1 ; + ( c ) The function F ( r e sp . F ) can be extende d t o a me r o

mo rphic func tion o n R( s ) :::' l having no pole s exc ept ( r e sp . exc ept p o s s ibly) fo r a s imple pole a t 5 = 1 with r esidue c+ > 0 ( r esp . c ) . T hen

a (whe r e c

=

n

=

(n

cn + o(n)

0 if F i s holomo rphic at s

=

�

(0 ) ,

1).

One applie s this the o r e m to F( s ) and we take fo r F

+

=

_�

v

X (xv ) log (Nv)

_____ _

(Nv)

s

the s e r ie s d

�

log (Nv) s , ( Nv)

whe r e d i s the deg r e e of the giv en rep r e s entation

p;

this i s pos sible

J. -ADIC R E PR E S E N T A T IONS

1 - 29

X (x ) i s a s um of d c o mp l e x numb e r s of a b s o l u t e v a l u e v < d ; mo r e ov e r , t h e s e r i e s henc e X (x ) v

1,

s in c e

I

I

�

log ( Nv)

v ( Nv)

s

diff e r s f r om t h e l o ga r ithmic d e r ivative of

n

l

(Nv ) - s

____

1

_

by a function which i s holomo rphic for R( s ) > 1 / 2 a s we saw above . Hence by the W iene r -Ikeha ra the o rem we have � X (xv )log( Nv) = c n X Nv
+

(n

o (n)

--...;. co ) .

Con sequently, by the Abel summation trick ( d . [13 ] , p . 124 , p rop . I ) , � X (xv ) Nv
=

c n/ log n + o ( n/ log n) X

(n

--...;.

(n

--...;. co ) .

co ) ,

and in pa rticula r ,

� 1 = n/ l og n + o(nl log n)

Nv
( � X (x » / ( � 1) v Nv
we may

apply p r op o s it ion

2

to

--...;.

c

X

c o nc lude the

as

p ro of .

n

--...;.

co ,

q. e.

d.

CHAPTER II TH E GROUPS

�

Throughout this chapte r , K denote s an algebraic number field. We a s s oc iate to K a pr oj e c t ive family (S ) of c ommutative alge m braic gr oups over Q , and we show that each Sm g ive s rise to a s tr ictly compatible system of r ational 1 - adic repre s entations of K . In the next chapter , we shall s e e that all " locally algebraic " ab elian rational repr e s entations are of the form de s cribed here . § 1.

PRELIMINARIES

1. 1 .

The torus T Le t T R (G ) b e the algebraic gr oup ove r 0 , ob K / Q m/ K taine d from the multiplicative g r oup Gm by r e s triction of s calar s fr om K to Q , d . Wei! [43 ] , § 1 . 3 . If A is a c ommutative Q algebra , the points of T with value s in A form by definition the * multiplicative gr oup (K QiO Q A) of invertible elements of K � Q A. * In particular , T (a) K [K: oJ , the group T is a torus of If d dimens ion d j this means that the gr oup T / = T Xo Q obtained Q fr om T by extending the s c alar s from 0 to Q , is is omorphic =

=

=

ABELIAN 1 -ADIC REPR ESENTATIONS

II - 2 to G l X m Q

•

.

.

X G rn

Mor e pr e c is ely, let

l Q (d time s ) .

s e t of emb e dding s of K into Q ; each a E r extends phis m K � 0 � 0 , hen c e defin e s a morphism [a] : 0 The c ollec tion of all [a] 1 s g ive s the is omo rphism T I Q � Gm l Q X X G l . Mo r e ov e r , the [a] 1 s m Q o f the char ac ter g roup X (T ) = HomQ (T I Q ' Gm l Q ) of •

•

•

r

b e the

to a homomor

T I Q � Gm l Q . form a bas i s T. Note that

th e Gal o i s g roup Gal (Q I 0 ) acts in a natur al way on X (T) , viz. by p e rmuting the [ a ] 1 s . (F or the dic tionary betwe en tor i and Galois module s , s e e for instanc e T. Ono [ 21] . ) 1. 2 .

Cutting down T Let E b e a s ubgroup of K

*

=

T (0) and let E be the Z ar iski

c l o s ur e o f E i n T . U s ing the formula E

X

E

that E i s a n alg eb r aic s ubgr oup of T . Let T

E

=

E

X

E , one s e e s

be the quotient

g r oup T I E ; then T E is al s o a torus ove r O . Its characte r g roup X = X (T ) is the s ub g r oup of X = X (T ) c ons isting of tho s e charac E E n te r s which take the value 1 on E . If h. = IT [a] a denote s a aEr c har acte r of T , then X E i s the s ubgr oup of tho s e n IT a (x) a = 1 , for all x E E .

h. E

X for which

Exer c i s e s o that dim T = 2 . Let E b e the g r oup of units of K . Show that T is of diInens ion 2 (r e s p . 1) E if K i s iInag inary (re s p . r e al) . a . L e t K b e quadr atic ove r 0 ,

b . T ake for K a c ub ic field with one r e al place and one c om plex one , and l e t again E b e its g roup of units (of r ank 1) . Show that dim T = 3 and diIn T E = 1.

T HE GRO UP

II- 3

5

m

(F or more example s , s e e 3 . 3 .

)

1 . 3 . Enlarg ing g r oup s Let k be a fie ld and A a c ommutative alg eb r aic g r oup ove r k. Le t (* )

o �

Y � Y � Y � 0 2 l 3

an exact s e quenc e of (ab s tract) c ommutative group s , with Y 3 finite . Let

b e a h o m om o r ph i s m of Y 1 into the g r oup of k - r ational points of A. We intend to c on s truct an alg ebr aic g r oup B , tog e ther with a mor

ph i s m of algeb r aic g r oup s A � B and a homomorphism of

B (k) s uch that , (a )

Y2

into

the diag ram

is c ommutative , (b ) B is " unive r s al " with r e spect to (a) . The unive r s al ity of B mean s that , for any alg eb r aic g r oup B ' ov e r k and m o r ph i s m s A � B ' B r e p l a c e d by B ' ) , the r e f: B �

B'

,

Y2

� B ' (k)

exi s t s a

s uc h that the g iv e n m a p s

un ique

s uc h that

alg e

A � B'

(a )

is

true

b r a i c m o rph i s m and Y 2 � B (k)

(with

can

II - 4

AB E LIAN l - AD IC R E PR ES E NT A T IO NS

b e obtain e d b y c ompo s ing tho s e of B with f. (In other wo rds , B i s a pus h - out ov e r Y 1 of A and the " c ons tant " g r oup s cheme defin e d b y Y . ) 2 The uniquene s s of B i s a s s ur e d by its unive r s ality. Let us pr ove its exis tenc e . F o r each y E Y 3 let y be a r epr e s entative of y in Y . If y, y ' E Y 3 ' we have 2 y + y'

=

y+y ' + c (y , y' )

with c (y , y ' ) E Y 1 ; the c ochain c is a 2 - c oc ycle defining the exten s ion (* ) . L e t B b e the disj oint union o f c opie s A o f A, indexe d y by y E Y 3 . Define a g r oup law on B via the mapp ing s 7r

y, y

, : A X A , --+ A y y+y ' Y

g iven b y addition in A followe d by trans lation b y E (c (y , y ' ) ) . One then checks eas ily that

B

map s A --+ B and Y 2

--+

has the r equir e d unive r s al property , the B (k) b e ing define d as follows :

A --+ B i s the natur al map A --+ A

by -c ( 0 , 0 ) , Y 2 --+ B (k ) map s an e lement y + the imag e of

Z

in

o

z,

followe d by trans lation y

E

A

Y3 '

Z E

Y 1 onto

y Note that for any e xtens ion field k ' o f k we have a n exact

s e quen c e 0 --+

and a c ommutative diagram

A(k' ) --+

B

(k ' )

--+

Y 3 --+ 0 ,

T HE

GRO U PS

S

II - 5

m o � o

The alg ebraic g r oup

B

�

y

j,

1

�

A (k ' )

�

B (k '

)

� y3 �

i s thus a n extens ion o f the

b r aic gr oup Y 3 by A.

II

o.

c ons tant l ' alge -

Remarks

1) Let k' be an extens ion of k and A'

=

A X k' We may k apply the ab ove c on s tr uction to the k ' - algebraic g r oup A' , with r e s pe ct to the exact s e quence (* ) and to the map Y 1 � A (k ) � A ' (k ' ) . The g r oup B I thus obtaine d is c anonically is omorphic to B X k ' k this follows , for ins tanc e , fr om the explic it c ons truction of B and B' . 2 ) We will only us e the ab ove c on s tr uction when char (k)

and A is a torus .

The enlarge d g r oup

B

=

0

is then a " g roup of multi

plicative type " ; thi s means that , afte r a s uitable finite extens ion of the gr ound field, B b e c ome s is omorphic to the pr oduct of a torus and a finite abelian group . Suc h a g r oup is uniquely determined by its char acte r g r oup X (B ) = Homk" (B / k" G / k' ) , which is a Galois m module of finite type over Z . Here X (B ) c an b e de s c r ib e d as the -*

set of p air s (tP , X ) , whe r e tP : y 2 � k is a homomorphism and X E X (A) is s uch that tP (Yl ) = X (Yl ) for all Yl E Y 1 . Note that this g ive s an alternate definition of B . Exer c is e a) Let k ' b e a c ommutative k - alg eb r a , with Spe c (k l

and 1 ) .

)

k'

f. 0 ,

and

c onne c ted (i . e . k ' c ontain s exactly two idempotents : 0 Show the existenc e of an e xa c t s equenc e :

11 -6

AB E LIAN L

- AD IC

RE P R E S E N T A T IO NS

) ---7 B (k ' )

---7 Y 3

---7

0 ---7 A (k '

b ) What happens when

Spe c (k ' ) i s not c onne c te d ?

§ 2 . C O NS T R UC T IO N 2. 1.

0

AND S

T

OF

m

m

IdMe s and idMe s - c la s s e s

We define d in Chapter I , 2 . 1 the s et � of finite plac e s of the K co numb e r fie ld K . Let now � K b e the set of e quivalence clas s e s of ar chim e dian ab s olute value s of

and

�

co

K

.

If

v

E

�

K

if

v

E

and let

E

�

�

K

co

K

.

�

K

be the union of

�

K

denote s the c ompletion of K with v we have Kv = R or K = C , and K v v For v E � K ' the gr oup of units of Kv

the n K

r e s p e c t to v . F or v is ultram e tr i c

K,

i s denote d b y U . The idMe g r oup I of K is the s ubgr oup of v * IT Kv c on s i s ting of the fam ilie s (av ) with aV E Uv for almost

VE � K

all v ; it i s given a topology b y de c r e e ing that the s ubgr oup (with the pr oduct top ology) U

b e ope n .

We emb e d

K

*

v

into I b y s ending a

E K

*

onto the id� le *

(a ) , whe r e a = a for all v. The topology induc e d on K is the v v * dis c r e te topology. The quotient g r oup C = II K is c alled the id�le -

c las s g r oup of or W e il [ 44] . )

K.

(F or all this , see Ca s s e l s - F r �hlich [6 ] , Lang [1 3 ] ,

THE

GRO UPS

Let S

S

S

II - 7

m

be a finite s ub s e t of m = (m )

w e me an a fam ily E LK

v

If

v

If

E is the gr oup of units of

and

m

YE S

LK .

Then by a m odulus of s uppo r t

whe r e the mV are inte g e r s � 1 .

is a modulus of s upp ort S , we let U J.

V, m

denote

th e c onnecte d c omponent of K: if v E L � , the s ubg r oup of Uv c ons isting of tho s e u E Uv for which v (l -u) if v E S , and U > m v v if v E L - S . The gr oup Um = IT U is an o p en s ubgr oup of I. K v v ,m The subgr oup E n Um E m i s of finite index in E . (C onv e r s e ly , by a the orem of Chevall e y ([8] , s e e als o [24] , n O 3 . 5 ) eve r y subgr oup o f finite index i n E c on -

tairis an Em *

Let Im

II K Um

=

for a s uitable modulus b e the quotient II Um

CI (Image of Um

m

.

}

and Cm

the quotient

in C ) . O ne then has the exact s e quence

*

l � K / E m � Im The gr oup Cm

=

K , let Em

� Cm

is finite ; in fac t , the imag e of

henc e c ontains the c onnected c omponent D of

� l . Um

C,

in

C

i s open ,

and the g r oup

C I D is known to b e c ompact (s e e [13 ] , [44] ) . More over , any open s ubgr oup of I c ontains one of the U m

j e c tive limit of the

C

m [6 ] ) ,

I

I S ,

CI

D i s the pr o s . Cla s s field the ory (d. for ins tanc e henc e

g ive s an is omor p-hism of CI D = � lim Cm ab of the maximal abe l ian extens ion of K . onto the Galois g r oup G Cas s els - FrOhlich

Remark A more clas s ic al definition of C m

i s a s follows . Let IdS be the gr oup of fractional ideals of K p r ime to S , and Ps the s ub -

gr oup o f pr iric ipal ideals

,m

(y ) , whe r e y is totally pos itive and

AB E LI A N

II - 8

'{ '=-

V

E

'( b e l ong s to U

1 rrlO d . m (i . e . �

CD

K

)·

Le t C l

m

= Id I P S S, m 1

F o r each a

v

E

U

V ,m

�

a = IT v V v1 S

if

v

E

E

---»-

P

S , ilt

a

- AD IC R E P R ES E NT AT IO NS

for all v

S

�

�

Id � C l S m

and

l�

E

S and

We hav e the exac t s e quenc e :

Id ' choo s e an idHe S

in 1 m = I I Um hotnotnorphistn g : IdS � 1 m a c Otntnutative diag ratn T h e itnage of

V, m

1

v ia

v

)

1.

a

=

=

av if

(a

v

) , with V

E

�

K

- s.

depends only on � . W e th en g e t a One che cks r e adily that g e xtends t o

� Id � C l � l S m Ji g -I,::!: J, l � K / Em� Im � C m � l P

s ,m

and that f: Clm � Cm i s an i s otnor phistn ; henc e C m c an be iden tified with the ide al cla s s g r oup tnod m (and this show s ag ain that it is finite ) . 2 . 2 . The gr oups Tm and S m W e ar e now in a p o s ition to apply the g r oup c onstruc tion of 1 . 3 . We take for exact s equenc e

(�, )

the s e quenc e � Im � Cm � l

and for A the algebraic g r oup T = T I E wh e r e Em m m before , T is the torus define d in 1 . 1 , and (G RK I Q m I K )

is as E m

i s th e

II- 9

T HE GRO UPS S m

Z ar i ski c l o s ur e of Em in T , d. 1 . 2 . The c on s tr uc t i on of 1 . 3 now yields a O - algebraic gr oup Sm and a gr oup homomorphism with an alg ebr aic morphis m Tm � Sm £:

I

m

� S (0) . m

The s equenc e

is exact (Cm be ing identifie d with the c or r e s ponding c ons tant alge br aic group) and the diag r am

(** ) 1 � Tm (0 )

�

Sm (0 )

�

Cm

�

1

is c ommutative . Remark Let m ' be another modulus ; a s s ume m ' :: m , i . e . > m if v E Supp ( m ) . From the in Supp ( m ' ) :J Supp ( m ) and mv' v clus ion Um , e Um one de duc e s maps T m , � Tm and

� Im whence a morphism S m � Sm Henc e the S m ' s ' ' form a projective s ystem ; the ir limit is a p r oalgebr aic gr oup ove r 0, Im

extens ion of the pr ofinite gr oup CI D Exe r c is e s

=�

Cm

by a torus .

*

1) Let E m (0 ) be the Zar iski - c1osure of Em in K = T (0) . Show that the kerne l of £ m II Um � Sm (Q ) is the imag e of E m (0) � II Um

AB E LIAN 1. - ADIC R E PR ES E NT AT IO NS

II - l O 2)

Le t Hm '

Sm ,

be the kernel of

Im

�

Sm

where

a) Show that Hm , 1 m is a finite s ubgroup of and that it is contained in the image of E m ' b ) Construct an exact sequence (cf. Exer . 1 )

(S m ' (Q ) )

1.

-adic representation with values in Sm Let m be a modulus , and let L be a prime number . Let E : I � 1 m � S m (Q) be the homomorphism defined in 2 . 2 . Let T � S m be the algebraic morphism T � Tm � Sm by taking points with values in Q L defines a homomorphism 2 . 3 . The canonical

1T :

'

1T

IT K , the group T (Q L ) can be identified with vl L v * * K L IT Kv , and is therefore a direct factor of the idHe group I. vl l Let pr L denote the projection of I onto this factor. The map

S ince K � Q L

=

=

a

1

=

1T

1

0

pr 1 : I � T (Q L )

�

Sm (Q 1 )

is a continuous homomorphism. LEMMA -

a

L and

E

coincide on K

:::.;

This is tr ivial from the commutativity of the diagram

("�':' )

of 2 . 2 .

THE GROUPS S m Now, let E 1 : I

�

.u. - u

Sm (Q 1 ) be define d by

(* ** )

E ia)

E

wr ite a 1

I,

-1

)

-1

i. e . (If a

= E (a) a 1 (a

E 1 = E . a1

the £

1

- c omponent o f

1 (a)

= £

a. Then

-1 (a) 1r 1 (a 1 ) ) .

*

By the lemma , £ 1 is tr ivial on K an d , henc e , define s a map s ince Sm (Q 1 ) i s totally dis c onnecte d (it is an C � S (Q 1 ) m ; 1

- adic Lie gr oup ) , the latter homomorphism i s trivial on the c on c

recalled that C / D ab may be identified with the Galois gr oup G of the maximal ab elian extens ion of K . So we end up with a homomorphism 1 Gab - adic r epre s entation of K with i. e . with an � Sm (Q 1 ) ' 1: values in Sm (d. Chap. I, 2 . 3 ) . This r e pr e s entation is rati on al in the s en s e of Chapte r I, 2 . 3 . nec te d

ompo ne n t D o f C .

W e have alr eady

£

More pre cis ely, let v I Supp ( m ) , and le't fv I b e an idHe which is a uniformizing parameter at v , and which is equal to 1 everywher e els e ; let F = £ (f ) b e the image o f f in Sm (Q ) . With the s e nota v v v tions we have : E

PROPOSITIO N

a) T h e repr - e s entation

s entation with value s in S m

S .1

b)

=

E

1

{ v i Pv

=

E

1

•

"

GQ.b �

S m

(Q 1 ) is a rational r epr e -

i s unramifie d outs ide S upp ( m ) U S

.1

}.

1

'

whe r e

II - 1 2

AB E LIAN 1 - ADIC R E PR E S E NT A T IO NS

c)

(d . Chap . I , P r oof.

2. 3)

S1

i s e qual t o

(r e s p . . ab ln G

F

th en th e F r ob e n ius e l e ment

'

F

v

E

Sm

v £ ' l

(Q ) .

It i s kn own that th e c l a s s f i e l d i s om o r ph i s m

K

map s

Supp (m ) U

�

!! v

ab C / D =::.,.. G

U )

onto a den s e s ub g r oup of the d e c omp o s ition v ab . . . ). ln G g r oup of v (r e s p . onto the lne r h a g r oup of v ,, and th at a unifo r m i z ing e l em ent f of K i s mapp e d onto the v v F r ob e n iu s c l a s s of v . v

'. -'

1

If v S upp ( m ) and 1 , 1= 1 (a) = 1 , henc e

p cr v th i s p r ov e s b ) .

�) ,

a

E

Uv

£ 1 (a ) =

F or s uc h a v ,

the n E

1

an d

we have

and �) follows fr om �) .

CORO LLARY

c ompatible

1

-

The r e pr e s entations

E1

= 1;

(a) £

if m o r e ov e r

i s unr am i f i e d a t v ;

1

£ (f ) = E (f ) = l v v

form

a

F

v

;

hence

s y s tem of s t r i c tly

- adic r e pr e s entation s with value s i n Sm

W e als o s e e that the exc e ptional s e t of th i s s y stem i s c ontaine d in S upp ( m ) ; for an example whe r e it is diffe r ent fr om Supp ( m ) ,

s e e Exe r c i s e 2 .

R emark By

x � 7T

l

c on s truction ,

(x

-1

E

1 :

I�

S m

(Q 1 )

) on the open s ubgr oup U

1 ,m

i s g iven b y

=

IT Uv ,m

vl l

of K

*

l

ont a i n s 7Tl ( U ) C T (Q l ) C S (Q l ) ' and 1 ,m m m open s ub g r oup of Sm (Q 1 ) ' Th i s open s ub g r oup map s onto C

H e nc e ,

Im (E 1 )

r emarke d ab ove . 1m (£ 1

)

c

The s e

p r o p e r t i e s imply , in

is Z a r iski - dens e in

S

m

par ti c ul ar , that

i s an

m

as

1 1 - 1 -'

T HE GRO UPS Sm Exe r c i s e s (1 )

Le t

K

and

=

S upp ( m )

Em

fJ

•

= {l} ,

Cm

= { l } , henc e * = Q , S (Q) = G and S (Q 1 ) = Q1 . m m m b ) Show that I i s th e d ir e c t p r o du c t of its s ub g r oup s he nc e any a E I may b e wr itt e n as

= S m

Q

Q,

Show that

a)

Tm

=

*

*

a = u. 'Y Show that ,

if

a

=

u

E

lIn

'

'Y

1m

*

E Q .

(a ) , one has p

£

(a)

=

= sgn (a ) IT p

'Y

00

P

v

P

(a ) P

Show that

c)

P1

and

(a) =

'Y '

-1

a1

F =p . P

d ) S h ow th at Chap .

I, 1. 2.

p1 c o

m)

in c i d e s

with the char acter

X

1

of

= { 2 } and t:n

= 1 . Show that the 2 gr oups F. , Cm , Tm , Sm coinc ide with tho s e of Exe r c ise I , hence In that the exceptional set of the c or r e sponding s ys tem is empty. (2 ) L et

2. 4.

K

= Q , Supp (

Linear repr e s entations of Sm We r e c all fir s t s ome well known fac t s on r e pr e s entat ion s .

a)

Let k

be a f i e l d of char acte r i s ti c

0 ;

let

H

be

an

affine

AB E LIAN 1 - AD IC R E PR ES E NT A T IO NS

II - 14

c ommutat iv e alg e b r a i c

g r oup

b e th e g r oup of c h a r ac te r s of c h a r ac te r s of

on

X (H)

ov e r

k.

Let

(of d e g r e e 1 ) .

H

=

) , G Hom {H k /k m/ k He r e we w r i t e the

X {H )

T h e g r oup G = Gal (k/

multipl ic ative ly .

k)

ac t s

X {H) . L e t A b e the affine alg eb r a of

. Ev e r y e l e m e nt X E /k inv e r t ib l e e le m e nt of A . Henc e , one of

H

the ac tion of

�

�A

is th e g r oup alg ebra of

G - h o m om o r ph i s m if

A = A

by linear ity , a homomorphism

a : k[ X {H) ] whe r e k[X {H) ]

let

k be the k X {H) c an be identifie d with an

H , and

G

X {H)

ove r k .

This is a

is define d by

s (�a X ) = �s (a ) s (X ) for a E k and X E X (H) . It is well - known X X X (line ar independenc e of cha r acte r s ) that a is inj e ctive . It is b ij e c tive if and only if

H

is a g r oup of multiplic ative type (cf. 1. 3 , re

mark 2 ) . Henc e we may ide ntify

k[X {H) ]

with a sub alg ebra of A .

b ) Let V be a finite - d imens ional k-ve ctor s pac e and let

be a l ine ar repre s entation of

H

(this is always the c a s e if

is of multiplic ative type ) .

to q,

in

its

H

into V . As s ume q, i s s emi - s imple We a s s ociate

trac e

Z [X (H) ] ,

whe re n (q, ) i s the multiplic ity of X in the de c om _X pos ition of X ove r k .

T HE GRO UPS

S

11 - 1 :>

m

e

(h ) = T r ( q, (h » for any point h of H (with value in any q, c ommutative k - alg eb r a) . Let Re Pk (H) be the s e t of i s omorphism clas s e s of linear s e mi - s imple r ep r e s entations of H. If k is an ex l tens ion of k , then s c alar extens ion fr om k to k define s a map We

have

Re P k (H)

---+

ReP

k

1

(H /

1

) which is e as ily s e e n to b e inj ective . k 1

that an eleme nt of Re P k

1

(H

/k

c an b e defined ov e r

)

k , if it

We s ay is in the

1

image of this map .

e cP define s a b ij e c tion betwe en and the s et of elements e = � n X of Z [X (H) ] which

PROPOSIT ION Re P k (H) s atisfy:

1

-

The map q,

�

X

(a) e is inv ar i ant by G (i . e . n S E

G, X

X

E X (H» .

=

----

n

s (X )

for all

(b ) n > 0 for e v e r y X E X (H) . X -

The inj e c tivity of the map cP � e

i s well -known (and doe s cP not depend on the c ommutativity of H) . T o p r ove s urj e c tivity , c on (i) s ider fir s t the c a s e whe r e e has the form e = � X e1 ) whe r e X

Pr oof.

is a full s e t of diffe r ent c onj ugat e s of a c har acte r X

i s the s ubgr oup of G fixing X

e =

(* )

The fixe d field k

X

X

�

If

G (X )

then

s E G/ G (X )

o f G (X ) in

X (H) .

s (X )

k i s the small e s t s ubfield of k s uc h

a s a r e pr e s en tation of de gr e e 1 of One gets , by r e s tr ic t i on of s calar s to k , a r e pr e s entation

that X E A � k H/

k

,

E

X

Con s ider X

¢

of H

of de g r e e

is e qu a l

to

The

e.

fac t that any f

1.

AB E LIAN

II - l 6

e

[k X : kJ .

s u r j e c t ivit y of ¢ � e

s at i s fy in g

(a)

-

In o r de r th at

it is ne c e s s ar y and s u f f i c i e nt

and (b ) i s

¢l

E

Re P

that

e

k

a

1

¢ s um of e l e m ents of the

(H /

k

A i»

¢ E 1

tr ace e

of ¢ ¢ now follows f r o m th e

One s e e s e a s ily that the

o r m (':' ) ab ove .

CO RO L LAR Y

- AD I C R E P R E S E N T A T IO NS

1

k

)

c an b e de f ine d ove r

k

l

b e l ong s to

k,

A .

(c ) We return now to the g r oup s Sm PROPOSITION 2 ¢ E ReP

k

(i )

1

(S ¢

-

Let

be an extens ion of k and le t l The following proper tie s ar e equivalent:

m I k1 ) . c an

be

k

d e f in e d ov e r

k ,

(ii ) F or eve ry v . Supp ( m ) , the c oeffic ients of the character

i s tic polynomial ¢ (F ) b e l ong to k , v (iii) The r e exis ts a s et l: of plac e s o f k o f dens ity 1 Chapter I , 2 . 2 ) such that T r (¢ (F » ) E k for all V E l: v P r oof. The implications (i)

==:>

(ii )

==:>

(d .

•

(iii) ar e trivial . T o p r ove

(iii) ===> (i) we ne e d the following lemma . LEMMA - The s e t of F r ob enius e s F , V E l: , is dens e in Sm v the Z ar iski to pology . Pr o of. numb e r .

Let X b e the s e t of all Fv I s , V E l: , and let Let

x CSm

Z a r i s k i t o p o l o g y (r e s p .

(r e s p .

x 1 C Sm (Q 1 »

1 - adic topology) .

1

th e c l o s u r e of It i s c l e a r th at

fo r

be a pr ime

X

i n the

T H E GRO UPS

S

II - 1 7

m v

O n th e o th e r han d , C eb ota r ev ' s the o r e m (d .

C h a pte r I ,

= Im (£ ) (d o 2 0 3 ) . T h e s e t Im ( £ J. ) , J. J. ev e r , i s Z ar i s k i d e n s e in S (d o R e m a r k in 2 . 3 ) . H e nc e X m wh i c h pr ov e s the l e m m a .

2 . 2 ) im p l i e s that

L e t u s n o w p r ove that in

{ J.

a

how-

X

=:>

(i i i )

(i) .

Let

e

be

A {i) k k , whe r e A i s the aff ine alg eb r a o f H 1 } b e a b a s i s of the k - v e c to r s pac e k , w ith

l

=

J.

=

S m

the trac e of Le t S / k m

a

o

=

1

fo r

s ome

a

W e h av e e ;.. = L: � � 1. hence (� e A ) 'I' a o a a with f o r all h E H (k ) . T ake h = F T r ( (h ) ) = e ;.. (h) = L: � (h ) 1. v a a 1 'I' l S inc e F V E L: v b e ong s to H (k ) we have �a (F v ) E k for all a s in c e Tr ( (F ) ) E k , we g e t � (F ) = 0 for all a 1= a By the v a v o l e mma , the F ' s , V E L: , are Zariski- dens e in H; henc e � = a v a

in d e x

for

a 1=

a

c or ollary

and

e

= �

a to P r o p o s i t i on 0

o

b e l ong s to

A

and (i ) follows from the

1.

Exe r cis e Show that

the character s of

to the homomorphisms tie s : (a) (b )

X :

I

�

_*

Q

X

(x) = 1 if x E U m F o r each embe dding a

S m

c o r r e s pond in a one - one way

hav ing the following two proper -

of

K into Q

te g r al numb e r n (a) s uch that X (x)

fo r all x e K

*

=

IT

ae r

n a) a (x) (

ther e exists an in-

AB E LIAN 1. - ADIC R E P R E S E NTAT IO NS

II-IS

2. 5 .

1 - adic r e pr e s entations as s o c iate d to a l ine ar repr e s entat ion of Sm

1) The 1 - adic c a s e Y

Le t

1.

b e a finite - dirrl e n s i onal 0 -ve ctor s pac e and

1

q, : S

m / 01

a line ar r e p r e s e nta t ion of S

m /0

� G Ly

1

Y

L

in

1

•

Th i s define s a

homomorphism q, : Sm (0 ) � G Ly (0 ) L L 1

=

Aut (V )

L

which is c ontinuous for the L - adic topolog i e s of tho s e g r oups . ab B y c ompos ition with the map £ : G � Sm (0 1 ) defin e d 1

in 2 . 3 , w e get a map

= q,

q,

1

i . e . an ab e l ian

1

0

£

L: G

ab

� Aut (V

b ) Let

V E

Then q, i s unr am ifie d a t 1

of

A.

' 'I' 1. Sm

E

Aut (V

(Q )

l

)

)

,

- adic r e pr e s entation of K in Y 1 .

PROPOSI T ION - a) The r epr e s entation q,

FV

L

�k '

Vi

with v

,

i s s emi - s imple . Supp ( m ) and

p

v

fo

1.

the c o r r e s ponding F r obenius e lement

i s e qual to q, (Fv ) ,

defined in 2 . 3 .

1

whe r e F

v

de n ot e s the ele ment

II - 1 9

T HE GRO UPS S

m

c ) The r epr e s e ntation

2. 3)

is a g r oup of multiplicative type , all its r e p r e s en -

S

Sinc e

I,

m

tations can b e b r ought to diag onal form on a s uitable extens ion of the g r ound field; henc e a) . As s e r tion b ) foll ow s fr om 2 . 3 , and as s e rtion c ) follows fr om P r op o s ition 2 of 2 . 4 . Remark Let us identify rp 1

with the c o r r e s ponding homomorph i s m of

the idMe g r oup I into Aut (V 1 ) . K e r (rp ) 1

d)

Let rp

e)

T

:

c ontain s

T/ If

U

1 ,m

= TT

vl 1

U

of T (Q 1 ) '

v,m

Then v ,m � GL y

Q1 x

if v

U

-

and rp : S m o

�

r epr e s entation
1

be

o

GL y

e ach p r ime numb e r

the

v

1= 1 .

b e l ong s to the op e n s ub gr oup

one has

E

1 .

The r ational c a s e Let n ow Y

Q

Supp ( m ) , p

b e defin e d b y c ompo s ing

1

T he s e pr ope r tie s follow r e adily fr om tho s e of

2)

t

o

a

a

f inite

dimens ional

ve ctor s p a c e ove r

l inear r e pr e s e ntation of

S

For

m

we may app ly the p r e c e ding c on s truc tion to

: o/ 1

S

m

I Q1

�

G Ly

1

'

whe r e

Y1

=

Y

o QI)

Q1 ;

1.

AB E LIAN

II - 2 0 we then g e t an T HEORE M

1.

q, 1.

-adic r e pr e s entation q,

:

- ADIC R E PR ES E NT A TIO NS G

ab

Aut (V 1. )

�

.

fOrIn a s t r i c tly c om p atible sys tem o f r ation al ab e li an s e mi - s imple r e pr e s e ntation� . It s exce ptional s e t is c on taine d in Supp ( m ) . 2 ) For e ac h v 1 Supp ( m ) the F r ob enius element of v -

1) T h e

1.

(q,

)

q,

(F ) of Aut (V ) . 0 v 1. such that q, 1. 3 ) Ther e exis t infinitely many pr ime s

with r e s -pe c t to the sys tem Q 1.

i s diagonalizable ove r

1.

is the e lement

0

-

•

The fir s t two as s e r tions foll ow dir e c tly fr om the pr opo s ition ab ov e . T o pr ove the thir d one , note fir s t that the r e exists a finite e xtens ion E of Q ove r which q,

b e c ome s diagonalizable . If 1. is o a p r ime numb e r which s pl its c ompletely in E , one c an emb e d E and this s hows that q, 1. is diag onalizable . As s e r tion 3 ) now follows fr om the well -known fac t that the r e exis t infinitely many such into

Q

1

v

1

(thi s is , for instanc e , a c on s equenc e of Cebotar ev ' s the o r e m , cf. Chap . I , 2 . 2 ) . Remark The Frobenius e lements q, (F ) o v us ing the homomorphism q,

o0

£:

E

Aut (V

0

)

c an al s o b e define d

I � Sm (Q ) � Aut (V ) . 0

Note that the ir e ig envalue s g ener ate a finite extens ion of inde e d they are c ontaine d in any field ove r which in diag onal form .

q,

o

Q ;

c an b e b r ought

T HE GRO UPS Sm

II - 2 1

Exe r c i s e s 1) Let o : Sm and let

1

b e a l ine ar r e pr e s entat ion of Ebt '

b e a p r im e nurn.b e r . a) Show that the Z ar i ski closur e o f 1m ( 1 ) is the alge

b r aic gr oup o (S m ) . (Us e the fact that 1m (E 1 ) is Zariski dens e in d. 2 . 3 . ) Sm b ) Let �m b e the Lie alg ebra of Sm and <1> 0 (!m ) b e its imag e by , i . e . the · Lie alg eb r a o f 0 (S m ) . Show that the Li e o (Us e the alg eb ra of the 1 - adic Lie g r oup 1m ( 1 ) i s <1> 0 (! m ) � Q 1 fac t that

is open in Sm (Q 1 ) , cf. 2 . 3 . ) a) Show that the r e exi s ts a unique one - d imens ional r e p -

1m (E1 )

2) r e s entation

N' S • m

such that

N (F )

=

Nv

E Q

*

� G

m

for all v

�

Supp ( m ) .

b ) Show that the morphism T induc e d by the norm map fr om c ) Show that the

1

K

to Q

�

Sm

�

G

m

i s the one

i s omorphic to the r e pr e s entation Y 1 (fL) define d in Chap . I , 2. 6.

N is

- adic r e pr e s entation define d by 1.

2.

Alternative c onstruction

Let <1> 0 : Sm the map

E :

I�

�

G

�

b e as in 2 . 5 . o

If we c omp o s e

o

with

S (Q ) defined in 2 . 2 , we obtain a homomorphism m

4> 0 0

E : I � G Ly (Q ) =

o

Aut (V ) ' o

AB E LIAN 1. - AD IC R E PRES E N T AT IO NS

II - 2 2

C onve r s e ly :

Le t f : I PROPOSIT IO N - -exists a rP o : Sm

�

G Ly

�

Aut (V

0

)

b e a hom om o r ph i s m .

s uch that rP

o

0

o

E =

The r e

f if and only if the

following c onditions ar e s atisfie d : (a) The ke rnel of f c ontain s Um (b ) The r e exis ts an algeb r aic homomorphism s uch that I/I (x)

=

f (x) for eve ry x c:

Mor e ove r , such a rP

o

K

*

=

T (Q ) .

i s uni q ue .

Pr oof. The ne c e s s ity of the c onditions (a) and (b ) is tr ivial . Con ve r s ely, if f has pr ope rtie s (a) , (b ) , it define s a homomorphism II

Um

�

Aut (V 0) . On the other hand , s inc e f and

the morphism

1/1

is equal to 1 on

Z ar iski - c 1osur e Em

Em

=

This m e ans that

K

1/1

::'

n

U

1/1

ag r e e on

K

*

m ' hence on its fac tor s thr ough

By the univer s al pr ope r ty of S- m (cf . 1 . 3 and 2 . 2 ) , the map s � GL I I tIn y define an algebr aic morphis m o 0 -

and one checks eas ily that

rP

o

has the r e quir e d

pr ope r tie s , and is unique . Remark S inc e

Um

is

open , prop e r ty (a) impl ie s that

f

i s c ontinuous

T HE

II - 2 3

GRO UPS S

m

with r e s p e c t to the d i s c r ete topolog y of Aut (V ) . C onver s ely , any o . c ontinuous h oznozn o r ph i s zn f: I � Aut (V ) i s tr iv ial on s ozn e U m ' o zn o r e ove r , the r e is a s znalle s t s uch m ; it is c a l l e d the c on duc tor of f. Exe r c i s e Le t m

b e a finite dizne ns i onal o F o r e ach v . Supp ( m ) l e t F be an elezn e nt of v

b e a zn o dulus and l e t Y

Q -vector s pac e .

Aut (V ) . AS SUIll e o (a ) The F I S c oznznute pairwi s e . v (b ) The r e e xi s ts an alg e b r aic zn o r ph i s zn that 1/; (0: ) =

TT Fvv (o: )

for 0: E

K

*

, 0:

=

e ach r e al plac e .

1

(znod

Show that the r e e xi s t s an alg eb r aic znorphiszn If>

s uch

m ),

o

:

for which the F r ob enius e leznents are equal to the

2. 7.

and 0: >

S m

�

F I S. V

GL

0

y

at

o

The r e al c a s e The pr e c e d ing c on s tructions ar e r e lative to a g iven p r izne

nUInb e r

1.

Let

Howev e r , the y have an a r c h izne de an analogue , as follows : 11':

T � S m

b e the c anonic al znap define d in 11'

co

T

(R)

�

2 . 3 , and l e t

Sm (R )

b e the c or r e s ponding hoznozn o r ph i s zn of r e al L i e g r oup s . T (R) = (K Cil R )

*

=

IT

VE�

co

K

*

K v

fac tor of the ide le g r oup 1 .

we c an identify T (R) L e t pr

co

S inc e

with a dir e c t

b e th e p r oj e c tion o n th i s

II - 2 4

AB E LIAN 1. - ADIC RE P R E S E NTAT IO N

factor ; the map

i s c ontinuous , and one checks as in 2 . 3 that * on K One may then define a map E

00

E

00

I

�

S

m

a

00

c o inc ide s with

E

(R )

by

One has

E

00

(a)

=

(a)

= E

t.c

1 if a E K ,

(a)a

(a - 1 ) .

00

hence

E

morphism of the id"ele clas s g roup C

::

may be viewed as a homo * II K into the r e al Lie g r oup

00

S m (R ) .

The main differ enc e with the " finite " c a s e is that E 00 is not tr ivial on the c onne cted c omp onent of C , henc e has no Galois g r oup inte rpretation. :

�

(R)

with a c omplex charac �' � G one gets a homomorphism C � C , i. e . te r S mI C ml C ' a G r O s s enchar akte r of K , in the s en s e of Heeke . It is eas ily s e e n When one c omp o s e s

E

00

C

Sm

•

that the characte r s obtain e d i n thi s way c o inc ide with the " GrO s s enchar akte r of type (A ) " of W e il (d . [ 3 5 ] , [41] ) , who s e o c onductor divide m

THE GRO UPS S m

II - 2 5

Exe r c i s e Le t e:

1 �

b e the map define d by

(R ) X

Sm E

ITsm (0 1 ) L

and the

co

a) Show that the imag e of e is c onta ine d in the s ubgr oup Sm (A) of Sm

(R ) X

ITsm (0 1 ) , whe r e A denote s the r ing of ad'ele s L

of 0 , and that e : I

�

Sm (A ) i s c ontinuous (for the natural topol

ogy of the ade l i z e d gr oup Sm (A) ) .

b ) Let 7[A : T (A) � Sm (A) b e the map defined b y 7[ : T � Sm Show that , if one identifie s T (A) with I in the obvious way , one has e (x)

= E

( x) 7[

A

whe r e E : 1 � Sm (0) C S

m

-1

(x )

(A) i s the map define d in 2 . 3 .

/ s. ] i s not open in 9n (A ) if

this g ive s an alte rnate definition of the c ) Show that e (I)

.

E

e m

F

[ Note that

{l} .

2 . 8 . An example : complex multiplic ation of abelian varietie s (W e give here only a b r ie f s ketch of the the ory , with a few in dications on the proofs . For more detail s , s e e Shimur a - T aniyama [ 3 4 ] , Taniyama [3 5 ] , We il [41] , [4 2 ] and S e r r e - T ate [ 3 2 ] . )

Let A b e an abelian var iety of dim ens ion d defined ove r K . Let End (A) b e its r ing of endomorphisms and put K End (A) 0 = End K (A) @ O . K

II- 2 6

AB E LIAN 1 - AD IC R E PR ES E NT A TIO NS

Let

E

b e a numb e r field of deg r e e 2 d , and

b e an inj e c tion of E into End K (A ) . o h a v e " c om p l e x multip l i c ation " by E ;

The var ie ty

A

is then s aid to

in the t e r m in o l o g y of

Shimur a - T aniyam a , it is a variety of " type (C M) " . V

=

E1

=

Let 1 b e a p r ime inte g e r and define T 1 (A) and T 1. (A) (i Q as in Chapte r I , 1 . 2 . The s e ar e fr e e m odule s 1

1 ove r Z 1 and Q ' of ' r ank 2d. The Q - algebra End (A) acts on 1 K o V 1 henc e the s ame is true for E , and , by line arity, for E�

LE MMA

Q

Q 1. . One p r ove s e a s ily:

-

V1.

i s a fr e e E 1. -module of rank 1 .

P

1. : Gal (K / K ) � Aut (V 1 ) b e the 1 -adic repre s entation defin e d b y A. If s E Gal (K / K) , it is c lear that P 1. (s ) c ommute s Let

w ith E , henc e with E 1. . B ut the le mm a above implie s that the c ommuting algebr a of E 1. in End (V 1. ) is E 1. its e lf. Henc e , P 1. m ay b e identified with a homomorphism P

1. : Gal (K / K)

�

E

�

Let now T E b e the 2 d - dimens ional torus attached to E (a s T i s attach e d to K) , s o that T E (Q 1. ) = E , and P i take s value s

�

in T (Q i ) . E

T HEOREM 1 - (a) The sy stem o f r a t ional

(p

) is a s tr ictly c ompatible s y s tem 1 1. - adic r e pr e s e ntation s of K with value s in T (in the E

Il - 2 7

T HE GRO UPS S

m

s e n s e of Chap . I ,

2 . 4) .

(b ) T h e r e is

a

rrlOdulus m a n d
s uch tha t att a c h e d to

p1

i s th e imag e by cf. 2 .

S

m

m

.p

�

a

T

rrlOrphism

E

of th e c an o n i c al s ys te m

3.

Mo reove r , the r e s tric tion of

plic itly:

to Tm

(E

1

)

c an be g iven ex-

Le t t b e the tang ent s pac e at the origin of A. It i s a K-vector space on which E acts , i. e . an (E , K ) -b imodule . If we view it as an E -ve ctor space , the ac tion of K i s g iven b y a homomorphism * j : K � En d E (t) . In particular , if x E K , det Ej (x) i s an element * * * of E ; the map de t j : K � E is clearly the r e s tr iction of an E algebraic morphism 6 : T � T . E T HEOREM 2 - The map ma p T

�

Tm

�

6:

T

�

Sm � T • E

T

E

c o inc ide s with the comp os ition

Examp le If A is an elliptic curve , E is an imag inar y quadr atic field , and the ac tion of E on the one - dimens ional K -vector space t de fine s an embedding E

�

K.

The map de t j : K E norm re lative to this emb e dding .

*

�

E

*

is just the

Indications on the proofs of The orems 1 and 2 Part (a) of The orem 1 is p r oved as follows : Le t S denote the fin ite s e t of v E E K whe r e A has " bad r e duction" . If v 4 S , and

AB E LIAN 1 - ADIC R E PR E S E NT A TIO NS

II- 2 8 1

P v ' one shows e a s ily that p 1 is unr amifie d at v (the c on ve r s e i s als o tr ue , s e e [ 3 2 ] ) ; m o r e ov e r the c o r r e sponding F r obenius e lement F may b e identifie d with the Fr obenius endomo rphis m v, P1 F of the r e duc e d var iety A B ut F c ommute s with E in v v v E nd (A ) 0 and the c ommuting alg eb r a of E in End (A ) 0 is E its e lf v v * (d . [ 34] , p . 3 9) . Henc e F (Q) and this implie s b e long s to E = T v E (a) 1=

"-

"-

•

Theo r e m 2 and part (b ) of The o r e m I are le s s easy; they ar e pr ove d , in a s omewhat diffe r ent for m in Shimur a - T aniyama [ 3 4 ] (s e e al s o [ 3 2 ] ) . Note that one c ould exp r e s s the m (as in 2 . 6 ) by s ayin� that the r e exists a homomorph i s m f:

I

�

E

;�

(wh ere

I

d e n ote s , a s

us ual , the g r oup of id�le s of K) having the following p r op e r tie s : (a) f is tr ivial on s .

U m

for s ome modulus

m

with s upport

(b ) .!!. v + s , the imag e b y f o f a uniformizing par ame te r at v is the F r ob enius element F E E v * (c ) g X E K is a pr inc ipal id�le , one has f (x) = det j (x) . E � ",.

This is e s s entially what is pr oved in [ 3 4 ] , p . 148 , formula (3 ) , exc ept that the r e s ult is expr e s s e d in te rms of ide al s ins te ad of id� le s , and det j (x) is wr itten in a diffe r ent form , namely E !/; II IT NK/ K * (x) ex " ex

Remark Another po s s ible way of p r ov ing The orems 1 and 2 is the fol lowing : Let 1 b e a prime inte g e r dis . tinct fr om any of the pv , v O n e th e n s e e s that th e Galo i s -m o dule

the s ens e

of Chapte r I II ,

1.

2

(ind e e d ,

V1

E

S.

i s of Hodg e - T ate type in

the c or r e s ponding l o c a l m o dul e s

THE GRO UPS Sm

II- 2 9

are a s s o c iate d with 1 - divis ible g r oup s , and one may apply T ate ' s Hence p 1 is " l ocally alg eb r aic " (Chapter III, loc o c it . ) , and us ing the the orem of Chapte r III , 2 . 3 one s e e s it define s a morphism q, : � � T E · One has q, 0 £ 1 = P 1 by c onstruc tion; the s ame is tr ue for any p r im e numb e r 1 ' , s inc e q, 0 £ l ' and P l ' have the s ame F r obenius e le ments for alm o s t all v . Thi s prov e s part (b ) o f The o r e m 1 . A s f o r The or e m 2 , one us e s the explic it the o r em [ 3 9] ) .

form of the Hodg e - T ate dec ompos ition of V 1 as g iven by Tate [ 3 9] , c omb ine d with the r e s ults of the Appendix to Chapter III. '

§ 3 . S TRUC T URE OF Tm

AND AP PLICAT IONS

3 . 1 . Structur e of X r:r:m ) If w i s a c omplex place of Q ,

the c ompletion of Q with

r e spect to w is is omorphic to C ; the de c ompos ition gr oup of w i s thus c yclic of orde r 2 ; its non - tr iv ial e lement will be denote d by (the " Frobenius at the infinite plac e w " ) . Th e c ' s are c onw denote the ir c onjugacy c las s . (B y j ug ate in G = Gal eQ / Q) ; let C c

w

00

a the orem of Artin [1] , p . 2 5 7 , the elements of

C

non -trivial elements of finite or de r in G . )

are the only

00

Let X (T ) be the char acte r group of the torus T , cf. 1 . 1 ;

we

wr ite X (T ) additively and put Y eT ) = X (T ) S z Q . We dec ompo s e Y as a dir ect s um Y = y O @ Y @ y + of G - invar iant s ub s pace s , as follows (cf. Appe ndix , A. 2 ) G o Y = Y = { y E Y I gy = y

Y - = {Y E Y l cy = -y

for all g

for all c

E

C

E

00

G} , }

AB E LIAN 1 - ADIC R E PR ESENT A TIONS

II-3 0

and Y + is a G - invar iant s uppl e m e n t to

Y

y+

0

E& Y

-

eas ily that i s unique , d . App e nd ix , loc o c it. Mor e expl ic itl y , if a E r i s an emb e dding of

m

Y;

K

one p r ov e s

into

Q ,

let

X (T ) be the c o r r e s p o nding char ac te r of T ; the [al ' s , a E r , o form a bas is of X (T ) and g . [a] = [g . a] if g E G . The spac e Y is gene r ated by the norm e l e ment L: [aJ , and its G - invar iant [a]

E

-

supplem ent is Y $' Y any char acte r

X

X =

+

aE r { L: b [a] l b E a aE r a

=

b

=

o}

Henc e ,

E X (T ) c an b e wr itten in the fo rm aL:

aE

r

[a] +

L:

aE

a, b E

a

r Q,

b ( a] , a L: b

a

=

0, a + b

(In particular , we s e e that da E Z whe r e d Y

aE r a L:

Q ,

=

Z aE . [K: QJ . ) The s ub s pace

c an now b e de s c r ib e d a s follow s Y

=

{ L: b a [ a J l b a

all

C

E Q , L: b a E

=

ca

=

-b

a for

C 00 and a E r } .

On the othe r hand , the p r oj e ction tion of X (Tm ) into X (T ) ; we identify thi s inj e c tion . PRO POSITIO N - X (Tm ) � Z Q

0, b

=

yO ED Y -

Thi s follows fr om Appe ndix ,

A.

2.

T�

Tm defi n e s an inj e c X (Tm ) with its imag e unde r

.

T HE GROUPS Sm

CORO LLAR Y

II- 3 1

- The char ac ter g r oup X (T

1

finite index o f X (T )

n

CORO LLAR Y 2 - I f

X

2a

E

In

fac t , g iven

(Y E

1&

C

E

=

2a + b + b

C

a

3 . 2 . T he m o rp h i s m j

*

Y ).

a E

and

00

:

G

ca

m

=

�

r ,

we have

(a+b ) + (a+b a

'Iln

We h ave s e en that any char acte r

the form X

i s a sublatti c e of

m

-

X (T ) i s wr itten in the form (* ) . then m

Z .

2a

o

::

a :E a e

r

[ a] +

X

:E

a E

E

Z . ca) E

X (T

m

)

c an

be

wr itt e n in

b [ a] r a

= 0 , 2a E Z . H e n c e X � 2 a d e f ine s a homo morphism j : X ( T m ) � X (Gm ) = Z and we obtain by duality a mor ( : Gm � Tm phism of algebr aic g r oups If q, o : Sm � GLy

with a , b

a

E

Q , :E b

a

J.

*

o

, we obtain by c OIll p os ition with j a m morphi sm of alg ebr aic gr oups G � GLy This repr e s entation m is a repr e s entation of S

o (i) define s (and i s defined by) a g rading Y = :E Y of y o of G III o iE Z o r e c all that G acts on Y ( i ) by means of the c h a r ac te r III i E Z = X (G ) . III (n) . Y We s �y that Y is homogene ous of degr e e n if Y o o 0

=

0

AB E LIAN 1. - AD I C R E P R E S E N T A T IO NS

II- 3 2

Remark For r e pr e s entat ions c orn ing from the

1. - adic hom ology

H �, (X ) of a pr oj e ctiv e s mooth var iety X , the g r ading defined ab ov e should c oincide with the natur al one : H ..... (X )

�

=

.._

. 1

H. (X ) . 1

Exe r c i s e Le t N : Sm � G m be the morphism define d in Exe r c i s e 2 2 is � � � of 2 . 5 . Show that N o j : Gm � Sm � G Show that m n any morphism S m � G is e qual to £ N whe r e £ is a charac m ter of em with value s in { ±. l } and n E Z . 2 ) Let
.

,

As s ume
=

dim Y

a) Show that dh i s even (apply Exe r c . 1 to �

G ). m b ) P r ov e that the r e ex is ts on

de t (
quadr atic form Q s uch that

Y

o

o

a pos itive definite

Q (p (x) y) = N (x) d Q (y) for any y

E

Y

o

and any x

E

Sm (Q ) . [ Let H b e the kerne l of

N: S m � G Us ing the fact that H (R) i s c ompac t , prove the m exis tence of a pos itive definite quadr atic form Q on Y invar iant �_

by H ; then note that S m 3. 3.

0

i s gene r ated by H and j (G ) . ] m ....

Structur e of Tm We ne e d fir s t s ome notation s : Let H

c

b e the clos e d s ubgr oup of G

by C ro (cf. 3 . 1) .

=

Gal (Q / Q) gene r at e d

The r e is a unique c ontinuous homomorphism

THE GR O U PS S {+ l}

for all c E C . Inde e d the 00 e: i s c le ar , an d one p r ov e s its exis tenc e by taking the re s tr iction to H c of the hom om o r ph i sm G --:. {2:. 1 } a s s o c iate d with an im ag inar y quadratic extens ion of Q . W e let H = Ker (£ ) . The g r oup s

£ :

H

�

II - 3 3

m

c unic ity of

-

s uch th at

e:

=

( c)

-1

are c l o s e d invar iant s ub g r oup s of G , and (H : H ) = 2 . c c Le t now K b e , a s before , a fi nite e xtens ion of Q ; we ide ntify

H and H

it with a s ubfi e l d of Q ; l e t G The field

s ubg r oup of G .

K

= Gal (0. 1 K) be the cor r e s pondin g K i s totally real if and only if all the

e l ements c of C 00 act tr iv ially on K ,

l ' •

e if and only if G K Henc e , the r e e xi s t s a maxim al totall y r e al s ubfield c ontains G c K of K , who s e Galo is g r oup i s = G We let K l b e o K Hc the field c or r e s ponding to G

K H.

•

We have

.

and As s hown b y W e il (ef . [ 4 7 ] , p . 4 ) the fie lds K

o and K l are c l o s e l y relativ e t o K . Inde e d , if

c onne cted to the g r oup s

Tm in 3 . 1 , we X = � b [a] i s an e lement of the gr oup denote d by Y a for all c E C b If h = c . . . c , this give s have b n ca - a 00 l = (-l)�

a

= E (h )b

and b y c ontinuit y the s ame holds for all h this :

PROPOSITION yO

K

-

The norm m ap define s -

an

a E

H

c

One de duc e s fr om

is omorphism of the spac e

r e l a tiv e t o K onto the space Y relative to K. K 1 l

11 - 3 4

AB E LIAN l - ADIC R E PR ES E N T A T IO NS

Mo r e pr e c i s ely, if

X

1

X 1

the imag e of

= �

b [ a ] b e l ong s to al l

Y�

,

1

whe r e

by the norm map is

is the r e s tr ic tion of a to K . It is clear that this map l l is inj e c t ive . C onve r s ely, if X = � b l a ] belong s to y� , we s aw a = £ (h)b ab ove that b for all h E H , henc e b = b for ha a c ha a h E H and of c our s e al s o for h E H . G . Thi s shows that b a K depends only on the r e s tr ic tion of ' a to K , and henc e that X l b e l ong s to the imag e of the norm map . whe r e a/ K

CORO LLAR Y

-

to each othe r .

The tor i T m

attache d to K and K

l

are is ogenous

attached to K . The r e remain s to de s c r ib e the tor i Tm l There ar e two c a s e s : i s one (1) K = K · In this c as e , we have Y = 0 and T o l m dimensional , and is omo rphic to G m Inde e d , if X = � b [a ] b e long s to Y , and c E C , we

have b c Y

E

ca

G . K

= o.

=

He

-b

=

a

a

(cf. 3 . 1) but al s o b

ca

=

b

G . H. This s hows that b K a

a

= 0

s ince

00

for all a , henc e

(2) [ K : K 0] = 2 . T h e fie ld K i s the n a totally imag inary l l quadr atic extens ion of K (and it is the only one c ontaine d in K , as o one checks r e adily) . In this c a s e Y is of dimens ion d = [ K o : Q] i s (d+l) - dimens ional .

T HE GRO UPS S

II - 3 5

m

Mor e pr e c i s e ly , the s pa c e Y attache d to Kl is 2 d - diITlens ion a l and the invo lution a o f K c or r e s ponding t o K de c oITlpo s e s Y l o in two e ig enspac e s of diITlens ion d each; the s pac e Y is the one c or r e s ponding to the e ig e nv alue -1 of a . Thi s is prov e d by the s aITle arguITlent as ab ov e , onc e one r eITlarks that all c

E

C

co

induc e

ReITlark In thi s last c a s e (which is the ITlost inte r e sting one ) , the torus Tm is is og enous to the product of G by the d- diITlens ional torus ITl kernel of the norITl ITlap fr oITl K to K l o 3 . 4 . How to c OITlpute F r obenius e s Le t

be a l inear r e p r e s entation of Sm of degr e e n. B y extending the g r oundfield , the r e s tr iction of

(n (i ) a

E

Z) .

We s ay that X . i s pos itive if all the n (i ) 1 s ar e > O . Let 1 a v + Supp ( m ) , and let F E S m (Q) be the c or r e s ponding Frobenius v e leITlent, d . 2 . 3 . Sinc e Sn = Sm / Tm i s finite , ther e exists an N . mte g e r N > I s uch that F E T (Q) . If Ev is the priITle ide al m v N of v , this ITleans that ther e e xi s t s {)( E K.... with .P = ({)( ) , v {)( == 1 ITlod m , and {)( > 0 at all r e al plac e s of K . ...

1

AB E LIAN

ll - 3 6

PRO POSITIO N 1 - The e i g env alue s of n (i) = , n) . IT (a ) cr (i = 1 , X . (a ) 1

cr cr

•

•

ar e the numbe r s

.

FNv

This is tr ivial by c on s truc tion , b e c aus e unde r T (Q)

�

- ADIC REPR ES E NTATIO NS

T m (Q ) .

a

is the image of

(F

-

are { pv } - units (i . e . The e i g- e nvalue s of 41 v ) the y are units at all plac e s of Q not div iding p ) . v CORO LLARY 1

CORO LLARY 2 - Let z l ' . . . ' Z b e the e ig envalue s of 41 (F v ) , n N Le t w b e a place o f Q dividing p , indexe d s o that z . = X . (a ) . -v 1 1 Then w (z . ) = I: n (i) . normalized s o that w (pv ) = v (pv ) = e v 1 --

N We have w (z . ) 1

=

w (IT cr (a) cr E r

n (i ) cr

) =

I:

cr E r w. cr =v

cr

n (i )w o cr (a) , and

cr E r cr

w . cr (a )

= 0

if

w o cr (a )

=

N

if

wo Wo

cr

cr

F

v

=

v,

N s ince (a ) = Ev . Henc e the r e s ult . COROLLARY 3

41 1 : Gal cKl K )

ciate d to 41.

-

�

Let 1 b e a pr im e numb e r and let Aut (Vl ) b e the 1 - adic r epr e s entation of K as s o

Then 41 1 i s inte g r al (d . Ch. I, 2 . 2 ) if and only if all the charac te r s X . oc c ur r ing in 41 are pos itive . 1

P r oof of C o r ollar y 3 . As s ume fir s t the

X .

1

'

s ar e pos itive .

Let

v • Supp ( m ) and let z ' . . . ' z b e the c o r r e s p onding e igenvalue s of l n

THE GRO UPS S

II - 3 7

m

F

a s in C o r ollar y 2 . C or ollar ie s l and 2 s how that the w (z 1. ) ar e v pos itive for all valuations w of Q ; henc e the z . ar e integr al ove r 1 Z . Henc e the 4> l ' s ar e integ r al. C onve r s ely, as s um e 4> 1 I S integ r al for s ome 1 . The r e exi s ts a finite s ub s e t S ' o f 1: K , c ontaining Supp ( m ) ) such that if v t s ' , the e ig e nvalue s of 4> (F v ) ar e integral. Choo s e a pr ime number p

which s pli t s c o mp lete ly in K and i s such that p == p im p li e s v v � S ' . Let w b e a valuation of Q div iding p . The valuation s w . a,

a E

r , ar e

pairwi s e ine quivalent.

a

Let

E

r ; and let v b e

t h e normal i z e d valuation of K e quivalent to w o a s o that ==

w o a for s om e

X, >

Let z " ' " z b e the e igenvalue s of n l 4> (F ) . By Cor ollar y 2 , w (z . ) = x'n (i) . S ince the z . are integ r al , a 1 1 v thi s shows that the n (i ) ' s ar e all po s itive . x'v

O.

a

PROPOSI T IO N 2

�

Le t X

=

T

E

-

Let v

X (�

i

Supp ( m ) and le t X

be a characte r of

) b e the r e s tr iction of X

to

1ln

and let

j (X T ) b e the inte g e r define d in 3 . 2 . Then , for any archime dian ab s olute value w of Q extending the u s ual ab s olute value of Q , i

we have

P r oof.

If x = a

1: [ a ] + 1: b [ 0] a s in 3 . 1 , we have a aer ae r

w (X (F ) ) N = w (X (F N ) ) v v

=

IT w .. a (a ) a

a

.

IT w o a (a ) a

b

a

,

AB E LIAN 1 - ADIC RE PR ES E NT A TrO NS

II - 3 8

and

IT W O (a) a O" 0"

= W (N (a»

a

remains to s how that x =

= Nv 0

0"

0

c = w,

w

b

iN/ 2 , whe r e

0"

(d .

B ut y henc e

x

2

c = c

W

0,

c O" b = IT w o c oT (a) T T 0"

0"

we have y = x,

Le t

3 . 1) . Sinc e b + b

b

It

= 2a.

is e qual to 1 .

IT w oO" (a) cO"

we have x. y = 1 with y =

w

= Nv

IT W cr(a)

b e th e I I F r ob e nius I I attache d to

and , s inc e

aN

= I,

=

and x = I ,

s ince x > O . Exe r c is e s 1 ) Check the pr oduc t formula for the e ig envalue s of th e cP (F ) . v (Us e Cor . 1 and 2 to P r op . 1 and P r op . 2 . ) 2 ) Show that P r op . 2 and Cor . 1 and 2 to Prop . 1 de termine

the e igenvalue s of the cp (F ) ' s up to multiplic ation by r o ots of unity . v 3 ) (Gene r alization of C or . 1 to Pr op . 1) . Let (p 1 ) be a s tr ictly c ompatible s y s te m of r ational 1 -adic repr e s entat ion s , with

exc eptional s e t S (d . Chap . I, 2 . 3 ) . the e igenvalue s of F

v, P1

,

1 f:. p

v

'

Show that , for any v E �

K

- S,

are p - units . v

APPENDIX Killing ar ithmetic gr oups in tori

A. 1 .

Ar ithmetic g r oup s in tor i Let A b e a l ine ar algebraic gr oup ove r Q ,

and let

s ubgr oup of the g r oup A (Q ) of rational po ints of A.

r be a

Then r is

s aid to b e an ar ithme tic s'ubgr oup if for any algeb raic emb e dding

THE GRO UP S

II- 3 9

m

A C G L (n arb itr ary) the g r oups r and A (Q) n G L (Z ) ar e c om n n mensur able (two s ubg r oup s r l ' r 2 are s aid to b e c o mm ensurable if r l n r is of finite index in r and r ) . It i s we ll -known that it 2 1 2 s uffic e s to che ck that r and A (Q) n GL (Z ) ar e c ommensurable for n one embe dding A C GL n . ----

Example Le t K b e a n umb e r field and let E b e the gr oup of units of K . Then E i s a n ar ithmetic s ubgr oup o f T = R / � ) . K Q m If

T i s a torus ove r Q ,

let T

O

b e the inte r s e ction of the kernels

of the homomorphisms of T into G O

m

The torus T is s aid to b e

"anis otropic if T = T ; in te r m s of the char ac te r g r oup X = X (T ) this means that X has no non - z e r o e l e m ents which are left fixe d by G = Gal (Q / Q) . T HEOREM - Let T b e a torus ove r Q , and let r b e an ar ithmetic o s ub g r oup of T . Then r (\ T is of finite index in r , and the quo O O tient T (R ) /r () T i s c o mpac t .

This is due to T . Ono ; for a pr oof of a m o r e g ene r al s tatement (" Godement ' s c onj e ctur e " ) s e e Mos tow - T amagawa [18] .

CORO LLARY - Le t T b e a torus ove r Q ,

and let r b e an ar ith

metic s ubgr oup of T . If T is anis otr op ic , the n T (R) / r i s c o m pact.

II - 4 0

AB E LIAN

1

- ADIC R E PR ES E N T AT IO NS

Exe r c is e Let T b e a torus ove r

with char ac te r group X.

Q,

a) Show that T (Q ) = Hom

Gal

- * (X , Q )

_ �c

b ) Let U b e th e s ubg r oup of Q g eb r a ic units of Q .

.

who s e e lement s are the al -

Le t r = Hom

Gal

(X , U ) .

Show that r is a n ar ithmetic s ubgr oup o f T (Q) and that any ar ith metic s ubgr oup of T (Q) i s c ontained in r A. 2 . Killing ar ithm etic s ubgr oups Le t T b e a torus ove r g r oup; put Y (T ) = X (T ) � Z Q

Q,

and let X (T ) be its characte r

Let

A

be the s e t of c las s e s of

Q - ir r e duc ible r e pr e s e ntations of G = Gal {Ol Q) thr ough its finite quotient s . For e ach

h.

E A,

let Y

Xo

b e the corre sponding is otypic

s ub -G -module of Y , i . e . the s um of all s ub -G -module s of Y is omorphic to

o

Xo .

O n e has the dir ect s um dec ompos ition

whe r e 1 is the unit r e pr e s e nta1 : on of G ; let Y b e l the sum o f tho s e Y whe r e for all the infinite 'rob enius e s c E C Xo 00 + (d . 3 . 1) we have Xo (c ) = - 1 ; let Y b e the s um of the other Y ' Let Y

We h ave

= Y '

Xo

T HE GRO U P

S

II-41

m

y O = yG = { y e y ! gy = y y-

=

y = Y

Note that

{y e Y ! cy

o

=

-y

if and only if

for all for all

g C

E G} E

C }, 00

is ani s otr opic .

T

��

and H = { l , c } , then , s inc e T (R ) = Hom (X (T) . C ) , H we s e e that T (R ) i s c ompac t if and only if Y = Y If

c E

C 00

PROPOSITION - Let r be an ar ithmetic subg r oup of the torus T , and r its Zariski c lo s ur e

(d.

1 . 2) .

Then:

(* ) [Sinc e the torus T /r is a quotient of T , we identify Y (T / r ) with a s ubmodule of Y (T) . ] Pr oof. Suppo s e fir st that Y is ir r e duc ible , i . e . that T has no p r oper s ubtori and is 1= o. If Y =

yO,

then T is is omorphic to

This shows that Y (T i f )

=

G

Y (T ) , hence

m

(* ) .

and

h enc e

If Y

=

-

r

Y ,

is finite . then T (R)

i s compact. Sinc e r is a dis c r e te subgroup of T (R ) , it is finite . Hence Y (T i f ) = Y (T ) and (* ) follows . =

Y + , then T (R) i s not c ompact . C ons e quently , r is infinite s ince T (R ) / r is c ompact by Ono ' s the or em. Hence r is If Y

an algebr aic subgr oup of T of dimens ion > 1 . Its c onne cted c om ponent is a non - tr ivial subtorus o f T . This shows that henc e Y (T/r)

=

O.

Henc e again (* ) .

r=

T,

II - 42

AB E LIAN l - ADIC REPRESE NTAT IO NS The g en e r al c a s e follows e a s ily fr om the ir reduc ible one ; for

ins tanc e , choo s e a torus T '

to T which s plits in dir e c t product of i r r e duc ible to r i and note that r is c ommen s urable with th e imag e by T '

�

T of an ar ithme tic s ubgr oup of T .

Exe r c i s e Y

Let y by G .

E

Y.

Define Ny as the me an value of the tran s form s of

a Pr ove that N is a G -line ar pr oj e ction of Y onto y O , hence Ker (N) = Y

C HAPTER III LOCALLY ALGEBRAIC AB E LIAN R E PRESE NTATIONS

In

this Chapt e r , we define what it m e ans for an ab el ian

1 -adic

repr e s entation to be locally alg ebraic and we p r ove (c!. 2 . 3 ) that s uch a repr e s entation , when r ational , c om e s fr om a linear repre s entation of one ()f the g r oup s S

m

of Chapt e r

II .

When the g r ound fie ld is a c ompo s ite o f quadr atic extens ions of

Q , any r ational s em i - s imple

1 - adic r e pr e s e ntation is ip s o fac to

locally alg eb r a ic ; this is p r oved in § 3 , as a c on s e quenc e of a r e s ult on tran s c endental numb e r s due to Siegel and Lang . In the local c as e , an ab elian s e mi - s imple r e p r e s entation is loc ally alg eb r aic if and only if it has a " Hodg e - T ate de c ompos ition " . This fact , due to T ate (C oll e g e de Franc e , 1 96 6 ) , is proved in the

Appendix , tog eth e r with s ome c omplements . §l.

T HE LOCAL CASE

1. 1 . Definitions

Le t p be a p r ime numb e r and K a finite e xtens ion of Q

let T

=

RK I

Q

(G p

ml

p

K ) b e the c o r r e s ponding algebr aic torus over

AB E LIAN l - ADIC REPRES E NTATIO NS

III - 2 Q

(d .

Weil [43 ] , Chap . I) . L e t Y b e a finite dim e n s i onal Q p -ve ctor s pace and de note , as us ual , b y GLy the c o r r e s ponding linear gr oup ; it is an algebr aic p

g r oup ove r Q , and GLy (Q p ) = Aut (V ) . p ab Let p : Gal (KI K) � Aut (V ) b e an abelian p - adic repr e s en tation of K in y , whe r e Gal {K I K)abde note s the Galois g r oup of the * ab is the maximal ab elian extens ion of K . If i : K � Gal (K I K) c anonic al homomorphism of local clas s field the ory

(d .

for ins tanc e

C a s s e ls - F r �hlich [6 ] , chap . VI , § 2 ) , we then get a c ontinuous homo * morphism p o i of K = T (Q ) into Aut (V) . P

D EFINI TIO N - The repr e s entation p is s aid to b e locall y alg eb raic if the r e is an algebraic morphism r: T � GLy such that * -1 P 0 i (x) = r (x ) for all x E K clo se e nough to 1 . Note that , if r : T

�

GL y

s atis fie s the ab ove c ondition , it

is uni q ue ; this follows fr om the fact that any non -empty open s e t of * K = T (Q ) i s Zariski dens e in T . We s ay that r is the alg ebr aic P

morphism as s oc iat e d w ith

p.

Exampl e s and dim Y = 1 , s o that p is g iven by a p ab c ontinuous homomorphism Gal (Q I Q ) � U whe re U is p P P P the g r oup of p - adic units . It is e a s y to s e e that the r e exi s ts an elem ent li E Z s uch that p o i (x) = xII if x is clos e enough to 1 . p The repr e s entation p is locally algebraic if and only if II b elong s 1)

to Z .

T ake K = Q

This happens for ins tanc e when

y

= y (f.L) ,

cf. Chap . I, p 1 . 2 , in which c a s e II = -1 and r is the canonical one - dimens ional r e p r e s entation of

T =

G lQ m

p

LO CALLY ALGEBRAIC R EPRESE NTATIO NS

III - 3

2 ) The ab e l ian r e pr e s entation

as s oc iate d to a Lub in - T ate forrnal g r oup (ef. [ 1 7 ] and [ 6 ] , Chap . VI , § 3 ) is locally alg ebraic (and al s o of the form u

�

u

-1

on the ine r tia g r oup ) .

PROPOSITION 1 - Let p : Gal (K / K) ab � Aut (V) be a loc a lly alge b r aic ab e l ian r e p r e s entation of K. The r e s tr iction of p to the ab ine rtia s ub g r oup of Gal (K / K ) i s s emi - s imple . Let us identify the ine r tia s ubgroup of Gal (K/ K)

ab

with the

gr oup U K of units of K. By as s umption , the re is an open s ubgroup U ' of U and an algebr aic morphism r of T into GLy s uch K -1 that p (x ) = r (x ) i f x E U ' . L e t W be a sub -vector spac e o f Y stable by

p (U ) ; it i s then s table by p (U ' ) , hence by r (T ) . But K eve ry linear repr e s entation of a torus is s e mi - s imple . Henc e , the r e

exists a pr oj e c to r T . I f w e put 1T '

=

1T : Y � W which c ommute s with the action of 1 -1 p (S ) 1T p (s ) , we obtain a p r o 1: (U K : U' ) S E UK / U '

� W which c ommute s with all p (s ) , S E U K ' q . e . d. Conver s ely, let us s tart from a repr e s entation p who s e re

j e ctor 1T' :

Y

s tr ic tion to U K is s emi - s imple . If we make a s uitable larg e finite extens ion E of Q ' the r e s tr ic tion of p to U K may be br ought p into diag onal form , i. e . is g iven by c on tinuous characte r s *

X

: U � E , i=l , . . . , n . We as sume E larg e enough to c ontain i K all c onj ug ate s of K , and we denote by r K the s et of all Q - ern

b e dding s of K into E . Rec all (ef . chap . II , 1 . 1 ) that the [ a) ,

a

E r K ' make a b a s i s of the char acte r g roup X (T ) of T .

PROPOSITION 2 - The repr e s entation p is loc ally algebraic if and only if the r e exis t integer s n a (i ) s uch that

AB E LIAN l - ADIC REPR E S E NTATIO NS

III - 4

x . (u) = TT 1

aE r K

a (u)

- n (i ) a

for all i and all u c l o s e enough to 1 . The n e c e s s ity i s tr ivial . C onve r s ely, if the r e exis t such in n (i) of te g e r s n (i ) , the y define alg eb r a ic charac te r s r . = TT ( a ] a a

1

T , henc e a l ine ar r e pr e s entation r of T / E . It i s c lear that ther e -1 is an ope n s ub g r oup U ' o f U ' such that p (u) = r (u ) for all K

Henc e it r emains to s e e that r can b e defined ove r Q p (c f. chap . II , 2 . 4 ) . B ut th e tr a c e a = � r . o f r (loc . � ) is 1 r s uch that a (u) E Q for all u E U ' . S inc e U ' is Zariski - dens e in p r T , this implie s that e is " defined ove r Q " hence that r r p c an b e defin e d ove r Q (loc . c it . ) , q . e . d. p

u

E

U' .

-

Extens ion of the g r ound field Let

K'

b e a finite extens ion of

K,

p'

and let

b e the r e

s tr ic tion of the g iven r e p r e s entation p to Gal (K I K ' ) . Then p '

i s locally algebr a ic if and only p is ; mor e ove r ,

if

this is s o , the

as s oc iate d algeb raic morph i s m s r: T ar e s uch that r ' K'

N Q r , K' / K

, : T' � K / K from K ' to K .

and

norm

=

N

-+

G Ly , r ' : T '

-+

G�

whe r e T ' i s the torus a s s o ciate d with

T i s the algebraic mo rphism define d b y the

All this follows e a s ily fr om the commutativity of the diag r am

LO C AL L Y ALGEBR AIC R EPRESENTAT IONS

III - 5

i

K'

':c

--+

ab Gal {K / K ' l

and fr om the fac t that the kernel of N K , for the Z ariski topology .

/

K: T '

--+

T i s c onnec ted

Exe r c i s e Give an example o f a loc ally alg ebr aic abel ian p - adic repre s e ntation of dimens ion 2 which is not sem i - s imple . 1 . 2 . Alternative definition of " l oc ally algeb r a ic " via Hodg e - T ate module s Let us r e c all fir s t the notion of a Hodg e - T ate module § 2 ) ; here

K

[ 2 7] .

i s only as s umed to be c omple te with r e s pect to a dis

c r ete valuation , with pe rfe c t r e s idue fie ld k and char (K ) char (k) = p . Denote b y C the c ompletion of

(d

�

= 0,

of the algebraic closur e

K

K.

T h e gr oup G

=

Gal (K /

K)

acts c ontinuous ly o n

K.

This action

extends c ontinuously to C . Let W be a C -ve ctor spac e of finite dimens ion upon which G acts c ontinuously and s em i - linearly acc or d ing t o the formula s (cw)

s (c ) . s (w)

(s

E

G, c

E

C and w

E

W) .

G � U b e the homomorphism of G into the g r oup p U = Z of p adic units , define d by its action on the p v_ th r oots p p of unity (d. chap . I , 1 . 2 ) : Let

X :

=

*

III - 6

AB E LIAN 1 - ADIC REPR ES E N T A TIO NS s (z ) Define for every i W

i

=

{w

=

E

E

z

X (s )

if s

E

G and zP

v

=

1.

Z the s ub s pa c e w i sw

=

i X (s ) w f o r all s

E

G}

i o f W . This i s a K -vecto r s ub s pac e of W . Le t W (i) = C (8) K W This i s a C -v e ctor s pa c e upon which G acts in a natur al way (i . e . i b y the formula s (c � y) = s (c ) � s (y) ) . The inclusion W � W extends uni q uely to a C -l ine ar map a . : W (i ) --:> W , which c om 1

mute s with the ac tion of G .

PRO POSITION (Tate ) - Let .u W (i) be the direct s wn of the W (i) , i

E

ab ove .

Z.

Le t

Then

a

a:

.u W (i) --:> W b e the sum of the

is inj e ctive .

a. I

1

s defined

F or the p r o of s e e [ 2 7] , § 2 , p r op . 4 . i CORO L LAR Y - T h e K - spac e s W (i

E

Z ) are o f finite dimens ion .

They ar e line arly independent ove r C . DEFINITION 1 - The module W i s of Hodg e - T ate type if the homo morphi s m

a:

11 W (i ) --:> W i s a n i s omorphism. iE Z

Let now V be as in 1 . 1 , a vector space ove r Q , of finite dimen p s ion . Let p : G � Aut (V ) b e a p - adic repr e s entation . Let W

=

C Q!)

Qp

V

and let G act on W by the formula

III - 7

LO CALLY ALGEBRAIC REPRES E NT AT IONS 5

(c

� V) = 5

(C )

� P

(s ) (v ) ,

5 E

G, c

E

C,

V E V.

DEFINIT IO N 2 - The r e p r e s entation p i s of Hodge - T ate ty p e if the C - spac e W = C Q9 Q V i s o f Hodg e - T ate type (d . def. 1) . p E x aInple

Let F b e a p - divis ible gr oup of finite he ight (d . [ 2 6 ] , [3 9] ) ; it . ) and V = Q Qg T . The gr oup G let T b e its T ate Inodule (lo -c . cp acts on V , and Tate has pr ov e d ([3 9] , C or . 2 to Th . 3 ) that thi s Galois Inodule is of Hodg e - T ate type ; Inore p r e c i s e l y , one has W

=

W ( O)

ED

W (l ) , whe r e W

=

C

�

V as ab ove .

its T HEOREM (Tate ) - As sUIne K i s a finite extens ion of Q (i . e . p r e s idue field is finite ) . Let p : G � Aut (V ) b e an abe lian p - adic r e pr e s entatior: of (a)

p

K.

The following p r opertie s are equivalent:

is locally algebr aic (d. 1 . 1) .

(b ) P is of Hodg e - Tate type and its r e s tr ic tion to the ine rtia g r oup is s e Ini - s iInple . F or the proof, s e e the Appendix. § 2 - T HE G LOBAL C ASE 2 . 1 . Definitions We now go back to the notations of chap . nUInber field. Let

1

-

1

- adic

i. e .

K

denote s a

b e a pr iIne nUInbe r and let p : Gal (K / K )

be an ab elian

II ,

ab

-?

repr e s entation of

Aut (V 1 ) K.

Let

v E

�

K

be a plac e

AB E LIAN 1 - ADIC REPRESENTA TIO NS

III - 8

ab b e th e of K of r e s idue characte r i s tic 1 and let Dv C Gal (KI K ) c o r r e s p onding de c Olnpos ition gr oup . This gr oup is a quot ient of the ab (the s e two g roup s ar e , in fac t , is o l o c al Galois g r oup Gal (K 1 K l v v m o r phic , b ut we do not ne e d this he r e ) . Henc e , we get by c ompo s i tion an 1 - adic repr e s entation of Kv p : Gal (K 1 K )

v

v

v

ab � D -4 Aut (V 1 ) . v

DEFINI TION The repre s entation p is s aid to be loc ally alg eb r aic if all the loc al r e pr e s entations p , with P = 1 , are locally alge v v br aic (in the s en s e defined in 1 . 1 , with p = 1 ) . -

--

It is c onvenient to r eformulate this de finition , us ing the torus

0 -torus obtaine d fr om T by extend ing the g r ound fie ld fr om 0 to 1 0 . W e have 1

= K &? 0 . 1 1 Let I b e the id� le gr oup of K , cf. Chap . II , 2 . 1 .

whe r e K tion K i: I

�

*

� I, followe d by the c las s fie ld homomorphism 1 ab Gal (K I K ) , define s a homomorphism

* ab i : K l � Gal (K / K ) . 1

The inj e c -

III - 9

LOCALLY ALGEBRAIC REPRESENTAT IO NS

PROPO SIT IO N - The r e pr e s entation p is loc ally algeb r a ic if and only if the re exi s ts an alg eb r a ic morph i s m f: T / Q

1

� G LV

-1 s uch that p o i 1 (x) = f (x ) for all

X

E

1 *

K 1 c lo s e enough to 1 .

(Note that , as in the local c as e , the above c ondition determine s f unique ly; one s ays it is the algebraic morphism as s o c iate d with p. )

Sinc e K I1O Q Q 1 = TT K vl 1 v T/

Q1

=

,

we have

TT Tv

vl 1

is the Q 1 -torus define d by Kv v tion follows fr om this de c ompo s ition.

whe r e T

cf. 1. 1.

The propos i-

Exe r c i s e Give a cr ite r ion for loc al algebr aic ity analogous to the one of P r op . 2 of 1 . 1 .

2 . 2 . Modulus of a locally algebr aic abe lian r e pr e s entation ab Let p : Gal (K I K ) � Aut (V 1 ) be as ab ove ; by c ompos ition with the clas s field homomorphism i : I � Gal (KI K) ab , p define s

a homomorphism p o i : I

�

Aut (V 1 ) .

AB E LIAN 1 - ADIC REPR ESENT A TIO NS

III - I O

We as s wne that p is loc ally alg ebr aic and we denote b y f the a s s oc iate d alg ebraic morphism DEF INITIO N P

-

Let

is de fine d mod

m

I Q1

�

G LV

•

1

b e a modulus (chap . II , 1 . 1 ) . (or that

m

T

One s ays that is a modulus of definition for p )

m

if (i ) p .. i is tr ivial on U V, m (ii ) P I) i l (x) = f (x - l ) fo r x (Note that IT vl l

In

U

v, m

is

an

E

p 1= 1 . v

when ---

IT

vl l

U

V, m

ope n s ubg r oup of K

�

=

T

/ Q1

(Q l ) . )

order to prove the exis tenc e of a modulus of definition , we

ne e d the following auxiliar y r e s ult: PROPOSITIO N - Let H b e a Lie gr oup ove r Q 1 (re s p . R ) and let a be a c ontinuous homomo rphis m o f the id�le gr oup I � H. *

(a) If p 1= 1 (re s p . p 1= (0) , the r e s tr iction o f a t o Kv - v v * i s equal to 1 on an open s ubgr oup of K . v * (b ) The r e s tr iction of a to the unit group Uv of Kv is equal to I for alm o s t all v ' s . Part (a) follows fr om the fact that K

*

is a pv - adic Lie gr oup v and that a homomorphism of a p - adic Lie gr oup into an 1 - adic one is loc ally equal to 1 if p 1= 1 . T o prove (b ) , let N b e a ne ighb orhood of 1 in H which c on tains no finite s ub g r oup e xc ept { l } ; the existence of s uch an N is c las s ical for r e al Lie group s , and quite easy to pr ove for l - adic one s . By definition of the idMe topology , for alm o s t all v ' s . B ut (a) shows that , if

a

(U )

is c ontained in N p 1= 1 . the g r oup v v

LO CALLY ALGEBRAIC REPRESE NTAT IONS i s finite ; hence

a (U )

v

a (U )

v

III - ll

= { l } for alm o s t all v ' s ,

q . e . d.

CORO LLAR Y - Any abe lian 1 - adic r e pr e s e ntation of K is unr am i fie d outs ide a finite s e t of plac e s . This follows fr om (b ) applied to the homomorphism induc ed by the g iven r e pr e s entation , s in c e the the ine r tia s ubg r oups .

a (U )

v

a

of I

are known to b e

Remark This doe s not extend to non - abel ian r e pr e s entations (even s olv able one s ) , cf. Exe r c i s e . -

PROPOSITIO N 2

Eve ry loc ally algebraic ab elian 1 - adic repr e s enta

tion has a modulus of definition. ab Gal (K I K ) � Aut (V 1 ) b e the g iven r e pr e s entation and f the as s oc iated morphism of T into G L Let X b e the V1 I Q 1 Let

p:

p '1= 1 , for which p is r amified; the v cor ollary to Prop. 1 shows that X is finite . By Pr op . 1 , (a) , we can s e t of plac e s

v

E

choo s e a modulus

l;

K

m

'

with

s uch that

p o i:

I

�

Aut (V 1 ) is tr ivial on all

, v E X . Enlarg ing m if nec e s s ar y , we c an as s ume that v ,m Henc e , m is a modulus of p o i 1 (x ) = f (x - 1 ) for X E IT U , v m Pv = 1

the

U

definition for

p .

R emark It is eas y to show that the r e is a s mall e s t modulus of definition for p ; it is calle d the c onductor of p .

IIl - l 2

AB E LIAN 1 - ADIC REPR ES E NTATIONS

E xe r c i s e ...

',

Let z ' . . . ' z ' . . . E K o F o r e ach n , let E b e the s ub l n n field of K generate d b y all the 1 n - th roots of the element 1 1 n-l zlz 2 · · · z n a) Show that E is a Galois extens ion of K , c ontaining the n 1 n - th r o ots of unity and that its Galois group is is omo rphic to a s ub g r oup of the affine

group (� � ) in GL (2 , Z /

1 nZ) .

b ) Let E be the union of the E I S . Show that E is a Galois n e xtens ion of K , whos e Galois g r oup is a clos e d s ubgr oup o f the affine g r oup r e lative to Z 1 . c ) Give an example whe r e E (and he nc e the c o r r e s ponding

2 - dimens ional 1 - adic repr e s entation) is r amifie d at all plac e s of K . 2 . 3 . B a c k to S

m

Let

m

b e a modulus of K and let

be a line ar r e p r e s entation of S

m

/Q

Let

1

ab

�

Aut (V 1 )

b e the c o rr e s ponding l - adic r e pr e s entation

(d .

chap . II, 2 . 5 . ) .

T HEOR E M 1 - The repr e s entation

LO C A LL Y ALGEBRAIC R EPRESE NTATIO NS f:

i s 4>

0

1( ,

whe r e

1(

T/ Q

1.

�

G LV

III - 1 3

1.

denote s the c anonical morphis m o f T into S

m

(ef. chap . II , 2 . 2 ) . This i s tr ivial fr om the c on s tr uction of

4> 1 a s 4>

(chap . II, 2 . 5 ) and the c or r e s ponding pr op e r tie s of 2. 3) . The conve r s e of The orem I is true .

£

1

'"

£

1 (chap . II ,

We s tate it only for the

c a s e of rational r epr e s entations : ab T HEOR EM 2 - Le t p : Gal (K I K) � Aut (V 1 ) be an abelian 1 -adic repr e s entation of the numbe r field K . As s ume p i s rational

(chap . I, 2 . 3 ) and is locally algebr aic with m

as a modulus of de

finition (cf. 2 . 2 ) . Then, the r e exi s t a a -vec tor s ub spac of e V0 V 1 ' with V 1 = V 0 Q!)a Q 1 ' and a morphism rP o : Sm � G LV o of a - algebr aic gr oup s such that p is e qual to the 1 - adic r epr e s en tation rP

l

as s oc iate d to rP o (ef. chap . II , 2 . 5 ) .

(The condition V 1 structur e " on V 1 ' ef. Pr oof.

Let r: T p .

c iate d with p

(>

/a

1

�

=

V0 @

a

Q

B o u rb a k i

GL

V1

�

means that V 0 s a " Q r Alg . , chap . II , 3 ed. )

1

b e the algebr aic morphism as s o -

We have

i (x)

=

-1 r (x ) for x

•

E

K n

�

Um

=

IT vi i

U

V ,m

III-14

l - ADIC

AB E LIAN

Define a map

rj;:

I

�

Aut (V L ) by rj;

whe r e x

L

is the

d i ate ly that rj;

R E PR E S E NTATIO NS

(x)

=

p o i (x) . r (x )

L

th c omponent of the id� le x. One che cks imme L-

i s tr iv ial on

Henc e r is triv ial on

Um J•

=

Em

and c o inc ide s w ith

K'" n

b r aic morphi sm rm : T m / QL

�

Um

on

K

�'<

and factor s through an alge

GLy

of the Q 1 - algebraic gr oup S m / 1 Q

r

. By the unive r s al property

L

(ef. chap . II , 1 . 3 and 2 . 2 ) ,

the r e exists an algebraic morphism 41: S

m / QL

�

G Ly

L

with the following propertie s : (a) The morphi sm T

m /

Q

1

�

S

m

/

Q

1

L

G Ly

L

is rm

(b ) the map I � Sm (Q L ) .l...,. Aut (V L ) is rj; . It is tr ivial to check that the L - adic repr e s enta tion 41 1 attache d to 41 as above coinc ide s with p . Ind e e d , if a E I, we have (with the notations of chap . II)

=

-1 rj; (a ) 4J (tr L (a L ) )

= p o i (a )

III - I S

LO CALLY ALG EBRAIC REPR ES E NT ATIONS s inc e q,

0 7r 1

=

r b y (a) ab ove .

Henc e q, 1

=

p ; the fac t that p is r ational then impl ie s that q, V

c an b e defined ove r Q (chap . II , 2 . 4 , P r o p . ) ' and th is g iv e s and q, , q . e . d . o

o

Remark The s ub s pace V 0 of V

c on s tructe d in the pr oof of the 1 the o r em is � unique ; howev e r , any other choic e g iv e s us a s pac e whe r e cr i s an automorphism of V 1 c ommuting of the form crV 0

with p .

To a g iven V o c o r r e s ponds of c our s e a unique q, .

CORO LLAR Y 1 - For each prim e numbe r l ' the r e exis ts a unique (up to is omorphism) pl '

l'

- adic r ational semi - s imple r e pr e s entation

of K, c ompatible with

p.

It is ab el ian and loc ally algebraic .

The s e r epr e s entations form a s tr ictly c ompatible system 2 . 3 ) with exc eptional s e t c ontained in Supp ( m ) .

b e r of l '

Pl

'

(d .

chap . I,

For an infinite num -

c an b e b r ought in diag onal fo rm .

P r oof. The unic ity of the P l ' follows fr om the the or e m of chap . I ,

2 . 3 . F o r the exi s tenc e , take P l ' to b e the q, l ' as s oc iate d t o q, a s i n chapter II , 2 . 5 . The r emaining as s e r tion follows from the pr opo

s ition in chap . II , 2 . 5 . COROLLAR Y 2 (v � Supp ( m ) , p

-

The e ig envalue s of the Fr ob enius e lements F f:.

1 ) gener ate a finite exte ns ion of Q .

v This follows fr om the c or r e s ponding prope r ty of q, 1 chapter II, 2. S , Remark 1:

'

v, P

d.

III - l6 2. 4.

AB E LIAN l - ADIC REPR ES E NTATIO NS A mild g e n e r al i z at ion Mo s t r e s ults of th is and th e p r e v ious Chapte r may be e xtende d

to the c a s e whe r e we take fo r g r ound fie ld of the l ine ar r e p r e s enta tion a numb e r field E (in s tead of Q) .

Mo r e pr e c i s e l y , let

X,

be a

b e the x' - adic c omple tion of E . The x, notion of an E - r ational x' - adic r e pr e s e ntat ion of K ha s b e e n de finite plac e of E and l e t E

fine d in chap . I , 2 . 3 , Remark.

Let

p: Gal (K / K) � Aut (V )

x,

b e such a r e pr e s e ntation , and a s s ume p is ab el ian.

Let 1 b e th e

s o that E x. c ontains Q 1 . As in 2 . 1 , w e s ay that p i s loc ally alg ebr aic if the r e exists an alg eb r aic r e s idue char a c t e r i s tic of x' ,

morphism f: T / � GL V X, E X, s uch that p o i 1 (x)

K

�

=

-1

f (x ) for x

E

*

K 1 c l o s e enough to 1 (note that

= T (Q l ) is a s ub g r oup of T (E » As in 2 . 3 , one pr ove s that x, s uch a p c om e s fr om an E - line ar r e pr e s e ntation of s ome S m (and c onve r s e ly) .

2. 5.

The function field c a s e The ab ove c on s tr uc tions have a (r ather e lementary) analogue

for function fie lds of one var iable ove r a finite fie ld: Let K b e s uch a fie l d , and let p b e its char acte r is tic . If

m

is a modulus for K (i . e . a pos itive divis o r ) we define the subgr oup Um

of the id� l e g r oup I as in chap . II , 2 . 1 , and we put

LO CALLY ALGEB RAIC REPRESE NT ATIO NS rm = I/

III -l 7

U

m

The deg r e e map de g : I � Z is tr ivial on exac t s e quenc e

U

m

,

hence defin e s an

l � Jm � r � Z � l . m One s e e s e a s ily that the g r oup J m i s finit e ; mor e over , i t may b e interpreted a s the g r oup of r ational points of the " g ene r aliz e d "

Jacob ian var iety define d b y m " . If rm

denote s th e c ompletion of

with r e s pe c t to the topology of s ubgr oups of finite index, it is "'known (c las s field theory) that Gal (K / K) ab � lim r . � m _ ab Let now p : Gal (K / K) � Aut (V 1 ) b e an abe lian 1 -adic rm

r epr e s entation of K, with

1 1=

p. One pr ove s as in 2 . 2 that the r e

exists a modulus m s uch that p i s tr iv ial o n Um that p may b e identified with a homomorph ism of

i. e .

such

Aut (V ). Mo r e over 1 PROPOSITION - A homomorphism � : rm � Aut (V 1 ) can be ex" tended to a c ontinuous homomorphism of rm if and only if there

exists a lattic e of V 1 which is stable b y

The ne c e s s ity follows from Remark

p (f'

1

m

).

of chap I, L L The

s uffic iency is clear . Note that , as in the numb e r field c as e , we have Frobenius elements and we can define the notion of r ationality of an repr e s entation . THEOREM -

An

ab elian 1 -adic r e pr e s e ntation

1

- adic

A B E LIAN 1 - AD IC REPR ESE NTATION S

III-18

of K is r ational if and only if '( E

r .

T r q, ( '( ) b e long s to

a

for ever y

m

If v

�

Supp ( m ) , and if f v is a uniformiz ing paramete r at v , the image F of f in rm is the Frob e nius e lement of the v v Henc e , if T r q, take s r ational value s on r Galois g r oup r m m the char ac te r i s tic polynomial of q, (F ) has r ational c oeffic ients for v all v � Supp ( m ) an d q, i s r ational . "

�

T o prove the c onve r s e , n ote fir s t that C eb otar ev ' s the o r e m (Chap . I , 2 . 2 ) i s valid f o r K , i f one us e s a s omewhat we ake r de finition of e q uipar tition . Henc e , the F r obenius elements F

are v and , fr om this ,

1\

dens e in rm In particular , the y g e ne r ate f'm one s e e s that T r p (y ) b elongs to s ome numb e r field E .

We can

then c on struct an E - l inear r e pr e s entation q, : r � GL (n , E) with m the s ame tr ac e a s p , and the the o r e m follows fr om: LE MMA - Le t r b e a fin ite l y g e ne r ated abel ian group , and q, : r � GL (n , E) a l ine ar r e pr e s entation of r ove r a numb e r field E.

Let

�

r , which is dens e in r for the top

be a s ub s e t of

olog y of s ubgr oups of finite index. As s ume that all

'( E

I: .

The n

Tr q, (y )

E

a

for all y

E

T r q, (y ) E

a

for

r .

Pr oof of the lemma . S inc e q, (r ) is finitely generate d , the r e is a

finite S of plac e s of E s uc h that all the elements of q, (r ) are

S - integ r al matr ic e s . If

l'

is a pr ime numbe r not divis ible by any

elem ent of S , the imag e of q, (r ) in GL (n , E � 0 1 , ) is c ontained i n a c ompac t s ub g r oup of G L ( n , E � a 1 , ) j henc e q, e xt e n d s by

LOC AL LY ALGEBRAIC REPR ESE NT AT IONS c ontinuity to

.'"

/\

q, : r

�

Q!)

GL (n ; E

III-1 9

0l ' )

"

whe r e r is the c omple tion of r for the topology of s ubgroups of ,.. ,... ,. finite index. S in c e � i s de ns e in r , it follows that T r q, (y ) ,.. b e long s to the adhe r en c e 0 1 , of 0 in E 0 0 , for eve ry y,. E r 1 Henc e , if y E r , we have T r q, (r )

E

E

nO

l'

=

0,

q . e . d.

Exe r c is e s 1 ) Let q, : s entation of

rm

�

Aut (V 1 ) b e a s emi - s imple l -adic repre -

Show the e q uivalence of: ,.. (a) q, extends c ontinuous l y to rm r m

(b ) F or every y E

r m

the e igenvalue s of

units (in a s uitable extens ion of 0 1 ) , (c ) The r e exi s ts y E rm , with

deg (y )

I: 0 ,

q, (y ) ar e s uch

that the e ig envalue s of q, (y ) are units . one has Tr q, (y ) E Z 1 . 2 ) Let q, : rm � Aut (V 1 ) be a rational 1 -adic repr e s entation of K . Show that, for alm o s t all p r ime nwnb e r 1 ' , the r e i s a r ational l ' -adic repr e s entation of K c ompatible with q, . Show that (d) F or eve ry y E ,..

this holds for all for all y E of

r m

l ' I:

r m

p if and only if the following pr operty is valid:

, the c oeffic ients of the characteris tic polynomial

q, (y ) are p - integer s .

.

lll - � U

AB E LIAN 1 - AD IC REPRES E NTAT IO NS

§ 3 . T HE C ASE O F A CO MPOSITE OF Q UADRAT IC FIE LDS

3 . 1 . Statem ent of the r e s ult The aim of th is § i s to pr ove : THEORE M - Le t p b e a r ational , s emi- s imple , 1 - adic abe lian repr e s entation of K. As s ume (':' ) K i s a c ompo s ite of quadratic exten s ion s of Q . Then p is locally algebr aic (and henc e s terns fr om a linear r e pr e s entation of s ome S m

d.

2. 3).

This appl ie s in particular when K = Q o r when K i s q uadr atic over Q . Remarks 1 ) An analogous r e s ult holds for E - rational s em i - s imple ab el ian "- - adic repr e s entations

(d.

2 . 4) . 2 ) It is q uite likely that c ondition (* ) is not ne c e s s ary. But

p r oving thi s s e em s to r equire s tr onge r r e sults on tr ans c endental numb e r s than the one s now available . 3 . 2 . A c r ite r ion for local algebraic ity ab PROPOSITIO N - Let p : Gal {Kj K)

�

Aut (V 1

)

be a rational s em i -

s imple 1 - adic abelian repre s entation of K. As sume that the r e N exists an inte g e r N � 1 s uch that p is locally alg ebraic . Then p i s loc ally alge br aic .

III - 2 1

LO CALLY ALG E B RAIC REPRESE NTA T IO NS P r oof. S inc e p is s emi - s imple , it c an be b r ou g ht in d i a g o n a l ove r a finite e xte n s ion of Q 1 ' he nc e g ive s r i s e to a fam i l y

form

of n c ont inuous char acte r s 1/1 . C --. Q* whe r e i" K 1 ' C i s the id� le - c la s s g r oup of K , and n = d im . V Le t K N I N = 1/1 n b e the c o r r e s p ond ing c h a r a c t e r s o c c u r r in g in X l = 1/1 1 , , X n N N Sinc e p i s locally algeb r a ic , to e ac h X c o r r e s p o nd s an p lg alg ebr aic char ac t e r X � E X (T ) of the tor us (h e r e we i d en t i fy X (T ) with Hom (T/ O ' G / O ) ' s in c e 01 i . a l g e b r a ic al l y m I l (i ) n lg is of the for m IT (0) a , wh e r e r i s the clos ed) . Each X �1 { I/I ' l

.

.

•

, I/I } n .

.

•

;

O'E

s e t of emb e dding s of K into Q 1 X .

1

for all x E K LE MMA

-

�

X

(x) =

r

d . Cha p .

'

II,

1g - 1 �1 (x ) = ITa ex)

-n

1 . 1. One has a

(i )

c l o s e enough to 1 .

All the inte g e r s n (i) 0'

!?r N.

,

I

� i � n,

O"C

r ,

ar e

divis ible

Pr oof of the lemma

th . . 0 O * c onta m mg no N - root f be an open s ubgroup 0 f I unity except 1 , and let m be a m odulu s of K s uch that 1/1 1. (x) E U , n ; the e x i s t e n c e of s uch an fol m for all x E Um and i = 1 , Let

U

•

.

.

lows fr om the c ontinuity of 1/1 1 ' . · · ' 1/1 n . W e take m lar ge enough s o that : N a ) It is a modu lus of defin ition fo r p b ) p is unram ifie d at a ll v E SU p p e D! ) ' an d the c orre sponding F r obenius elements F v , p hav e a c h a r a c t e r i s t i c polynomial with

L

AB E LIAN

III- 2 2

-

ADI C REPRES E NTAT IO NS

rational c o effic ient s . Le t

�

b e the ab e l ian e xte ns i on of K c o r r e s ponding to the *

open s ub g r oup K Um

of the id� l e g r oup I .

and let L b e a finite

Galo is e xtens ion of Q c ontaining K m C h oo s e a pr ime numbe r p wh ich i s dis tinct from L . i s not d iv i s ib l e b y any plac e of Supp (m ) . and s plits c ompletel y in L .

Le t v b e a plac e o f K dividing p .

and let fv b e an idHe wh ich i s a unifor m i z ing eleme nt at v and i s equal to 1 e l s ewh e r e . The fac t that v splits c ompletely in K m (s inc e it do e s in L) impl i e s that f

v henc e (b y c la s s -field the o r y? b e lon g s to p r ime ideal Ev i s a p r inc ip al ide al

a

po s iti v e at all r e al plac e s of K .

is the norm of an idHe of K m :,� ; this m eans th at the K U rn

(a ) .

with

a ==

1 m od . m and

N (f ) and y = X . (f ) the s e ar e s o that y = x 1 v v th e F r obenius e lements of 1/1 . and X . r elative to v . By definition 1 1 alg of X we have i Le t x

= 1/1 .

1

•

y

whe r e

a

al g

= X .

1

(a )

=

IT

(1

(a )

n (i ) (1

is a s ab ove . -

Hence y belong s to the s ubfield L of Q 1 c or r e s ponding to L (this field is we ll defined s inc e L is a Gal ois extens ion of Q ) . Mor e ove r . if w K.

(1

is any plac e of L s uch that w

we have (as i n chap . II. 3 . 4 ) :

w (y) (1

=

(1

0 (1

induc e s v on

n (i ) . (1

As s ume now that n (i ) i s not div i s ible by N. Then x . which i s (1 an Nili - root of y . do e s not b e l ong t o L . Henc e the r e i s a _

LOCALLY ALG EBRAIC R E PR E S E NTATIO NS non - tr ivial N

th

' y - root 0 f un it

z

-

s uch that x and

ov e r L , and a for tior i ove r Q . a l of

III - 2 3 zx

are c onj ug ate

Sinc e the c h ar ac te r i s tic po lynom i -

has r ational c o e ffic ients , an y c onj ug ate ove r Q of an ,p e ig e nvalue of F v p is ag ain an e i g envalue of F H e n c e , the r e v, p , e xi s t s an inde x j such that

Fv

t/J J. (fv )

= z.

x =

z .

t/J . (f ) . 1

V

':'

*

B ut f E K Um and all t/J . are tr ivial on K and map U into v m J the open s ubg r oup U we s tarte d with . He nce z = t/J . (f ) . t/J . (f ) - 1 1 V J v b elong s t o U , and this c ontradic ts the way U has b e e n chos en . Pr oof of the pr opos ition

Sinc e the n (i ) are divis ible by N, the r e exi s t q, . with q,

� = X �lg

1

a

If x E K

E

X (T )

*

.t ' we have :

lg -l N q, 1. (x ) = x � (x - l ) 1

=

X 1. (x )

=

t/J . (x) 1

N

th if x is clo s e enough to 1 . Henc e q, 1. (x)t/J 1. (x) is an N - r oot of unity when x is c l o s e enough to 1 , and , by c ontinuity , it is equal to 1 in a ne ighb ourhood of 1 . Henc e , the r e s tr iction of p .', -1 to K � i s locally equal to q, , whe r e q, is the (a lgebraic ) repr e s entation of T defined by the family (q, " ' " q, ) The repr e s enta l n ' tion q" a pr ior i defined ove r Q .t ' c an be defined ove r Q .t (and even ove r Q) ; this follows , for ins tanc e , from the fac t that the

family (q, ' . . . ' q, ) is s table unde r the action of Gad a / Q ) , s inc e l n alg . alg the family , . . . , X n ) is . (X 1 Henc e p is locally alg eb r aic , q . e . d.

III - Z 4

ABE LIAN 1 - AD IC REPRESE NTATIO NS An

3. 3.

auxiliary r e s ult on tor i In [15 ] , Lang p rove d that two exponential functions exp (b l Z ) , e xp (b Z z ) , b , b Z E C , wh ich take algebraic value s for at leas t 3 l O - l ine ar ly independent value s of z , ar e multipl ic ative ly dependent : th e r atio b l / b Z is a r ational numb e r . Thi s had al s o b e e n noticed by S ieg e l . Lang prove d th e following 1 - adic analogue : 1

- Le t E be a fie ld c ontaining 0 1 and c omple te fo r a r e al valuat ion extend ing the valuation of 0 1 . Le t b l , b Z E E PROPOSIT IO N

and let r b e an additive sub g r oup of E . As s um e : (1 )

r i s of r ank a t l e a s t 3 ov e r Z

n ( Z ) The expone ntial s e r ie s exp (z ) = � z / n ! c onve rge s ab s olute ly on b r and b Z r . l (3 ) For all z E r the elem ents exp (b z ) and exp (b Z z ) are l algebraic ove r 0 and b Z are line ar ly de p endent ove r b e l ong s to 0 if b Z f. 0) . Then b

l

0

(i . e . b / b Z

F o r the pr o of , s e e [15 ] , Append ix , or [3 0] , § l . W e will apply thi s r e s ult t o tor i , taking for E the c omple tion

of 0 . We ne e d a few definitions fir s t : 1 a/ Let T b e a n n - dimens ional torus ove r

0 ,

with char acter

g r oup X (T ) . As b efor e , we identify X (T ) with the g r oup of mor We s ay that T i s a s um of one phisms of T into G /E m/ E · dimens ional tor i if the r e exi s t one - dime n s i onal s ubtor i T 1. of T , �

�

s uch that the s um map T X . . . X T � T is s urjective l n (and henc e ha s a finite ke r ne l ) . An equivalent c ondition is :

1

i

n,

LO CALLY ALGEBRAIC REPRESE NTAT IO NS

III - 2 5

X (T ) @ a i s a d ir e c t s um of one - d imens ional s ub s pac e s s tab l e � Gal (a l Q ) . bl

Let f be a c ontinuous homomorph i s m of T (Q 1 ) into E

*

We

s a y that f i s loc ally algeb r aic if the r e is a ne ighb ourhood U of in the 1 - adic Lie g r oup T (Q ) ' and an elem ent 4> E X (T ) s uch l that f (x) = 4> (x) fo r all x E U. We s ay that f is almost loc ally al gebr aic if the r e i s an inte g e r N > I s uch that f N is loc ally al g eb r a ic .

c l L e t S b e a finite s e t of pr ime numb e r s , and , for each pE S , let W b e an ope n s ubgr oup of T (Q ) ; denote b y W the fam ily p p (W ) P pE S ' be the s e t of elements x E T (O ) who s e imag e s in W T (0 ) be long to W for all pE S th is i s a s ubg r oup of T (Q ) . p

Let T (Q)

P

With the s e notation s , we have :

PROPOSIT IO N 2 - Let f: T (Q l )

phism. As sum e :

�

E

*

b e a c ontinuous homomor -

(a) The r e exis t s a family W = (W ) S s uch that f (x) is P pE algebraic ove r a for all x E T (0) W . (b ) T is a s um of one - dim ens ional tor i . Then f is almost locally algeb r aic . Pr oof. i) We s uppos e fir s t that T is one - dimens ional , and we de note by

X

a gener ator of X (T ) . If

X

is invar iant by *

Gal (0/ 0) , T

and T (Q) � Q If not , Gal {O l 0 ) acts on m X (T ) via a gr oup of order 2 , c o r r e s ponding to s ome quadratic is i s omorphic to G

AB E LIAN 1 - ADIe REPRESE NTAT IO NS

llI - 2 6

extens ion F of

Q ;

X

the char acte r

T (Q ) onto the gr oup F

�

define s an is omorphism of

of elements of F of norm 1 . In both c a s e s ,

one s e e s that T (Q) i s an ab e l ian gr oup of infinite r ank (for a more precise r e s ult, see Exe r c is e be low ) . On the othe r hand, each quo tient T ( Q ) / W is a finite ly gene r ated ab e l ian g r oup of r ank ::. 1 . p p is finite ly gene r ate d , and th is implie s that Hence T ( Q ) / T (Q ) T (Q )

W

is als o of infinite r ank. w S inc e T (Q l ) is an l - adic Lie gr oup of dimens ion I , it is

loc ally is omorphic to the additive group Q 1 . This means that the r e exists a homomorphism e: Z 1

�

T (Q l )

which is an is omorphism of Z 1 onto an open s ubgr oup of T (Q 1

).

By c ompos ition we get two c ontinuous homomo rphisms X

0

e: Zl

B u t a n y c o n tin u o u s h o m o m o r p h i s m o f Z 1

�

into

E E

*

*

is loc ally an exponential . This implie s that , after replac ing Z l by 1 m Z 1 if

ne c e s s ary , the r e exis t b , b 2 E E such that l f

0

e (z ) = exp (b z ) , l

X

0

e (z ) = exp (b 2 z ) ,

with ab s olute c onve rgence of the exponential s e r ie s . Le t now r b e the s et of ele ments z E Z 1 s uch that

e (z ) E T (Q ) W . S inc e T ( Q 1 ) / e (Z 1 ) is finitely gene r ate d , and T (Q) w is of infinite r ank , r is of infinite r ank. If ZE r . e (z )

III - 2 7

LOCALLY ALGEBRAIC REPRESE NT A T IO NS

henc e f o e ( z ) i s algeb r a ic ove r Q ; the s ame W ' * " e ( z ) s ince X map s T (Q) e ithe r into Q or into

b e longs to T (Q ) is true for

X

F�

Propos ition 1 then shows that b / b 1 2 N is r ational . This means that s ome inte gral powe r f of f , with N � 1 , is loc ally equal to an inte gral powe r of X , henc e f is the g r oup

de fine d ab ove .

almo s t loc ally alg eb raic .

ii) Gene ral c a s e . W r ite T = T . . . T whe r e T , . . . , T ar e n l l n one - dimen s ional s ubtori of T . Since X (T ) S Q i s the dir e c t s um of �

Q ,

for all i , the r e s tr ic tion f of f to T (Q 1 ) is almo s t loc ally algebr aic . B ut we may i i of T . (Q ) s uch that choo s e open sub g roups W . 1, p 1 P W , we then s e e that ...W C W . If we p ut W . = (W . ) n, p p 1 I, p 1 , p pE S £ . t a k e s alg eb raic value s on T . (Q) h en c e is almost locally al l W. 1 the X (T 1. )

i t is enough to show that ,

•

1

ge b r a i c b y i) ab ove .

q.

e . d.

Remark If one could suppr e s s c ondition (b) from Prop. 2. all the r e

s ults of this § would extend to arb itrary numb e r fie lds .

This would

be pos s ible if one had a s uffic iently s tr ong n - dimens ional ve r s ion of P r op . 1 ab ove ; the one given in [3 0] . § 2 doe s not seem s tr ong enough (it r e quir e s de ns ity prope r tie s which ar e unknown in the c a s e c on s ide r e d he r e ) . [83 1 .1

�

[ Th is h as b e e n d o n e b y W al d s c h m id t : s e e [ 6 3 1 .

Exe r c i s e Let T be a non - tr ivial torus ove r Q . Show that T ( Q ) i s the direct s um o f a finite g r oup an d a f r e e ab e l ian gr oup o f infinite r ank .

AB E LIAN 1 - ADIC RE PRES E NTATIO NS

IIl - 2 8 3. 4.

P r o of o f the the o r e m We g o b ack to the notations and hyp othe s e s o f 3 . 1 . -

p : Gal (K / K)

ab

-+

Let

Aut (V 1 )

be a rational , s emi - s imple , ab elian 1 - adic r e pr e s entation of

K.

E is the c omple tion o f Q 1 ' as i n 3 . 3 , w e may b r ing p in diagonal form ove r E . This g ive s r is e to a family (ttI " " , r/J ) of n l ab (henc e als o of the idHe g r oup c ontinuous characte r s of Gal (K / K ) * I) into E he r e , n = dim . V 1 . * * th Let f ' : K 1 -+ E b e the r e s tr iction of r/J . to the 1 - c ompo 1 l * * nent K 1 of 1. Note that K 1 = T (Q 1 ) , whe r e T is , as us ual , the torus defined b y K (chap . II, 1 . 1) . If

LEMMA - The torus T and the homomorphism f i s atisfy the as s umptions (a) and (b) of P r op . 2 , 3 . 3 . V e r ification of (a) Let S b e a finite s e t of pr ime s , with E

K

,

/: 1 ,

Pv � S , the r e pr e s entation v , and the char acte r i s tic polynomial of F

v

I:

1

Pv

v,

p

4 S, p

s uch that

if

is unr amifie d at

has rational c oeffi-

PE S , P r op . 1 of 2 . 2 shows that the r e exists an open * subg roup W of K = T (Q ) s uch that r/J . (W ) = 1 . Le t p p p 1 p * W = (W ) T (Q) ' S ince x E K , we have r/J 1. (x) = 1 , and let x E W P pE S when x is ident ifie d with an id� le of K . On the other hand, let us c ients . If

s pl it the idHe x in its c omponents

LOCALL Y ALGEBRAIC REPRESE NTAT IONS

III - 2 9

ac c o r ding to the de c ompo s ition of I in I *

,�

= K

':'

�:::

�I..

X K

00

�

�I...

X K

;X

I' .

IT K P and �'

l' is the r e s tr ic te d pr oduct pE S ,:� of the K v , for v E :E K , and p 1= I. , p # S . ) The re lation v v 1/1 . (x) = 1 , together with 1/1 . (x ) = f . (x) , g ive s I. 1 1 1

(He r e K

= (K

@ R)

f . (x) 1

K

-1

=

S

=

1/1 . (x 1

B y c ons truction , we have Henc e : f 1. (x)

1/1 .

1

00

)1/1 .

1

(x ) S

= + 1/1 .

1

(x )I/I . (x ' ) . S 1 =

1 and it is clear that 1/1 . (x

(x ' )

1

00

) = + 1.

-1

B ut , for each v E :E K , with P v � S , p 1= I. , we know that the v e igenvalue s of F are alg eb r aic ; henc e , if f is an id� le which v, P v is a uniformi z ing element at v , and is equal to 1 e l s ewhe r e , 1/1 . (f ) is algeb r a ic . If a (v ) is the valuation of x , at v , we have : 1 v 1/1 .

1

hence

1/1 .

1

(x '

) =

IT 1/1 1. (fV ) a (v )

(x' ) and f 1. (x) are algeb r aic and we have checked (a) .

Ver ification of (b ) . Sinc e K is a c ompos ite of quadr atic fie lds , it is a Galois extens ion of Q, and its Galois g r oup G is a pr oduc t of g r oup s of orde r 2 .

The char acte r g r oup X (T ) of T is is omorphic to the

III - 3 0

AB E LIAN 1 -ADIC R E PR ESE NTAT IO NS

re gular repr e s entation of G , and it i s clear that X (T)

@

Q s plits

as a dir e c t s wn of one - dime n s i onal G - s table s ub s pac e s (each c or r e s pond to a char acter of G ) . Henc e T i s a swn of one - dimen s ional tori . End of the p r o of of the theorem Us ing p r op . 2 of 3 . 3 , we see that e ach f . is almost loc ally 1 algeb raic . Henc e the r e is an inte g e r N > 1 s uch that the f N . are 1 N locally algeb r aic . This implie s , c f . 1 . 1 , that p i s locally alge b r aic , henc e (ef. 3 . 2 ) that p its e lf i s loc ally alg ebr aic , q . e . d. Exe r c i s e As s ume that K i s a c ompo s ite o f q uadr atic fie lds . Let X be a G r O s s enchar akte r of K and s uppo s e that the value s of X (on the ide als pr ime to the c onductor ) ar e algeb raic numbe r s . Show that

X

is " of type (A) " in the s en s e of W e il [ 41] .

method than a!:; ove , with E replac e d by C . ) in a finite extens ion of Q , s how that

X

If

(Us e the s ame

th e value s of

is " of type (A o ) "

.

X

-+

lie [ no

a s s u m p tio n o n K is n e c e s s a r y , th a n k s to [ 8 3 ] . J

APP ENDIX Hodg e - T ate de c ompo s itions and loc ally algebr aic r e pr e s entations Let K be a field of char acte r is tic z e r o , c omplete with re s p e c t t o a dis c r e te valuation and with perfe c t r e s idue field k o f charac te r is tic p > O . In thi s Appendix w e deal with Hodge - T ate de c ompo s i tion of p - adic abe lian repr e s entations of K .

LO CALLY ALGEB RAIC RE PR ESE NTAT IO NS Se c t ions A l and

III 3 l -

g ive invar ianc e pr ope rtie s o f the s e de

A2

c ompos itions unde r g r ound fie ld extens ions . Spe c ial char acte r s of Gal (K / K) ar e defined in A4 ; they ar e clos ely c onne cted b oth with Hodge - T ate module s (A4 and AS ) and local algeb raic ity (A 6 ) . The pr oof of T ate ' s the o r e m (c f. 1 . 2) is g iven in the las t s e c tion. AI . Invar ianc e of Hodg e - T ate de c ompos itions Le t C be the c omple tion of K (cf. 1 . 2 ) ; the gr oup Gal (K / K) ac ts c ontinuous ly on C .

Le t

X

b e the characte r of

Gal (K/ K )

into the g r oup o f p - adic units define d in chap . I, 1 . 2 . Let K ' / K be a s ubextens ion of K / K on which the valuation -; of K is is a finite extens ion of an unrarnified

dis c r e te ; this means that K' one of K .

A

Le t K '

denote the clo s ur e of K '

in C .

Let now W b e a finite dimens ional C -ve c tor space on which Gal (K/ K) acts continuous ly and semi - linearly (s e e 1 . 2 ) . As befor e , � n n we denote by W (r e sp . W K ' ) the K - (re sp . K ' - ) vector space defined by n n W = {w E W I S (w) = X (s ) w (r e s p .

w� ,

= {W E

w i s (w)

=

X

n (s ) w

s ' Gal (K/ K) }

for all

for all s

E

Gal (K / K " ) } ) .

n n Identifying the ' Let W (n) = C <» W and W (n) ' = C �K' W K ' K module s W (n) and W (n) ' with the ir c anonical imag e s in W, we p r ove

THEOREM I - The canonic al map

"

K ' - i s omor phism.

"

K' � W K

n

�

W

n

K'

is a

AB E LIAN 1 - AD IC R E PRES E NTAT IO NS

III- 3 2

CORO L LAR Y 1 - The Galois modul e s W (n) and W (n) ' Inde e d , The orem 1 shows that W

n

and

w�,

ar e e qual .

gener ate the

s ame C -vector sub s pac e of W . The Galo i s module W is o f Hodge - T ate type ove r " K if and only if it is s o ove r K ' . CORO LLARY 2

-

Proof of The o r em 1 Note fir s t that r e plac ing the action of Gal (K / K) on W by -i (s , w) � X (s ) sw , i E Z , j u s t chang e s W n to W n+ i This

sh ifting pr oc e s s reduc e s the p r oblem to the case n = 0 ; in that n n c as e , W (r e s p. W ) i s the s e t of elements of W which are K' invar iant under Gal (K / K) (r e s p . unde r Gal (K / K ' ) ) . Note als o o that the inj e ctiv ity of 1< , � W � w�, is triv ial , s inc e we know O that C Q!) W � W is inj e c tive (cf. 1. 2 ) . K O n the othe r hand , an e a s y up - and - down argum ent shows that one c an as s ume K ' / K to be e ithe r finite Galois or unr amified Galo is . In both cas e s , s inc e Gal (K / K ' ) acts tr ivially on w�, we have a s emi- line ar action of Gal (K ' / K) on W � , . When K ' / K i s finite , it is well known that this implie s that W o is K' O gene r ate d by the e lements invar iant by Gal (K ' / K ) , i. e . by W (this i s a non - c ommutative analogue of Hilb e r t ' s " The o r em 9 0 " cf. for in stanc e [ 2 9] , p . 15 9) .

•

Let now K ' / K be unr am ifie d Galois and let G be its Galois g r oup . ...

0'

Let

,..

A

0'

0

denote the r ing of inte ge r s of K ' . A

Let

0

A

be an

(i . e . a fr e e 0 ' - s ubmodule of W K ' of the s ame K' r ank as W � , ) . Since G acts c ontinuously on W � , ' the s tab iliz e r in G o f A i s open, henc e o f finite index, and the latt ic e A has - lattic e of W

LO CALLY ALGEBRAIC REPRESE NTAT IO NS

III - 3 3

The s um A 0 of the s e tr ansforms is

finite ly many transfor m s .

. Let e , . . . , e b e a b a s I s of A N l

invar iant by G.

0

.

Le t s

E

G.

Then

N

s (e . ) = � a . . (s ) e . 1 I J i =l J

a. . E IJ

A

0' A

and the matr ix a (s ) = (a . . (s » b e l ong s to G L (N , 0 ' ) . W e have IJ a (st) = a (s ) s (a (t » ; this m e an s that a i s a c o ntinuous l - c o c yc le on

G a'

with value s in

'"

GL (N, 0 ' ) .

R e c all that two s uch c oc yc l e s a and

a r e s aid to b e c ohomologous if the r e exists b E =

that a ' (s )

b

-1

a (s ) s (b ) for all s E

G.

G L (N , 6 , )

s uch

This i s a n e quivalenc e

r e lation on the s e t of c o c yc le s and the c o r r e s p onding quotient s pac e

" H 1 (G , GL (N , 0 ' » . In fact :

i s de note d b y

LEMMA

If (G, GL (N, 0 ' »

-

= {I} .

As s uming the lemma , the pr oof of the the orem is c onclude d a s follows .

A

Sinc e a (s ) i s c ohomolog ous to 1 ,

GL (N, 0 ' )

s uch that b

=

a (s ) s (b ) for all s E o define a new b a s i s e i , · · · , e N of W ' by

b E

the r e exis t s

G.

If b = (b . . ) , IJ

K

e ! = � b . .e . . J IJ 1 i =l U s ing the identity b

G,

invar iant unde r tiv ity of

K'

GO

K

W

O

=

a (s ) s (b ) , one s e e s that e i , . . . , e N ar e

henc e b e l ong to W �

W

O

K'

O ; this p r ove s the surj e c

AB E LIAN l - AD IC REPRES ENTATIO NS

III - 3 4 P r oof of the 1enuna Let

7r

�

'"

0' .

b e a unifo rmiz ing e lement of

Filte r th e r ing

G L (N , O ' ) b y mean s of A = { a E A I a == 1 mod 7r } . We g e t n AI A � G L (N , k ' I k) , whe r e k ' I k is the r e s idue field extens ion l of K' I K. Mor e ove r , for n ::: 1, the r e is an is omorphis m A I A + -.. M (k ' ) , whe r e M (k ' ) is the additive gr oup o f N X N n n k N N matr ice s with c o efficient s in k ' . The l emma follows now fr om the triviality of H1 (G , G L (N , k ' ) ) and H1 (G , � (k' ) ) , whe r e now k ' i s A

=

n

endowe d with the dis c r e te topology (s o this is ordinar y Galois

c ohomolog y ,

C£.

[ 2 9] , p. 15 8 - 15 9) .

A 2 . Adm i s s ible characte r s Let G morphis m .

=

Gal (K1 K) and let q, : G

�

K

*

b e a c ontinuous homo -

DEF INITIO N - The characte r q, i s s aid to b e admi s s ible (notat ion : q, -.. 1) if the r e e xi s ts x E C , x F 0 , s uch that s (x) = q, (s )x for all s

E

G.

Remarks

1) The admis s ible characte r s form a s ub g r oup of the g r oup of * all char acte r s of G with value s in K ; if q" q, ' are two characte r s , -1 we wr ite q, -.. q, ' if q, q, ' -.. 1 . * 1 2) Let H (G , C ) b e the fir s t c ohomolog y gr oup of G with * value s in C (c ohomology b e ing defined by c ontinuous c ochains , a s * in AI) . A c ontinuous char acter q, : G � K i s a l - c oc ycle , henc e * 1 define s an element q, of H (G, C ) . One has q, = q, ' if and only if q, -.. q, ' .

LO CALLY ALG EBRAIC REPRES ENTA T IO NS

III - 3 5

3 ) D e fine a n e w act ion o f G o n C b y m e an s of (s , c)

�

S

(c )

E G, C E C.

De note the C - G - module thu s obtain e d by C (
Then
ibl e if and only if C (
PROPOSITIO N I - Suppo s e the r e exis ts C E C
Then

of G .

....

�

s uch that

for s in s ome open s ubgr oup N of the ine r tia g r o up q,

i s adm i s s ible .

Let K ' I K b e the s ub e xte ns ion of KI K c or r e s ponding to

Pr oof.

N; it is a finite extens ion of an unr am ified one . O

Le t W = C (q, ) ,

as

�

W , ) b e the s ub s pac e of W c on s i s ting of e lement s invar iant b y G (r e s p . b y N) . B y hypothe s is , O W , is 1= O . Henc e , b y AI , T h e o r e m 1 , we al s o have W 1= 0 ,

in Remark 3, and let W

(r e s p .

�

and this means that q, is admi s s ible , Le t n o w U

q.

e . d.

c b e the g r oup of units of C , U

�

the s ub g r oup o f

-*

units c ong r ue nt to I m odulo the maximal ide al , and identify k with the g r oup of multiplicative r e pr e s entative s , s o that - �< I U = U X k , d . [ 2 9] , p . 44 . De fine the l ogar ithm map

c

c

log :

Uc -� C

by log (x) log (x)

=

=

0

00

�

n =l

( - 1)

n-l

n (x - I ) / n

if x E U

l C

This i s a c ontinuous homoITlorphisITl and e v e n a local is oITlorphism .

AB E LIAN

III- 3 6

L

- ADIC REPRES E NTATIO NS

Mo r e ov e r :

LE MMA - (a) log is surjec tive . X

(b ) The kernel of log : fJ.

whe r e

p

CD

fJ.

n p - th r oots of unity . for

i s the s e t of all

p

00

•

n = 1. 2

•

.

.

.

As s e r tion (a) follows from the fact that C i s algebr aic ally c l o s e d , hence that U

C

is divis ible .

On the othe r hand , if u l im . uP

N

= I , henc e ,

gr oup of U

�

E

U

�

is s uch that log (u) = 0 , uP

if N is lar g e enough ,

whe r e log is inj e c tive

N

one has

b e l ong s to a s ub -

(for in s tanc e the sub g r oup of

N 2 . elements x wIth x :: 1 mod p ) . Henc e up

=

I , and u

E fJ.

00

P

this implie s (b ) . W e now apply the log map to the c ohomology g r oup s of G with * value s in U , C , C , . . . (c ohomology be ing . as usual , define d by

C

c ontinuous c ochains ) .

Fir s t , s ince the valuation gr oup of C

is

Q ,

we have the e xact s e quenc e

1� U

�

c

C

*

-+ Q -+

1 .

* * 0 By T ate ' s the or em ([ 3 9] , § 3 . 3 ) one has H (G, C ) = K an exact s e quenc e

K

*

� Q � H

1

(G, U C )

� H

1

(G, C

*

) -+

0 ,

,

hence

LO CALLY ALGEBRAIC R E PR ESE NT A T IO NS

III - 3 7

or , e quivalentl y :

Let N

=

We hav e the exact s e quenc e

K e r (l og ) .

i H I (G , N) � H I (G , U

C)

X. I � H (G , C ) ,

whe r e X. i s induc e d b y the log . S inc e H I (G , C ) i s a K -v e c tor s pac e , the c ompo s ite X. 0 0 : Q f Z � H I (G, C) is 0 , henc e the r e is a un ique map

L:

s uch that

Loi

PRO POSI T ION 2

*

H1 (G, C ) � H I (G, C )

= x. .

-

The map L:

*

H I (G , C ) � H I (G , C)

i s inj e c tive .

U s ing the exact s equenc e s ab ov e , one s e e s it is e nough to I

pr ove that io j : HI (G , N) � H (G , C*) i s tr ivial. s ub g r oup of K*

-*

K

tr ivial , henc e al s o i o j , Le t now
G�

K

B ut N i s a dis c r e te -*

H (G , K ) , whe r e -* The o r e m 90 , HI (G , K ) is

henc e i o j fac tor s thr ough

is viewe d as a dis c r e te g r o up ; by

I

q . e. d.

*

b e a c ontinuous characte r .

S ince
i s c ompac t , it is c ontaine d in U K ' henc e in U ' and C log
H I (G , C ) i s * H I (G , C ) .

L
now

whe r e
1II - 3 8

AB E L IA N l - AD IC R E PR ES E NT A T IO NS

P R O P OS I T IO N 3 - T h e p r o p e r t i e s Th i s

- l and

I/>

LI/> =

0 are equivalent .

follows fr oITl the inj e c tivity of L.

COR O LLAR Y - If the r e exis ts a non - z e r o inte ger n such that n I/> - 1 , then 1/> - 1 . Inde e d , LI/>

=

!. L'i

n

n

=

O.

ReITlark

I Spr ing e r has prov e d that H (G , C) is of diITl ens ion l ove r K l (cf. T ate [ 3 9] , § 3 . 3 ) . Henc e , one can take for basis of H (G , C ) the eleITlent LX , whe r e X is the fundaITle ntal char ac ter define d in chap . I, 1 .

2.

In particular , for any

eleITlent c (q,) of

K

s uch that Lq,

=

1/> :

*

G � K , the r e i s an

c (q, ) LX ;

when

K

is

loc ally

c OITlpac t, this c (q, ) ITlay be c OITlpute d explic itly, s e e A6 , Exe r . A3 .

A

2.

c r ite r ion for loc al tr iviality

F r oITl now on , E denote s a s ubfie ld of K having the following prop e rtie s : (a) E c ontains Q c OITlpact) . .(b )

p

and [E : Q J p

< 00

(s o that E i s loc ally

K

c ontains all Q - c onj ugate s of E . p We denote by r E the s et of all Q - eITlbe dding s of E in K . p C ons ider a c ontinuous char acte r 1/1 :

-

Gal (K/ K )

�

E

*

with value s in E . For e ach cr E r E this give s a char ac te r * cr * * cr o r/; : G � E � K of G = Gal (K / K ) into K .

III - 3 9

LO C ALL Y ALG EB RAIC R E PR E S E NTAT IO NS

PROPOSI T IO N 3 (1 ) r./J

T h e following two pr o p e r tie s a r e equival ent :

-

i s e qual to 1 on an op en s ub g r oup of the ine r tia group

of G , (2 )

ao

(1 )

=>

r./J

1 for all

--

a

E r

'

E

P r oof

(2 ) i s tr ivial fr om th e r e s ul t of Al

(s inc e we know

that admi s s ib il ity c an b e s e e n on an o pe n s ubg r ou p of the iner tia g roup ) . (2 )

=>

(1 ) .

W e us e the l o g map defined in A2 .

take s value s in the g r oup U

E

of unit s of E ,

Note that r./J

henc e log r./J : G

�

E

Le t I b e the ine r tia g r o up of G ; the s ub g r oup n log r./J (1 ) of E i s c ompa c t , and h e n c e i s omorphic to Z for s ome p n . If W i s the Q - v e c t o r s ub s pa c e of E g e ne r ate d b y log r./J (I) , p we s e e that l og r./J (1 ) i s a latti c e in W , and d im W = n. Note that i s well define d.

s aying that r./J is e qual to 1 on

an

1

is

� E is a local E S upp o s e thi s i s not the c as e , i . e . s upp o s e that

e quivalent to s aying that log r./J (1) i s om o rph i sm) .

open ne ighb ourhood of I in

=

0 (s inc e log : U

Cho o s e a Q - l inear map f : E � K s uch that d im f (W ) = 1 ; p s uch a map obviously exis ts . B y Galois the o ry (indep e ndenc e of

n > 1.

char acte r s ) the s e t r exi s t

k

a

E

and we have

E

i s a b a s i s of Hom

Q

p

(E ,

K) . Henc e , the r e

K wi th

f o l og

r./J = � k a o l og r./J = � k l og (a o r./J ) . a a

AB E LIAN 1 - AD IC R E PR ES E NT ATIO NS

III - 4 0

But b y as s umption (and Prop. 3 of A2 ) , the additive l - c o c yc le log (a D rJ; ) : G fo r f .. l o g rJ;.

�

K is c ohom o l o g ous to O . Hence the s ame is true

N B ut we may as s um e (r e plac ing f b y p f ,

with N

larg e , if n e c e s s ary) that the r e exists a c ontinuous homomorphism F : U � U s uch that f o log = log o F . We then have E K log (F o rJ; ) = f o log r/J and h en c e (d . P r op . 3 0f A2 ) , F" r/J - l , L e . F o rJ; is admis s ible . But F o rJ; has now the pr ope rty that F o rJ; (I) C U is a p - adic Lie gr oup of dimens ion 1 (product of Z p K with a finite g r oup ) . This c ontr adicts a the orem of Tate ([3 9] , § 3 , Th . 2 ) , henc e the r e s ul t

A4.

Th e charac ter

.

X

E We keep the s ame hypothe s e s on K and E as in th e p r e v i o us ab s e ction . By c las s field theo r y , the g r oup Gal (E / E ) may be id e n "*

tifie d with the c ompletion E

of E

*

with r e s pe c t to the topology of

open s ubgr oup s of finite index. In particular , we have an exact s equenc e

wher e Z

-

IT Z 1 denote s the c ompletion of

t o p olo g y of s ub g r oup s of finite index

-

(d .

or Cas s e l s - F r �hlich [6] , Chap. VI , § 2 ) .

Z

with r e spect to the

for ins tanc e Ar tin - T ate [2]

LOCALLY ALGEBRAIC REPRESE NT A TIO NS

III - 4 1

Let now 7r be a uniformiz ing e lement of E. The imag e o f 7r ab in Gal (E / E ) g ener ate s a s ubgr oup who s e c l o s ur e is is oITlOrphic ,.. to Z , and this g ive s an is omorphism:

ab Gal (E / E ) � U E b e the proj e c tion as s o c iate d with this 7r de c ompos ition (the Galois extens ion of E c or r e s ponding to Ker (pr ) Let pr

:

i s the c ompos ite of all finite abe lian extens ions of E for which is a norm , cf. [6 ] . p. 144 - 14 5 ) . O n the oth e r hand , the inc lus ion E

phism Gal (K. / K)

�

X

E , 7r

(abbr . G

=

x-

1

7r

K define s a homomor

Gal (E / E ) , henc e als o a homomorphism rE : G

Define

�

7r

X

�

�

Gal (E / E ) ab

E ) to be the comp o s ite homomorphism ab Gal (E / E )

�

i U UE � E '

U E . Ob s e rve that the r e str iction of X E 1 to the inertia gr oup of G is x � r E (x - ) , and hence is indepen dent of the choice of 7r.

where i (x)

for x

E

PROPOSIT ION 4 - Let F be the Lub in - Tate fo rmal gr oup ( [17] , 7r s e e als o [6] . chap . VI, § 3 ) as s oc iate d to E and 7r. Let T be its

Tate -module , which is fr e e of r ank 1 ove r the r ing 0 E of integer s of E . The action of Gal {K. / K) on T is g ive n b y the character : G � U , defined ab ove . X E E

AB E LIAN l - AD IC R EPR ESE NT A T IO NS

III - 4 2

This follows fr om the main th e o r e m of [ 1 7 ] (s e e al s o [6 ] , Th . 3 , p . 14 9) . CORO LLAR Y - If E

=

7r

Q

and p c o inc ide s with the char ac te r X

=

th en the characte r X

p,

E

defin e d in chap . I , 1 . 2 .

Inde e d , the Lub in - T ate g r oup i s now the multi p lic ative g r o up G

m

and its Tate module i s the modul e

R emark If

* . 1 o c a11 y c ompac t , we may 1' d e nh' f y G ab to K K 1S .... an d th e X

c har acte r

E i s g iven b y

"* N ,, * K � E

whe r e N

T (f.I.) defined in ch ap . I , 1 . 2 . P

=

N

K/

E

pr

7r

�

•

U � U ' E E

i s the norm map .

[ This follows fr om the func

tor ial p r op e r tie s of the " r e c ip r o c ity law " of local c las s field the o r y . J

U

In

K

AS .

particular , the r e s tr ic tion of X to the ine rtia s ub g r oup E ab -1 (x ) . of G is x � N

K/ E

Character s a s s oc iate d with Hodge - Tate de c omp o s ition s Retaining the notation of the pr e vious s e c tions , let p : G � U

b e a c ontinuous homomorph i s m .

s pac e ove r E ; we make G ac t on V by (s , y)

Henc e V i s a G - m odule .

�

E

Le t V b e a one - dimens ional v e c toI

p (s ) y ,

Le t W

=

S

C 60 Q

E

G, y E V. V,

p

whe r e C

=

�

K as

III - 43

LO CALLY ALG E BRAIC R E PRESE NTAT IO NS

befor e . =

d a

x

This i s a d - dimens ional vector spac e ove r C, whe r e

[E : Q J . p

Eve ry element x o f E de fine s a C - endomorphism

of W by a

X

(!:

c.

l

�

y. ) I

=

!:

c.

I

� xy .

I

C .

,

1

E C , y. E V . 1

We g et in this way a r e pr e s entation of E in the C -vector space W ; note that th e action of ax c ommute s with the action of G. Let a E r and put E

Wa

=

{w l w

E

Each W LEMMA I - (a) --

s table by G .

W , a (w) x a

=

a(x)w

for all x E E } .

is a one - dimens ional C - ve c tor spac - e

(b ) W is the dir e c t s um of th e

(c ) For e ach i s omorphic to C (a o p ) .

a E r

E

'

W ' a

s,

a E r

th e Galois module

E

W

.

is

a -

[ For the definition of the " twiste d " module C (a o p )

s e e A2 , Remark 3 . J P r oof.

The as s e r tions (a) and (b) are c ons equenc e s

known fact that C p r oj e ction s C

�Q

�Q

P

p

of

the we ll-

E is a pr oduct of d c opie s of C ,

the

E � C be ing given b y the elements of r E .

For (c ) note that the s ame de c ompo s ition holds for V

K

=

K � Q

V,

s inc e

P

henc e for e ach

a E r

E

K

c ontains all the Q - c onj ugate s of E ; p

' the r e exists a

W

E W

a

c onta ine d in

V

K

.

AB E LIAN 1. - AD IC R E PRE S E NT A T IO NS

III - 4 4 =

F or s uch a w , s ay w s (w)

� k.

1

=

� k.

=

� k 1.

1

= a

= a

and this im.plie s that W

�

p (s ) 0

a

09

�

(k . E 1

y.

1

K , y . E V ) we have 1

s (y 1. ) p (s ) y .

1

w s ince w belong s to W

P (s )w

a

i s i s om.orphic to C (a o p ) .

If P I and P ar e two characte r s of G into K 2

t"

then we

if P and P == P I 2 c o inc ide on an open s ubgr oup of l 2 the iner tia g r oup of G .

shall wr ite P

THEOREM 2 - Let P , V , W b e as ab ove and , for each a E r E ' let

n

a

be an inte g e r . The following ar e equivalent : P =- IT

(i) n

(ii )

-

1

n

a oX E a

a

for all a E r E for eve r y a E r E the Galois -m.odule W a o p "'" X

(iii) n

a E rE

a

to C (X a ) .

a

is is om.orphic

[Re c all that X is the char acter define d in chap. I, 1 . 2 , and that X E is the one attache d to the s ubfield a E of K, as in A4 . a Note that, s ince X E r e s tr icted to the ine r tia gr oup depends only

on a E ,

a

(i) is m.e aningfu1 . ]

LO CALLY ALGEBRAIC R EPRESE NTAT IO NS

III - 45

CORO LLAR Y - V is of Hodg e - Tate type if and only if the r e exi s t n

a

E

Z

s uch that p _ - IT a a E rE

-1

'X

na

aE

This follows fr om (i ii ) and the fac t that W s um of the W ' s . a

=

C

�

V is the dire c t

Pr oof of The orem 2 We p r ove fir s t : LEMMA 2 - (a)

X

E

(b ) If

Pr oof.

Let

7r

-

X

a E rE

is not the inc lus ion map ,

b e a uniformizing par amete r of E , let F 7r ,

Lub in - Tate g r oup a s s oc iate d to E and module , and

aX

V 7r =

T

7r

�

Q

P

V 7r

S ince

7r

E

-

1.

b e the

let T 7r b e its Tate

is a one - dimens ional ve ctor X

: G � U E (d . A4 , P r op . 4 ) , the ab ove c ons tructions apply to V 7r and X . By a E the orem of Tate ( [ 3 9] . § 4 , C o r . 2 to Th. 3 ) , W = C � Q V has 7r 7r p a Hodg e - Tate de c ompos ition of the type s pac e over E , and G acts on V 7r thr ough

W 7r whe r e dim W

7r

(0)

=

= W (0) 7r

d - l , dim W ( 1 ) 7r

=

fine s c anonic al is omorphisms W 7r (0 )

ED

E

W (1 ) 7r

1 . Mor e pr e c is ely, T ate de =

C Ci> K Hom (t ' , K ) , whe re t ' E

AB E LIAN 1 - ADIC REPRES E NTAT IO NS

III - 4 6

i s the W (l ) = 7r

F (d - l) - dimen s ional tang e nt s pac e of the dual of 7r (C � Q V (f.L ) ) riJ t , whe r e t i s the one - dime n s i onal K P P

tang ent space to F , and i s th e

Q

7r

p

V (f.L )

P

-vec tor s pac e of dimens ion

1 define d in Chap . I, 1 . 2 .

Note that C

�Q

p

V (f.L)

p

is . i s omorphic t o C (X ) , henc e one gets an

i s omorph i s m

The s e i s omorphisms c ommute with t h e action o f E . S ince E acts on t by the inc lus ion map a E � K , this l: of W is W (I ) . Henc e , us ing show s that th e c omponent (W ) 7r

Lemma I , we have C (X )

Clj,

7r

7r

�

C (X E ) ' and this implie s X E -- X , whenc e (a) . On the o ther han d , the s ame ar gum ent shows that (W ) 7r a

•

a 1= a

'

ar e c ontained in the oth er fac tor W {O J of W

l hen c e C { a o x ) -- C {l) , E

7r

7r

{whe r e 1 s tand s , of c our s e , for the unit

charact e r } , and this prove s (b ) . W e now g o back t o the proof of The orem 2 .

The equivalenc e

of (ii) and (iii) follows from Lemma 1 . T o show (i) � (ii ) , note * -1 take s value s henc e a o X fir s t that X take s value s in aE in E

p =

1

*

aE

and the s ame is true for the characte r

aE

LO C A L L Y A L G E B R A I C R E PR ES E N T A T IO NS

Let

T E L

E

.

We have =

To a

T 1= a

IT

aE I

Tc a

-1

n

oX

E

a aE

2,

applie d to the field aE , we s e e that -1 - 1 l' f T o a is not the identity o n aE , L e . if
F r om Lemma -1

III - 4 7

for all By P r o p . 3 of A3 , this is equivalent to

T E L

P I ·� p ,

E

.

q . e . d.

Loc ally c ompac t c a s e

A6 .

We now add to all the previous as s umptions re garding E,

K

and

(L e . K is loc ally p ab c ompact ) . B y local clas s fie ld the o r y , we may then identify G I" t..c ab with U K ' the group with K and the ine r tia s ubgr oup of G

the as s umption that

of units of

K

is finite ove r Q

K.

Let T (re s p . T ' T ) b e the Q p - torus as s oc iate d to K E aE (re sp . to E , aE , whe r e a E I E ) , cf. 1 . 1 . The norm map from K

to

aE

define s an algebraic morphism

B y c OInpo s ition with

0'

-1

: T E a

r 0' = PROPOSITIO N 5 HOIn

1

AB E LIAN

III - 4 8

-

0'

-1

(a) r (u - 1 ) a (b ) th e r

0'

, ) alg ( T T E ·

�

(0'

REPRES E NTATIONS

T E ' this g ive s a InorphisIn

�N K/ E : T a = 0'

- ADIC

�

TE .

-1 0 X

(u) for all u E U ' K aE E r ) Ina k e a Z -b a s i s of E

(Note that (a) In ak e s s ens e , s ince U has b e en identified with K ab the inertia gr oup of G . ) A s s e r t i on (a ) f o l l o w s fr oIn th e r e Inar k

at

th e e n d of

A4 . On

the other hand , let X (T) and X (T E ) b e the char acter gr oup s of T and T (0') , 0'

r e spectively. The charac te r s [ s ] , s E r K (r e s p . E E r ) Inake a b a s is o f X (T ) (r e s p . of X (T ) ) ' The Inor E E

phisIn r

0'

:

T

�

T E define s b y transpos ition a hOInoInorphisIn

One checks eas ily that the effe c t of [,.] . T E r

E '

is :

X (r

A s s e r t ion

(b )

X (r

0'

) ([T] ) =

t h e n f o l l o w s fr oIn :

l;

s a= T

0'

) on the b a s i s

[s ]

LO CALL Y ALGEBRAIC R EPRES E NT AT IONS LEMMA - The e lements X (r a ) ' O' Hom (X (T E ) , X (T ) ) . Gal P r oof. let

E r

E

III - 4 9

' fo rm a bas is of

The inde pendenc e of the X (r ) is c lear . On the other hand , a

Hom Gal (X (T E ) , X (T ) ) b e s uch that

Gal (O / Q ) is equal to the identity on TE , we have p p a [T] = [T ] , henc e a
a

E

a

=

=

!:

n

O' E r

E

!:

ae r

a

!:

s a =T

[s]

n aX (r a ) ( [T ] ) . E

This pr ove s the lemma . PROPOSIT IO N 6 Let r: T � T E

-

p

(n

) , a e r E ' b e as be the morphism defined b y

Let

and

n r a

r = 1T

ae r

a

a

E

in

Th. 2 of

AS .

AB E LIAN l - ADIC R EPR ES E NTATIO NS

III - 5 0

The equivalent prope r tie s (i ) , (ii) , (ii i ) of Th. 2 are equivalent to : (iv) Ther e exists a n open sub gr oup u r o f the inertia s ub g r oup ab U s uch that r (u) p (u) = 1 if u E U r • K of G cr

-1

Inde e d , (iv) is just a r eformulation of (i ) , s inc e we know that -1 " X crE (u) = r cr (u ) i f u E U K .

COROLLARY - The following ar e equivalent : (a)

p

is loc ally algebr aic .

(b ) The Galois module V attached to

�.

p

is of Hodge - Tate

This follows fr om The orem 2 , c omb ine d with P r op. 5 and P r op . 6 . Exer c i s e s a) Let A = End

1)

morphisms of K ; if a x

E

E

Qp

(K) b e the spac e of Q - linear endo p

A , denote by T r (a) the trac e of a. If

the endomorphism y � xy of K . Show that , x A, the r e exists a unique element c (a) of K s uch that K

K , denote by u

for any a

E

T r (u 0 a) = T r ( c (a) ) x K/ Q x. K P

for all x

E

K.

b ) Show that the map c : A ---+- K s o defined is K - l inear K for both the natural structur e s of K -vec tor s pace on A.

LO CALLY ALGEBRAIC R E PR ESE NTAT IO NS

III - 5 1

c ) Le t e 1. b e a Q - b a s is of K and l e t e i' b e the dual p b a s i s , s o that T r / Q (e i e � ) = 6 . . Show that K lJ J p c (a) K d)

If

=

L: a (e . ) e �

L :) K and a C

Show that c (Tr K/ Q K

p

L

) =

l

i

(a .. Tr

E

l

if

a

E

A.

A , show that

) L/ K = c K (a) .

1.

e ) If K i s a Galois extens ion of Q c

K

(0')

let a

= 0 for every ab 2 ) Let r/J : G r/J

0' E

Gal (K/ Q p ) '

�

K

*

0' f:

show that p id. , and c (id . ) = 1 . K

b e a c ontinuous homomorphism , and

b e the Q - l inear endom orphism of K such that the diagram p u

K4

U

K

� log j, l og

a K .L K

i s c ommutative . Let Lr/J (r e s p . LX ) be the im ag e of r/J (r e s p . X ) in the one - dimens ional K -ve c tor s pace If (G , C ) , cf. A2 . Show that Lr/J = c . LX whe r e

c

(Check the fo rmula fir s t when K is a Galois

III - 5 2

AB E LIAN

1.

- AD IC REPR ESE NTATIO NS

-1 extens ion of Q and q, = a o x K ' a E G al (K (Q ) ' in which c as e p p -1 a = -a and c q, ) is g iven b y E xe r . 1, d . ) (a q, K In partic ular , q, is adm i s s ible if and only if c K (a q, ) = o .

A7 .

Tate ' s The orem W e r e c all the s tatement (c f. 1 . 2 ) ; he r e again , K is locally

c ompac t . THEORE M 3 - L e t V be a finite dim ens ional ve ctor spac e ove r Q p and let p : G al � Aut ( ¥) b e an ab elian p - adic r e p r e s entation of K . The following ar e equivalent : (1 )

(2 )

p

is loc ally alg eb r a ic

p

is of Hodge - T ate typ e and it s r e s tr ic tion to the ine rtia g r oup is s emi - s imple . Pr oof. We have alr eady r emarked

(d .

1. 1) that (1 ) implie s :

- The r e s tr iction of p to the ine rtia group is semi - s im p le . Henc e we may as s ume that (* ) holds . (* )

Let

7r

b e a uniformiz ing element of K , and let pr denote 7r ab the proj e c tion map of G onto its ine r tia g r oup U K as s oc i ate d to (d. A4 and Cas s els - F r Ohlich [6] , p. 144 - 145 ) . Define a new r epr e 7r . s entahon p' of G ab by p' p

p'

=

po

pr

7r

.

doe s not affe c t the local algebr aic ity (c lear ) , nor the Hodge - T ate pr ope r ty (this follows from A I , C or . 2 to Th . 1) . S inc e (* ) implie s that p ' is s emi - s imple , this me ans Replac ing

by

LO C A L L Y A LG E B RAIC R E PR E S E NT A T IO NS

tha t , afte r re plac ing

p

by

p' ,

III - 5 3

p

w e may as s ume that

s imple and even (b y an e a s y r e duc tion ) that it is s imple . E C End (V) be the c ommuting algeb r a of p .

Sinc e

p

i s s em i Let then

i s ab e l ian

is a c ommutat ive fie l d , of finite de g r e e ove r Q p , and V i s a one - dimens i onal ve c tor space ove r E ; the r e pr e s e nta -

and s imple , tion

p

E

is g iven b y a c ontinuous characte r

Let now

K'

b e a finite extens ion of

p:

K

G

�

....

E'"

wh ich is lar g e enough

to c ontain all the Q - c onj ugate s of E . Call (I ' ) and (2 ' ) the prope r p tie s c o r r e s ponding to (1) and (2) , whe n K ' is taken as gr oundfield instead of Th. 1 of (1 ' )

�

K.

AI ,

(2 ' ) ,

We know

(d .

1 . 1 ) that (1 )

�

(1 ' ) . By Cor . 2 to

we have (2 ) � (2 ' ) . Hence it is enough to pr ove that and this has b e en done in A6 (Co r . to Prop. 6 ) ,

q.

e . d.

CHAPTER IV t -ADIC REPRESENTATIONS ATTACHED TO E LLIPTIC CURVES

Le t K be a numbe r field and let E be an elliptic curve ove r K.

If 1. i s a prime number , let

be the cor r e sponding l -a di c r e p r e sentation of K , cf. chap. I , 1 . 2 . The main r e sult of thi s Chapter i s the dete rmination of the Lie algebra of the 1. -adic Li e g r oup G 1 = Im( p ) . T hi s i s ba sed on a finitene s s l · the o r em o f Safar evic ( 1 . 4 ) combine d with the propertie s o f locally algebraic abelian r e pr e sentations ( chap . III) and T ate ' s lo cal theory of elliptic cur ve s with non-inte gral modular invar iant (Appendix, AI) . The var iation of G 1 with 1 i s studie d in § 3 . The Appe ndix give s analogous re s ul t s i n the local c a s e ( i . e . when K i s a loc al field) .

A B E LIA N l - A DI C R E P R E S E N T AT IONS

I V- Z

PR E LIMINARIES

§ 1.

1 . 1 . Elliptic c ur ve s { d. Ca s s el s [ 5 ] , De uring [ 9] , Igusa ( 1 0 ] )

By an elliptic curve , w e m e a n a n abelian var iety o f dimen sion 1,

i. e. a c omple t e , no n s i ngula r , conne c te d curve of g e nus

1

with a

given rational po int P o , taken a s an or igin fo r the c ompo sition law ( and often wr itten 0 ) . Let E be such a c ur ve . It i s we ll known that E may be embedde d , as a non - s ingula r c ubi c , in the proj e c tive plane P / ' Z K in such a way that P be come s a " fl e x" ( one take s the p r oj e c tive o embe dding define d by the c o mplete linear s e ri e s containing the

divi s o r 3 . P ) ' In thi s embe dding , thr e e point s PI ' P Z ' P 3 have o sum 0 if and only if the divi s o r P + P z + P 3 i s the inte r s e ction o f E with a line .

l B y cho o s ing a s uitable c o o r dinate sy ste m , the

equati o n of E can be wr i tten in W ei e r stra s s form Y

Z

=

4x

3

- g x - g 3 Z

whe r e x, y a r e non - homog e ne o u s c o o r dinate s and the o r i gin P the point at infinity on the y - axi s . � =

g

3

Z

o

is

T he di s c riminant

- Z7g

Z 3

i s non- z e r o . The co effi cient s g ' g a r e dete rmine d up to the transfo r z 3 * 4 . . matlOns g z � u g z ' g t---+ u 6 g ' U ti> K j T h e mo d u l ar lnvarlant · 3 3 of E i s •

E L LIP TIC C UR YES

IV-3

if

Two elliptic curve s have the same j inva riant if and only they be come i somo r phic ove r the alg e braic clo sur e of K.

(All thi s r emains valid o ve r an ar bitrary field, exc ept that, when the characteri stic is 2 o r 3, the equation of E ha s to be written in the mo re general for m y

2

He r e again, 0 i s the point at infinity on the y - axi s and the cor r e s ponding tangent i s the line a t infinity . T he r e a r e corre sponding definitions fo r A and j , fo r whi c h w e r efe r t o Deuring [ 9] o r Ogg A:

[ 2 0] ; not e , howe ver , that the r e is a mi s p r int i n Ogg ' s fo rmula for 3 the coefficient of 13 sho uld be 8 instead of - 1 . ) 4 -

1 . 2 . Good r e duc tion Le t v e l; K be a plac e of the numbe r field K. We denote by ( r e sp. m , k ) the cor r e sponding lo cal r ing in K ( r e sp . its o -v v v maximal ideal , i t s r e sidue field) . Let E be an elliptic curve over K . One say s that E ha s good r e duction at v if one c an find a coo r dinate system in P 2 / K such that the cor r e sponding e quation f for E ha s coefficient in

0

v hence and its r e duction f mo d m de fine s a non- singular cubic E ( -v v an elliptic curve ) o ver the r e sidue field k ( in o the r words , the v di s c r iminant � f) of f must be an inve rti ble element of 0 ) . The v -

-

IV - 4

A BE LIAN l - ADIC R E PR ES E N T A T IO NS

curve E

i s called the r e ductio n o f E at v; it do e s not depend on v the cho i c e of f, provide d , o f c our se, that A ( f) = 0':' . v One can p rove that the a bove definition i s e q uivalent to the following one : ther e i s an a belian s c heme E the s en s e of Mumford [ 1 9 ] , c ha p .

VI,

over Spec(O ) , in v v who s e generic fiber i s E ; thi s --

s cheme i s the n unique , and i t s spe cial fibe r i s E . Note that E v i s v defined over the finite field k ; w e denote it s Fr obeniu s endo v morphi sm by F . v On either definition, one s e e s that E ha s good reduction fo r almo st all pla c e s of K . If E ha s good r e duction at a given plac e v, its j invariant

mv and its re duc tion J mo d -i s the j invar iant of the r e duc e d cur ve E . v The conve r s e i s almo st tru e , but not quite : if j belo ng s to

i s integral at v (i. e. be long s to -

0 ) v

0 ,

the re is a finite extens ion L of K such that E XK L ha s good r e duction at all the place s of L dividing v ( thi s i s the " potential v

good re duction" of S er r e - Tate [ 3 2 ] , § 2 ) . For the proof of thi s , s e e Deur ing [ 2 9 ] , § 4 , nO 3 . Remark The definitions and r e sults of thi s s e ction have nothing to do with numbe r field s . They apply to every field with a di s cr ete valuation. 1. 3 . Prope rtie s of Let the Galois

I

VI

r elated to goo d r eduction

T1

and VI by :

be a prime number . We define , a s in chap . I, 1. 2 ,

mo dul e s

ELLIPTIC C UR VES · whe r e E

in

I V- 5

i s the ker nel of i

n

E(K )

�

E(K ) .

W e deno te by Pi the co r r e sponding homomo rphism of G a l(K / K) into Aut( T i ) · Re call that E , T and V are of rank 2 1 i 1n n o ve r Z I 1 Z , Z and Q l ' r e spe c tively . 1 Let now v be a plac e of K , with p I: i and let v be some v extension of v to K ; let D ( r e sp . I) be the cor r e sponding de com po sition gr oup ( r e sp . ine r tia gr oup) , d. chap . I , 2 . 1 . I f E ha s good r e duction a t v , one ea sily s e e s that r e duction at v define s an i somorphi sm of E onto the cor r e sponding module fo r the r educ e d n 1 curve E . In parti cular , a r e unramified at v ( chap . v o f T corre spond s v, P1 1 to the Frobenius endomor phi sm F of E . Hence : v v

I , 2 . 1) and the Frobenius automorphi sm F

det( Fv and de t( l - F

v, P

1

)

' P1

) = de t( F ) = Nv v

= det( l - F ) = 1 - T r ( F ) + Nv

v

v

.....

i s equal to the nwnber of kv -points o f E v . C onye r sely : ,

"

"

CRIT ERION OF NERON - OGG -SAFARE VIC . If V1 i s unr amifi e d then E ha s goo at v for s ome 1 I: p v , -- d r e duction at v. Fo r the proof, s e e Se r r e -T ate [ 3 2 ] , § l . COROLLAR Y - Let E and E I b e two e lliptic curve s whi ch are i sogenous ( over K) . If one of them ha s goo d r e duction at a place v, the same i s true fo r the othe r one .

IV- 6

ABE LIAN 1 -ADIC REPRESENTATIONS

(Re call that E and E ' ar e said to be i so genous if the r e exi s t s a no n - trivial morphi s m E � E ' . )

T hi s follow s fr om the theorem, since the 1 - adi c r e p r e senta

tions a s s o c iated with E and E ' a r e i s o mo rphi c . R emark Fo r a dir e c t proof of thi s cor olla ry , s e e Koi zumi - Shimura [11] . Exe r ci s e Let S b e the finite s et o f pla c e s whe r e E doe s not have good r e duction.

If

v e E K - S , we denote by t the numbe r of k -point s v v of the r e duced curve E . v ( a) Let 1 be a prime number and let m be a po sitive -

integer . Show that the following prop e r tie s ar e equi valent : m for all v e E K - S , p I:. 1 . ( i) t v !! 0 mod 1 v ( ii) The set of v e E - S such that t ;: 0 mod 1 m ha s K v density one ( d. chap. I, 2 . 2 ) . m (iii) For doll s e Irn( P l ) ' one ha s de t(l- s ) a 0 mod 1 ( The e quivalenc e of ( i i) and (iii) follow s from C e botar e v' s •

density the orem. The implications ( i) e a sy . )

�

( ii) and ( iii)

�

( i) are

( b) W e take now m = 1 . Show that the prope r tie s ( i ) , (ii) , ( iii ) ar e equival ent to : (iv) The r e exi st s an e lliptic cur ve E ' o ve r K such that: E'

�

( a. )

Eithe r E' is i somo rphic to E, or the r e exi st an i s o ge ny

E of degree 1 . ( 13 ) The gr oup E ' (K) contains an element of order 1 . ( The implication ( i v) � ( iii) i s e a s y . Fo r t h e proof of the

conve r s e , us e Exe r . 2 of chap .

I, L L )

....,..

[ fo r m > 2 , s e e K a t z [ 6 4 J . J

ELLIPTIC C UR YES 1. 4 .

I V- 7

.,

Safa r e vi c., ' s theo r eIT1 It i s the following ( d . [ 2 3 J ) : -

Let S be a finite set of pla ce s of K . The s e t of i so IT1orphi sIT1 cla s s e s o f elliptic curve s ove r K , with good r e duction at all plac e s not in S , i s finite . T HEOREM

Sinc e i so genou s curve s have the saIT1e bad r e duction s et ( d . 1 . 3 ) , thi s iIT1plie s :

COROLLAR Y - Let E b e an elliptic cur ve ove r K.

Then, up to

i s oIT1orphi s IT1 , the r e a r e o nl y a finite nUIT1be r o f e lli p tic curve s which are K - i sogenou s to

E.

T o prove the theoreIT1, w e u s e the following crite r ion for good r e duction: LE MMA

-

of

place s of K c ontaining the divi sor s of 2 and 3 , and s uch that the ring O s of S - intege r s i s Let S be a finite set

p r inci pal. Then, an e lli p tic curve E defined over K ha s good r e duction out side S if and onl y if it s equation can be put in the Z 3 W eier stra s s for IT1 y = 4x - g z x - g 3 with gi 6 Os and * 2 3 � = g - 2 7 g e O s (the group o f unit s of O S ) . 3 2 Proof. The sufficiency i s trivial . To pr ove ne c e s sity, w e wr ite the curve E in the forIT1 Y e

2

= 4x

3

- g' x - g' 2 3

( *)

K . Le t v be a place of K not in S. Then, since the r e i s good reduction at v, and since the divi sor s of 2 and 3 do not be long

with g',

1

I V- 8

to

A B E LIAN l - A DI C RE PRES E N T A T IONS

the curve E c a n be written in the fo rm

S,

with g .

�

a unit in v thi s ring . U s ing the p r op e r tie s of the W eie r stra s s fo rm, the r e i s an .." 4 6 12. element u e K "O such that g 2. , v = u v g 2.' , g 3 , v = uvg 3' , �v = u �' ; v moreove r , a s we can take g . = g � fo r almo s t all v , w e s e e that 1, v 1 we can a s sume that u = 1 for almo s t all v � S . Sinc e the r ing O s v t,: i s princ ipal , the r e i s an e lement u Ei K with v(u) = v( u ) fo r all v -2. v f: S . Then, if we repla c e x by u x and y by u 3 y in ( �, , the 1,

v

in the lo cal ring at v and the di s c r imina nt

)

curve

E

take s the for m

6 12. 4 with g 2. = u g z ' g = u g 3 and � = u �I . Sinc e , by cons truction, 3 g e aS and � e O� , the lemma i s e stabli she d . i Proof o f the theorem . After po s sibly adding a finite numbe r of pla c e s o f K t o S , we may a s s ume that S contains all the divi s o r s of 2. and i s p r in cipal . 1£ E i s an elliptic cur ve de S fined ove r K ha ving good r e duction out s ide S , the above lemma 3 , and that the ring a

tell s us that w e can wr ite

E

in the fo rm

. * 3 2. with g i e a and � = g - 2. 7 g 3 e aS · But , s inc e we a r e fr e e to S 2. to< 12. to< 12. * i s a finite and since 0s l ( aS ) multiply � by any u E ( a S )

g r oup , we s e e that the re i s a finite s e t X C O � such that any elliptic

E L L I P T I C C UR YES

IV-9

c u r ve o f the a bo v e t yp e c a n b e w r i tt e n i n the fo r m ( ':' ) with g and

� e X.

B ut , fo r a g i v e n �,

i

the e quat i o n

r e p r e s e nt s an affi n e e lliptic cur ve .

e

Os

U sing a the o r em of Siegel ( g e n

e r ali z e d b y Mahl er and Lang , d . Lang [14] , chap . VII) , o n e s e e s that

thi s equation ha s only a finite number of solutions in O S ' Thi s fini she s the proof o f the the o r e m . R e ma r k The r e ar e many way s in which one can de duc e S afa r e vi c ' s the o r em fr om Siegel ' s .

The one we followe d ha s be en shown to us by

Tate .

§ 2 . THE GALOIS MODULES A T TAC HED T O E In thi s se ction, E denote s an elliptic curve ove r K. We a r e inte r e s ted i n the structur e of the Galois modul e s E de fine d in 1 .

3.

1n

, T l ' V1

2 . 1 . The i r r e duc ibility theor em Re call fi r st that the r ing End ( E ) of K - endomorphi s m s of E K i s eit he r Z o r of r ank 2 ove r Z . In the fir st ca s e , we say that E ha s " no compl ex multipli cation ove r K . " If the sam e i s true fo r any finite exten sion of K, we s ay that E ha s " no complex multiplica tion. " THEOREM - A s sume that E ha s no complex multipli cation ove r K .

A B E LIAN 1 - A DI C R E PR ES E N T A T IO NS

I V-IO

T he n :

( a ) V 1 i s i r r educ i bl e fo r all p r im e s 1 ; ( b) E i s i r r e du c i bl e fo r almo st all p r i m e s 1 . 1 W e ne e d the fo l lowing e l ementar y r e suIt: LEMMA - Let

E

be an e lliptic curve defined ov.e r K with

End ( E ) = Z . Then , if E I � E , E " � E � K - i s o genie s with K non- i somorphic cyclic ke rne l s , the curve s E I and E " a r e non i somorphi c o ver K. Proof. Let n l and n" be r e s p e c tive ly the o r der s of the ke r ne l s of EI

�

E and E "

�

over K, and let EI

E. Suppo se that E I and E " ar e i s omorphic

-->-

E " be an i s omo rphi s m . If E

transpo s e of the i s o g e n y EI nl , and hence the i so g eny

�

�

E I i s the

E , it ha s a c yclic ke r nel o f order

E �

E, obtained by compo sition o f

E � E I , E I � E " , E " � E , ha s fo r ke rnel a n extens ion of Z / n" Z by Z / nl Z. But, sinc e End ( E ) = Z , thi s i s ogeny mus t be K multiplication by an integer a , and its ke rnel mu st the r e fore be of the form Z / a Z X Z / a Z . He nce nl and n" divide a . Since 2 a = n l n " , we o btain a = nl = n " , a contra di c tion. Proof of the the o r em. ( a) It s uffi c e s to show that, if End ( E ) = Z , ther e i s no one K dime nsional Q sub spac e of V stable unde r Gal( K / K) . Suppo s e 1 1. the r e w e r e one ; i t s inte r s e ction X with T 1 would be a s ubmodule -

with X and T / X fr e e Z -modul e s of rank 1 . For n � 0 , 1 l n consider the image X(n) o f X in E = T / l T . Thi s i s a n l n submodul e of E whi ch i s cyclic of order 1 and sta ble by n 1 Gal( K / K ) . Henc e it co rr e spond s to a finite K - algebrai c subg roup of of T

E L LIPTIC C U R YES

I V - ll

E and one can d e fi n e the q uo ti e nt c ur ve E ( n) = E / X( n) . T he ke r n e l o f t h e i s o g e ny E � E ( n ) i s c y c l i c o f o r de r I n . T h e abo ve l e mma then s ho w s that the c u r ve s E ( n ) , n � 0 , v

.,

a r e p a i r wi s e non- i s omo r p hi c,

c o ntr adi c t i ng the c o r o l la r y to Sa fa r e vi c ' s th eo r e m ( 1 . 4 ) .

( b) If E i s no t i r r e duc i bl e , the r e e xi s t s a Gal o i s submodule l of E whi ch is o ne - dime n s ional o ve r F l . I n the same way a s X l a b o ve , thi s d e fi n e s. a n i s o g en y E � E / X who s e ke r nel i s c y c li c of 1 o r de r 1 . The above lemma show s that the cur ve s which co r r e s p ond to d i ff e r e nt

value s o f "

1

a r e no n - i s o m o r phi c ,

.,

and

one

again

applie s

the c o r o ll a r y to Safa r evi c 1 s the o r e m .

Rem a r k One can prove par t ( a ) o f the a bo ve the o r em by a quite diffe r ent method ( d . [ 2 5 ] , § 3 . 4 ) ; instea d of the S afarevic ' s theorem, o ne u s e s the proper tie s of the decomposition and ine rtia s ubg roup s of

Im(p ) ' 1

d.

Appendix.

2 . 2 . Determination o f the Lie algebra of G 1 Let G = Im(p 1 ) denote the ima g e of G al( K / K) in Aut( T 1 ) , 1. and l e t " C End( V ) be the Li e a l g e b r a of G " 1 1 .f:L1 THEOREM - If E ha s no complex multipli cation & 1

=

(d.

2 . 1) , then

End( V ) , i. e . G is open in Aut( T 1 ) · 1 1

Proof. The i r r e ducibility the or em of 2 . 1 s how s that, fo r any open subgroup U of G , V1 is an i. r r e ducibl e U - modul e . He nc e , V 1 1 i s an i r r edu c i bl e .& - mo dule . B y Schur ' s lemma , it follow s that the 1 commuting algebra &1 of & in End( V 1 ) i s a field ; since 1 dim V = 2 , thi s fi eld i s eithe r Q o r a quadratic extension of Q 1 " 1 1 then &1 i s equal to e ithe r End( V 1 ) , o r the subalg ebra If " I = Q l' .f:L1

I V - 12

A B E LIA N 1 - A DI C R E PR ES EN T A TIONS

S l ( V ) of End( V ) c o n s i sting of the endo m o r phi s m s with tr a c e 0 ; 1. 1. 2 but , in the s e c on d c a s e , the a c ti o n of .& o n A V would be trivial, 1. 1.

2

and thi s woul d c o nt r adi ct the fa c t tha t the Galo i s modu.l e s AV V ( /-I ) a r e i s om o r p hi c ( chap . I , 1 . 2 ) . 1. impo s s i bl e .

Henc e '&

1

=

1

a nd

s l ( V1 ) i s

Sup po s e now that '&1 i s a q uad r a t i c ext e n sion o f Q ' 1

Then

is a one - dimen s ional '&l - ve cto r spa c e and the commuting 1. a l g e br a of '&1 in E nd( V ) i s '&1 it s elf. Henc e '& i s containe d in 1 V

1 '& 1 ' and is abelian ('&1 i s a "non- split Cartan algebr a " of End( V 1 » ' Afte r r e pla cing K by a fini t e e xt e ns i o n ( thi s doe s not affe ct '& ' 1 d . c hap .

I , L l) , we may the n s uppo s e that G

itself i s abe lian . The

i s then s e mi - simple , abelian and rational. 1 i s , mo r e ove r , lo cally algebraic . T o s e e thi s , we fir st r emark that ,

1. - adic r epre s entation V It

1.

a t a plac e v dividing 1 ,

w e have v(j ) � 0 sinc e otherwi s e the de

composition group o f v in G 1 would be non - abe lian by Tate ' s the o r y ( d . Appendix, A . 1 . 3 ) ; henc e , aft e r a finite extension of K, we can a s s um e that E ha s g o o d r e du ctio n at all plac e s v dividing 1 ( d. 1 . 2 ) .

Let E ( l ) be the 1 - divi sible group attached to E at v :::: V1 ( E ( 1 » and thi s ( d . Tate [ 3 9] , 2 . 1 , example (a » . W e have V 1 modul e i s known to be of Hodge - Tate typ e (loc . cit. , § 4 ) . U s ing anothe r re s ult of Tate ( chap .

I ll ,

1. 2 ) , thi s implie s that the r ep r e

s e ntatio n V 1 i s lo cally alge braic , a s c laime d above . ( Thi s could al so be s e e n by using, in stead of the the o r y of Ho dge - Tate module s , the l o c a l r e sults of the Appendix, A2 . )

W e may now apply to V1 the r e s ults of chap . III, 2 . 3 . Henc e , the re i s , fo r each prime 1 ' , a r ational , abelian, s emi s imple l ' - adi c r epr e s entation W l ' compatible with Vl ' But Vl '

i s compatible with V ' and V 1 , i s s emi- s imple . Hence V , i s 1 1. i s om o rphi c t o W l ' ( d . c hap . I , 2 . 3 ) . But we know ( c h a p . Ill , 2 . 3 )

I V - 13

E L LI P T I C C U R YES

that we ma y c ho o s e l '

s u c h tha t W

l'

i s the d i r e c t s um of o ne -

dim e n s i o nal s ub s pa c e s s t a b l e unde r Ga l ( K / K ) . ir r e duc i bi l ity of V , 1 .& = E n d ( V ) , q . e . d . 1 1

T hi s c o nt r adi c t s the

H e nc e , we mu st have .& ' = Q and 1 1

Remark End( E X K) K i s the c o r r e sponding ima gina ry quadratic fi eld, one show s ea sily that If E ha s complex multiplication, and L

=

Q

�

"&1 i s the Car tan s ubalgebra of End( V 1 ) defined by L l split s if and o nly if 1 de compo se s in L .

=

Q QP L . It 1

Exe r ci s e s ( In the se exe rci s e s , we a s sume E ha s no complex multipli cation. Let S b e the set of place s v e

.E

whe r e E ha s bad

- S , we denote by F v the Frobenius endomor K phi sm o f the r educ e d curve E ; if 1 f. p , w e ide ntify F to the v v v c o r r e sponding automo r phi sm of T . ) 1 1 ) Let H { X , Y ) be a polynomial in two inde terminate s X, Y r e ductio n. If v

e

.E

K

with coefficient s in a field of chara cte r i s tic z e r o . Let V

be the H s e t of tho s e V € .E - S for whi ch H( T r ( F ) , N v) = O . If H i s not K the zero polynomial , s ho w that V ha s density O . (Show that the s e t H of g e. GL( 2 , Z ) with H( T r ( g ) , det(g ) ) = 0 ha s Haar mea sur e zero . ) 1 2 ) The eigenvalue s of F may be identified with complex v numbe r s of the fo r m -1 +1. 1'" 2 - TV

(Nv) e

cf. chap . I , Appendix A . 2 . Show that the s et of v for which cp v i s 2 2 a given angle cp ha s den s ity z e r o . ( Show that T r ( F v ) = 4(Nv) co s cp

and then u s e the p r e c e ding exer ci s e . )

A B E LIAN 1 - ADIC RE PR ESENTAT IONS

I V- 14

3 ) Let L = Q( F ) b e the field gene rated by F v . By the v v p r e c e ding ex e r c i se , L i s qua dr ati c imaginary exc ept fo r a set of v v of den s ity O . (a) Let 1 be a fixed pr ime . Let C b e a semi-simple commutative a lge b ra of rank 2 . Let X c be the s et of element s 5 e Aut (V 1 ) that the su ba l ge b r a Q [ 5 1 of E nd( V ) generated by s is i somo r 1.

1.

Show that X c i s open in Aut( V J. ) ' and show that it has a p hi c to C . non - empty inter se ction with e very open subgroup o f Aut( V 1 ) , in par ticular , with G J. . ( b) Show that F v e X c if and only if the fi eld L v i s qua dr atic and L � Q is i s o morphic to C . v 1. ( c ) Let J. , . . . , 1 be di stinct prime numbe r s , and choo s e fo r n l each an algebra C . o f the type conside r e d in ( a) . Show that the set 1 of v fo r whi ch F e X c . fo r i = 1 , . . . , n ha s dens ity > O . v 1 ( Us e the fact that the image of Gal(K/ K) in any finite pr oduct of the Aut( V1 ) is ope n ; thi s is an e a s y con sequence of the the o r em proved above . ) ( d) Deduce that, fo r any finite s e t P of prime num be r s , the r e exist an infinity of v s uc h that L v i s ramifi e d at a l l 1. e P . In pa rti cular , the r e ar e an infinite number of di s tinct fie lds L v . 2 . 3 . The i sogeny the o rem THEOREM - Let E and E ' be elliptic cur ve s over K, let

J.

be a

pr ime numbe r and let V ( E ) and V ( E I ) be the co r r e s ponding 1 J. J. - adi c r epr e s entations of K . Suppo s e that the Galoi s modul e s V ( E ) and V ( E I ) a r e i s omorp hic and that the modular invariant j J. 1 of E ( d. 1 . 1 ) is not an integer of K. T hen E and E' are K i s ogenous .

E L LI P T I C C UR VES

I V - IS

W e ne ed the following r e sul t : PROPOSITION - Le t E and E' be e lliptic curve s over following c onditions ar e e quival ent:

K.

The

(a) ThE' Galo i s modul e s V1 (E) and V1 ( E ' ) are i s omo rphi c

fo r all 1 .

( b) The Galo i s module s V 1 (E) and V1 ( E ' ) ar e i s omo rphi c

fo r one 1 .

( c ) If Fv and F v' a r e the Frobe nius e s of the r e duced curve s E v -and E'v , we ha ve T r ( F v) = T r ( F v' ) fo r all v whe r e the r e i s good reduction. ( d) Fo r a set of place s o f K of density one we have T r ( F ) = T r ( F v' ) . v Clearly ( a ) implie s ( b) , and ( c ) implie s ( d) . The implication

( c ) follow s from the fa ct that T r ( F ) i s known when V1 i s v known. T o pr ove ( d) � ( a) one r e marks fir st that the repr e s enta tions o f Gal( K / K ) in V/ E) and V ( E' ) have the s ame tra c e , by (b)

==:I>

1

.,

Cebotar e v ' s den s ity the o r em ( chap . I , 2 . 2 ) . Mor eover ,

V1 ( E) (and

al so V 1 ( E ' ) ) i s s e mi - simple . Thi s i s c lear if E ha s no complex multipli cation over K since V i E ) is then i r r e ducible ( 2 . 1 ) ; if E

ha s compl ex multiplication, it follow s from the Remark in 2 . 2 . Since

V i E) and V/ E ' ) are s e mi - s imple and have the same trac e , they

ar e i s omo rphic . Remark s 1)

If E and E ' have go od re duc tion at v, let t -

be the number o f k -points of E ( r e sp . E ' ) . v v v fo rmula s ( cf. 1 . 3 ) :

( r e sp . t ' ) v v We ha ve the

A B E LIAN l - A DI C R E P R E S E N T A T IO NS

IV-16

t

v t v'

==

1 - Tr(F

v } + Nv

- T r(F' }

=

v

+

Nv

Henc e conditi on ( c ) ( r e sp . condition ( d ) ) i s equivalent to saying that tv

=

v' s

t v' fo r all v whe r e the r e i s good r e duction ( r e s p . fo r a s e t of o f d e n s ity o ne ) .

2 } If E and E ' a r e K - i s o genou s , it i s clear that co nditions

(a) , ( b ) , ( c ) , ( d) are s ati sfie d . Proof o f the the o r e m . In view o f R emar k 2 ) above , i t suffi ce s to show t h a t t h e e q u iv a l e n t c o n d it i o n s ( a ) , ( b ) , ( c ) , ( d ) i m p ly t h a t t h e e l l i p t i c

curve s E and E' are i s ogenous when the modular invar i ant j of E

i s no t an inte g e r of K . L e t v be a pla c e o f K such that and let p be the char a cte r i s tic If j '

=

that v(j ' ) � 0 .

j ( E ' ) , we

fir s t

the r e sidue field k . v s how that v( j ' ) i s al s o < of

v(

O.

j) < 0,

Suppo s e

T he n, afte r po s s i bly r eplac ing K b y a finite

exte n sion, we may a s s um e that E ' ha s good r e duction at v. v

Then, if L

(d. 1 . 3 ) ;

f.

p , the Galoi s -module

but V / E) i s ramifie d at

v:

V (E' ) L

th i s follow s either from

the c rite rion of N e r on - Ogg - S af a r e v i c ( 1 . 3 ) or

fr

of the ine r tia gr oup given in the Appendix, A. 1. fa ct that V ( E ) and V ( E ' ) a r e i somo rphi c .

L

i s unr amified at

om the dete rmination

3.

Thi s contradi c t s the

L

Let now q and q ' be the elements of K v whi ch corr e s pond to j and j ' in Tate ' s the o r y ( d. Appendix A . l . I ) , and let E and q E q , be the co r r e sponding elliptic cur ve s ( lo c . cit) . Since E and E q have the same modula r inva riant j , the r e i s a finite extens ion K ' of K wher e they be c om e i somo rphi c , and we can do the same v fo r E' and E q " H e nc e , the T at e modul e s T p ( E q ) and T p ( E q , )

b e come i s omo rphic ove r

K' .

B ut , i n thi s c a s e

the i s o g e ny

I V- 17

ELLIPTIC CUR YES

the o r em i s tr ue ( d . Appe ndix A . l . 4 ) , i . e . the curve s E q and E q l , hence also E and E I , a r e K I - i s ogenous . Howeve r , if two elliptic cur ve s are i s ogeno u s ove r some exte n sion of the ground field, they a r e i so genous ove r a finite extension of the g r ound fi eld. We may thus cho o s e a finite ext e n s ion L o f K and an L - i s o g eny f

:

E� L

�

E I X L . We will s how that f i s automatically defined K s ove r K. For thi s , it s uffi c e s to s how that f = f fo r all s e Gal( K / K) , or , e quivalently , that V( f) : V ( E ) � V ( E I ) p P _ commute s with the a ction of Galoi s . Howe ver , if G = Gal(K / L) i s L the open subgroup o f G = Gal(K / K ) whi ch c o r r e sponds to L, then

V( f) commute s with the a ction of G . It is then enough to show that L Hom ( V, VI ) = Hom { V, VI ) . But V and VI are i somo r phi c a s G G L

G-module s . Hence we ha ve to show that End thi s i s clearly true ; in fa ct, G and G

( V) = End { V) . But G G L ar e open in Aut{ V) by the

L theorem in s e ction 4 , and he nce thei r commuting algebra i s r educed to the homothetie s in each c a s e , i . e. End Thi s complete s the proof of the the or e m .

G

( V) L

=

End { V ) G

=

Q . P

Remark It i s very likely that the theorem is true without the hypothe sis that j i s not inte gral. T hi s could be proved (by Tate ' s method [3 8]) if the following gene ralizatio n o f S afarevi c! ' s theorem were tr ue : given a finite sub set S o f E K , the abelian varietie s ove r K , of dime nsion 2, with polarization of degr ee one , and good reduction outside S , are in finite number ( up to i somo r phi sm) . b e e n p r o v e d b y F a lti n g s , s e e [ 5 4 ] , [ 5 6 ] , [ 8 2 ] . ]

� [ th is h as

IV-1S

A B E LIA N £ - A DIC R E PRESENTAT I ONS

§ 3 . VAR IA T ION OF G £ AND G £ WI T H £ 3 . 1 . P r eliminar i e s W e ke ep the no tations o f the p r e c e ding parag r aphs . Fo r each p r ime num be r 1 , we denote by p £ the homomo rphi sm

defined by the action of Gal ( K / K ) on T £ ' The p l ' s define a homomo rphi sm p : Gal( K / K )

�

n Aut ( T 1 ) 1

,

whe r e the p r o duct i s take n o ve r the s e t of all prime numbe r s . and G 1 = Im( p 1 ) C A ut ( T 1 ) , so th that G 1 i s the image o f G unde r the 1 p r oj ection map . Let G 1. be the imag e of G in Aut( E ) = Aut(T 1 i T 1 ) :::- GL( 2 , F 1 ) . 1 1 1 Let G

=

Im(p)

e n Aut( T 1 ) 1

( 1) The image o f G E.Y de t : n Aut( T ) ---+ n 1 �: * ( 2 ) For . Z 1 almo = st all , det( and de t(G ) G ) . 1 = F 1. 1 1 LEMMA

-

z;

i s open.

__

-

1 . 2 ) that det( P 1 ) : Gal(K / K ) ---+ Z l i s the character X giving the actio n o f Gal(K / K) on 1 n -th r o ot s 1 of unity . Hence det(G) C n Z ; i s the Galo i s gr oup Gal(K / K) , c whe r e K = 0 K i s the extens ion of K gene r ated by all r oo t s o f c c unity . Since one know s that Gal(O 1 0 ) = n Z * ( d . fo r instanc e [1 3 ] , 1 c chap . IV) it follows that det( G) i s the open subg r oup of n Z ; c o r r e sponding to the fi eld K n Q , henc e ( 1 ) . A s se r tion ( 2 ) follow s c W e know

( d.

chap . I ,

,�

�

�

ELLIPTIC CUR YES from

(1)

I V- 19

and the defi nition of the p r oduct topol o gy .

A s sum e now that E ha s n o complex multi plication . W e know ( d . 2 . 2 ) that each G l i s open in Aut( T 1 ) . T hi s do e s no t a priori imply that G its elf i s ope n . Howe ve r : PROPOSITION - The following prope r ti e s a r e e quivalent: ( i ) G i s open in n Aut( T 1 ) . 1 ( i i) � 1 == Aut( T 1 ) fo r almo s t all 1 . ( iii) � 1 == Aut ( E 1 ) fo r almo s t all 1 . (iv) G 1 contains S L( E 1 ) fo r almo st all 1 . plication ( iv)

==:»

==:»

==:»

==:»

(iv) a r e trivial . Im ( i ) fo llow s from the following gr oup - theo r etical

The impli cations ( i)

( ii )

( iii)

r e sult, who se proof will be g iven in s e c tion 3 . 4 below : MAIN LEMMA - Le t G be a clo s e d s ubgroup of n GL( 2 , Z ) and let 1 ..... G 1 and G 1 denote it s image s in GL( 2 , Z l ) and GL( 2 , F 1 ) a s above . A s sume : (a ) G 1 i s open in GL( 2 , Z l ) fo r all 1 . i s open . ( b) The image of G � de t : n G L( 2 , Z 1 ) � n ( c ) G contains S L( 2 , F 1 ) fo r almo st all 1 . 1 Then G i s open in n GL( 2 , Z l ) '

z;

R emark Fo r each integ e r n � 1 , l e t ..... of o r de r dividing n, and let G be n :::: / / nZ). K ) � Aut(E ) GL( 2 , Z Gal( K n ( i ) above i s equival ent to

E

b e the g r oup o f point s of E(K) n the image o f the cano nical map One s e e s ea sily that prope r ty

( i ' ) The index of G n in Aut( E n ) i s bo unded.

IV- 2 0

3. 2.

A B E LIA N L -A DI C R E P R E S E N T A T I O NS

The ca s e o f a non - i nte g ral

j

T HEOREM - A s s um e that the modula r inva riant j o f E i s not an inte g e r of K . Then E e nj o y s the equi valent properti e s ( i ) , ( ii ) , (iii) , (iv) o f 3 . 1 . Sinc e <

that v(j )

j

O.

i s not inte g r a l , w e c a n c ho o s e a p la c e

Let q be the element o f the lo cal field

c o r r e spond s to j by T ate ' s the o r y

( d.

v

of K

such

K

whi ch v Appendix, A . 1 . 1) and l e t E

be the c o r r e spondin g e lliptic curve over K . There i s a finite v e xte n s i o n K ' of K o ve r whi c h E and E a r e i s omorphic ; one q v can e v e n take fo r K ' e ither K o r a quadratic extension of K v . v Let v' be the valuation o f K ' whi ch extend s v; a s sume v' i s no rmali ze d s o that

* V' ( K I ) =

n

=

q

Z , and let v' ( q)

=

-

v' ( j )

W e have n > 1 . LEMMA 1

-

A s s ume 1 do e s no t divide n , and let I

be the v, 1 ine r tia s ub g roup of G 1 c or r e sponding to SOme extension of v to K . Then I l contains a tran s vection, i . e . an e l em ent who s e matr ix v, fo r a s uitable F 1 - ba s i s o f E . fo rm i s 1

(� �)

The

Thi s i s tr ue fo r the c ur ve E o ver K ' , d. Appendix, A . l . 5 . q ::: r e s ul t fo r E follo w s fro m the i somo rphi s m E , . l K ' E q/ K

LEMMA 2

-

Let H be a subg r oup of GL( 2 , F 1 ) which ac t s i r r e ducibly

o n F 1 X F 1 and which contains a tran svection. T h e n H contains

SL( 2 , F ) · 1

IV- 2l

ELLIPTIC CUR VES For any transve ction s

£

H , let

b e the unique one s dime nsiona l s ub s pa c e of F 1 X F 1 whi ch i s fixed b y s . If a l l such line s were the same , the line so defined would be stable by H, and D

H would not be i r r e ducibl e . Henc e the r e a r e transvections s , s ' such that D s " D s " If w e c hoo s e a s uitabl e ba s i s ( e , e ' ) of

e

H

F l X Fl , thi s means tha t the mat r ix fo r m s of s , s ' a r e s

,

=

(1

0)

1 1

The lemma follows then from the well known fact that the se two matrice s generate SL( Z , F1 ) . Proof of the the o r e m . --

Lemma 1 s how s that. fo r almo st all 1 , I

v,

l'

G , contains a transve ction. On the other hand. we 1 -know ( c f . Z . l) that G1 is ir reduc le for almo st all 1 . Applying le mma 2 to G 1 we then see that G 1 c o ntain s SL( E ) for almo st 1

and a fo rtiori

1;

all

�

hence we have ( iv) , q . e . d.

Remar k

s eems likely that the condition " j i s not integral" can be r epla c e d b y the weaker o n e " E ha s no complex multiplication. " It

�

[ y es : s e e [ 7 6 ] . ]

3. 3.

Num e r i cal example When E is given exp lic itly and ha s a

n n-i

o

nt e g r al j , one

may sometim e s dete rmine the finite s e t of l ' s with G 1 " GL( 2 , Fl ) . Take for instanc e K = Q , and E d e fin e d by the e quation : y

2

+x

3

2 +x + x = O .

Thi s i s the curve 3 + o f Ogg ' s li s t [ 2 0] ; its j invar iant i s Z 11 3 - 1 ,

A B E LIAN 1 - ADIC REPRESENTATIONS

I V- 2 2 i t s di s c r im inant i s

A =

4 - 2 3 , its " c onduc tor " i s 2 4 ( it i s 2 -

i s o g eno u s to the modular cur ve

c o r r e spo nding to the cong r uence 24 s ubgroup r ( 2 4 ) , d. [ 2 0 J ) . The exi stence of a non - tri vial 2 - i s o g eny o fo r E s hows that G 1 1:. GL( 2 , F ) fo r 1 = 2 ( G i s cyclic of order 2 1 2 and cor r e sponds to the quadratic fi e l d Q( .[:3 ) ) . But , for 1 1:. 2 , one ..... ha s G 1 GL( 2 , F ) . Inde e d , G ha s the following pr ope r tie s : 1 ... 1 a ) �et( G 1 ) = F ; , d. 3 . 1 . b ) G 1 contains a tran sve ction. T hi s follows fr om Lemma 1 J

=

and the fac t that n i s he r e e qual to 1 . .....

E

�

c ) G 1 i s irr educible . If no t, the r e wo uld be an iso geny E ' o f degree 1 ( de fined ove r Q) . The curve E ' would have

the same conduc tor 24 as E , henc e would b e one o f the curve s + + + 1 , 2 , 3 , 4 , 5 , 6 of Og g ' s li s t . But Ogg ha s proved that, fo r e a ch such c ur ve , the r e i s an i s o geny E ' � E of degr e e 1 , 2 , 4 o r 8 . T he map E � E ' � E would then b e a n endomo r phi sm o f E o f degr e e 1 , 2 1 , 41 or 81 , and thi s i s impo s sible fo r 1 1:. 2 sinc e End( E ) = Z . Now, using l emma 2 , one s ee s that prope rtie s a) , b) , c ) imply

Exe r c i s e

Pr o ve that G 1 = GL( 2 , F 1 ) fo r all 1 1:. 2 whe n K = Q and E i s an elliptic cur ve of c onductor 3 . 2 x., whe r e X. � 6 . ( U s e Ogg ' s + Table 1 . For X. = 5 , note that the curve s 7 and 7 be c ome

i s omor phi c over Q( i) , but a r e not i s ogenous ove r Q. For X. = 6 , + + u s e a s imilar ar gument, and ob s e r ve that the cur ve s 1 0 and 1 8 d o not have the s am e numbe r o f point s mod . 5 , he nce are not i s ogenou s ove r

Q. )

What hap p e n s whe n

X.

=

7, 8?

E L LI P T I C

I V- 2 3

C UR YES

P r o o f of the main l e mma o f 3 . 1

3 . 4.

W e ne e d fir s t a few l e mma s : LEMMA 1

= P S L ( 2 , F ) = S L( 2 , F ) / {±l } , 1 > 3 . Then 5 l l l 1 i s a s imple g r oup if 1 > S . E ve r y p r op e r s ubg r o up of 5 i s s o l vable -

Let S

1

or i s omorphi c to t h e a l t e r natin g g r oup A : the la s t po s s ibility S o c cu r s o nly i f 1 = + 1 mo d . S .

Thi s i s w el l known, d. fo r in stance Burnside [ 4 ] , cha p .

XX .

LEMMA 2 - No p r o p e r su b g r o up of 5L( Z , F ) map s onto P5L( Z , F ) . l 1 Thi s i s clear fo r 1 = Z , sinc e P5 L ( Z , F Z ) :: SL(Z , F Z ) . For 1 � 2 , s up p o s e the r e i s s uc h a p r o p e r s u b g r o u p X . W e would t h e n

have

=

SL( Z , F ) 1 and

thi s i s

{±1} X X ,

a b s u r d , s i n c e S L ( 2 , F ) i s g e n e r a t e d by the

.l

1 1 1 0 ( 0 1) and ( l 1) which are of order 1 ,

LEMMA 3

-

Let

X

e leme

nt s

henc e c ontaine d in X.

be a clo s ed subgr oup of S L( 2 ,

Z ) 1

who s e image

in 5L( Z , F .l ) i s SL( Z , F 1 ) . As swne l ? S . T hen X :: SL( 2 , Z ) . l W e prove by induction o n n that X map s onto SL( Z , Z I 1 n Z ) . Thi s i s true fo r n = 1 . A s sume it i s true for n , and let us prove it a b fo r n +1 . It i s enough to s how that, for any s = ( c d ) € S L( 2 , Z 1 ) whi ch i s congruent to 1 mod. l n, the r e i s x e X with n n+l W rite s :: 1 + 1 u ; since d e t( s ) :: 1 , one ha s x == s mod. 1 T r ( u)

==

0

mod . 1 .

But it i s e a s y to s e e that any such u i s congrue nt Z m o d . 1 to a swn of matrice s u . with u . :: O . Henc e , we may 1 1 Z a s sume that u :: O . By the induction hypothe s i s , the r e exi s t s y e X n - lu + L n v, whe r e v ha s c o e ffi c i e nt s in Z . Put s u c h tha t y = 1 + 1 1

I Y- 24 x =

y

1

A B E LIAN l - A DIC R E PRESENTATIONS .

W e ha ve : -l n + 1 ( 1 n u + 1 v) n 1 + ( 1 n - l u + 1 v)

x =

If n � 2 , it i s clea r that fo r n

x ==

Inde e d, since u2

= 1.

+

I

x

==

I

2 But (u + l v) :: l (uv +

I

=

+ + 1 nu mod . l n l . Thi s i s al s o true 0 , and u + L v == u mod. 1 , we have mod . l 2 .

2

1

Thi s s how s that

1

mod. 1 , henc e :

1( u + 1 v) :: l ( uv + vu)u

p r o ve s l emma 3 .

•

+ l u + ( u + l v) vu)

n 2 1 n-l ( ) ( 1 u + 1 v) + . . . 2

x�

I

2

==

0 mod . 1

2

since 1 > 4

•

n+l n + 1 u mod. 1 in all ca s e s , and

W e now consider a clo s e d subgroup G of X = n GL( 2 , Z ) 1 ha ving the pr ope r ti e s ( a) , (b) , ( c ) of the main lemma of 3 . 1 . Let S be a finite s et of prime s , and X s = n GL(2 , Z l ) . leS The image G of G by the proj e c tion X Xs i s S � open in X . S Repla c ing G by an open s ubgr oup if ne ce s sa ry, we can a s surne that e ach G l ' 1 e S , is c ontaine d in the group of ele ment s congrue nt t o I moq . 1 , henc e that e a c h G i s a pro - 1 - gr oup . Since 1 G i s a subg r oup of n G , it follow s that G i s pro - nilpotent S S 1 eS 1 ( p r oj e c tive limit of finite nilpotent group s) , henc e i s the product o f i i t s S ylow s ub g r o up s . Thi s s ho w s t ha t G S = lU G l ' and s i n c e G 1 s LEMMA 4

-

ELLIPTIC C UR YES

IV 25 -

o p e n in GL( 2 , Z l ) b y p rope r ty (a) , we s e e tha t G S i s o p e n in X . S B e fo r e we go fur the r , we introduc e s ome te rmino logy. Let Y be a profi nite group , and E a finite simple g r oup . W e say that E occur s in Y if the r e exi s t clo s e d subg roups Y , Y o f Y such that Y i s 2 1 I no rmal in Y 2 and Y 2 / Y 1 i s i s omorphi c to E . W e denote by O c c ( Y ) the s e t of clas s e s of finite simpl e non abelian group s o c c ur r ing i n Y . If Y

=

we have

lim . Y , and ea ch Y � Y i s surj ective , a a

�

Occ(Y)

= U

If Y i s an exten sion of Y ' and Occ( Y)

=

Oc c( Y ) . a

ylI,

we have :

Occ( Y ' ) U OCC ( ylI ) .

U sing the s e fo rmulae and lemm a 1 , one gets :

whe r e S 1

=

PSL( 2 , F 1 ) a s befo r e , and: Occ(S1 )

= tJ

O c c ( S1 )

=

O c c ( S1 )

=

O c c ( S1 )

=

if 1

=

2, 3

{A } if 1 = 5 S {S 1 } if 1 � ± 2 mod . 5 , 1 > 5 {SI }

=

{ S I ' A S } if 1

!!

± 1 mod . 5 , 1 > 5

Let now S be a finite s e t o f prime s so that 2 , 3 , 5

EO

S and

A B E LIA N l -A DI C R E P R E S E N T A T IONS

IV-26

1 f: S �

X =

� SL ( 2 ,

F ). l

Pr o p e r ty

( c ) show s that such a s e t exi s t s .

The group G c o ntain s n SL( 2 , Z I ) · I f.S a r tial product i s unde r s tood a s a subgroup of the full p roduct p

L E MMA ( T hi s

Gl 5

-

n G L( 2 , Z I ) · ) 1

It i s enough to show tha t G contains each S L( 2 , Z I ) ' 1 f: S . L e t HI = G n GL( 2 , Z ) . If 1 . S , the fac t that G 1 1 co ntains S L( 2 , F ) shows that S e O c c ( G ) hence S E:: Occ( G) . On the othe r 1 1 1 1 hand , G / H l i s i somo rphi c to a clo s e d s ubg roup of n GL ( 2 , Z I ' ) 1 ' 1: 1 hence S I � O c c ( G / H ) (we u s e the obvious fa c t that the s imple group s l S , p � 5 , ar e pairwi s e non i s o mo rphi c ) . Since p

we then ha ve S

e

-

1 O c c ( H l ) . Le t -HI be the image of HI in S L( 2 , F 1 ) ; the ke rnel of H � H a p r o - 1 - g roup, we have I being I O c c ( H l ) = Oc c (H 1 ) , henc e S I e O c c ( H ) . Hence H 1 map s onto l S 1 = PSL( 2 , F ) ' and, by lemma 2 , we have H = S L( 2 , F ) and, by 1 I 1 le mm a 3 , H1 = SL( 2 , Z 1 ) . Hence G contains SL( 2 , Z 1 ) . LEMMA 6 - The group G contains an open subgr oup of n S L( 2 , Z 1 ) .

be the proj e c tio J of G into S n G L( 2 , Z 1 ) and GS the p r oj e ction into the complementar y product 1eS Let S be a s in lemma 5 ; le t G

n GL( 2 , Z 1 ) · Let HS be G n n GL( 2 , Z 1 ) and 1 .S U: S H S = G n n GL( 2 , Z 1 ) ' s o that H C G ' H S C GS • One ha s canoni cal S S 1 .S i s omorphi sm s :

I V- 2 7

E L LI P T I C C U R VES

G /H S

S

�

G/ ( H X H ' ) S S

Lemma 5 shows tha t H S contains

�

G'S / H S'

n SL( 2 ,

1$S

Z ),

so that GS / H S

1

i s abelian. Henc e GS / HS i s abelian and H contains the adhe rence S ( G ' G ) of the c ommutator group of G . By lemma 4 , G i s open S S S S in n GL(Z , Z 1 ) . It is ea sy to see that thi s impli e s that ( G ' G ) S S 1eS contain s an open subgr oup of n SL( 2 , 1eS

Z

1

) (thi s follows fo r instanc e

fr om the fa ct that the der i ve d Lie alg e br a of gl

i s sl ) . Hence Z Z H S c ontains an open subg roup U of n SL( Z , Z 1 ) . U sing lemma 5 , 1

we then s e e that G contains U X n SL( Z , eS

n SL( Z , 1

1fS

Z ). 1

Z

1

) whi c h i s open in

End of the proof Consider the dete rminant map det : n GL( Z , 1

who s e ker nel i s n SL( Z , Z

1

).

Z ) � 1

n 1

Z

;

,

Hypothe s i s ( c ) means that the image of

G by thi s map is open and lemma 6 s how s that G n K e r ( de t ) is open

in K e r ( det) . Thi s i s enough to imply that G it s e lf i s open, q . e . d . Exe r ci s e s 1 ) a ) Gene ralize le mm a 3 t o S L( d , Z } fo r d > Z, 1 > 5 ( same 1 method) . b) Show that the only clo s e d subgroup o f SL( d , Z ) which 3 Z map s onto SL(d, Z / 3 Z } i s SL ( d, Z } itself. 3 c ) Show that the only clo s e d subg r oup of S L( d , Z } which Z 3 map s onto SL( d , Z / Z Z) is SL( d , Z Z } it s elf. Z } Let E be the unr amifi e d quadratic extens ion of Q Z ' and

A B E LIA N l -A DIC R E PRESE N T A T IO NS

IV- Z 8

its ring o f inte g e r s . Le t x � x b e the non tr ivial automo r E phi s m o f E . a) Show that 0 c ontains a pr imitive thi r d root o f unity z . E b) Show that 0 E c ontains an element u with u . u = - 1 ( take fo r instanc e u = ( 1 + 15) / 2 ) . 0

u ( x) that and

=

u

j3

j3

- linear e ndomo r phi sm s defined by = ux , whe r e z and u a r e a s in a ) , b) above . Show -1 -1 is of o rde r 3 , j3 o f order 4 , and j3 u j3 = u , s o that a

c ) Let z x, j3 (x)

a

generate

and

be the

Z

z

a non - a b e l i a n g r o up

G

of o r d e r

12.

d) Show that G i s c o ntaine d in SL( O ) :::: SL( 2 , Z ) and that Z E r e duction mo d . 2 define s a h o m o m o r p h ism of G onto 5L( Z , FZ ) ' (Hence lemma

3

3 ) Le t S

do e s not extend to the ca s e 1 =

9 S3 ) '

=

Z. )

SL( Z , Z / 9 Z ) , S3 = SL( Z , Z / 3 Z ) and

T he g r o up g i s i so mo rphic to a thr e e 9� Z dime nsional v e c t o r spac e ove r F3 . Le t x e; H ( 5 3 ' g) be the mology cla s s co r r e spo nding to the extension

g

=

Ker ( S

a ) Show that the re str i ction o f

x

to a 3 -Sylow

c o ho -

s ub g r o up

of

is z e r o ( no t e that S L( 2 , Z ) c o nta i n s a n e l e m e nt of o r d e r 3 , vi z . ( - 31 1 ) ) ' -2 b) Deduc e fr om a) that x = 0 , i . e . that the r e exi st s a s ubS

3

gr oup X of S 9 which is mapp e d i so mo r phically onto S 3 ' ( The inve r se im ag e of X in S L( Z , Z 3 ) is a non - tr ivial s ubg roup which i s

mapped onto S ; henc e l e mma 3 doe s not ext end to the ca se 1 3

= 3. )

E L LIPTIC C U R YES

I V- 2 9 A P P E N DIX La c a l

R e s ul t s

K deno te s a fi e ld wh i c h i s complete with

I n wha t fo llow s ,

re spe c t to a di s c r e te valuation v; we denote by 0 ( r e sp . by k) K the r ing of intege r s ( r e sp . the r e s i due field) of K ; we a s sume that k i s p e r fe c t and of c ha r a c t e r i s t i c p 1=

o.

L e t E b e a n e l l i p ti c cur v e over K and l e t

I.

be

a p r ime

number diffe r ent fr om the char a c t er i s t i c of K. Let T J. and V I. be the cor r e sponding Galois mo dule s ; we denote by G the ima ge of 1 Gal(K / K) in Aut( T 1 ) , an d by I i. the i n e r ti a s u b gr o up o f GJ. . The Lie algebra s '&1 = Lie ( G ) , 1,1 = Li e ( I ) ar e su ba lg e bra s of End( V ) 1 1 1 and we will deter mi ne them und e r suitabl e a s sumptions on K and v; note that, sinc e I

i.

is

an invar iant subgr oup o f G , it s Lie algebra - J.

(1 -i 1 i s a n ideal o f i:l.1 If j = j ( E ) i s the modular i nvariant of E ( d. 1 . 1) , we •

--

consider the c a s e s v( j ) < 0 and v(j ) � 0 s eparately .

A. 1 .

The C a s e

v(j )

<

O.

In thi s se ction we a s s um e that the modular invariant j of the e lliptic cur ve

A. I. I.

E

ha s

a

pole . i .

e . tha t

v(j )

< 0.

The e lliptic curve s o f T ate L

et

q

be an element of K with ...

v( q)

> 0 , and let

r

q

be the

di s c r ete subgr o up o f K gene r at e d b y q . Then , b y Tate ' s theory of ul trame tr i c t h e t a function s ( unp ubl i s h e d - but s e e Morikawa , Nagoya ....

IV-3 0

A B E LIAN l -A DIC R E PR E S E N T A T rONS

1 9 62 ) ,

the r e is an e l lip ti c curve

E

with the pr o p e r ty tha t , fo r a ny finite e xt e n s ion

K'

Math . J o u r n . ,

analyti c g r o up

K'

*I r

q

of E

with value s in q in the fo rm

q

K' .

T he e q uation de fining

K, E (K ' )

the

of

i s i s omo r p hi c to the g r oup

E

K

de fine d o ve r

o f po int s

q

q

c an be wr itte n

with b

2

3 n n = 5 � n q I (l-q ) n>l

the s e s e r i e s c onve r ging in

and b

K.

3

n 3 n = � ( 7 n 5 + 5 n ) q l 12 ( 1 - q ) n> l

,

of E

The modula r invariant j ( q )

q

is

give n by the u sual fo rmula 3 ( 1 + 4 8b ) 2 1 j ( q) = = - + 7 44 + 1 9 6 8 84 q + . . . n 24 q qn (l_q ) n>l a s e r ie s with inte gr al c o e ffi c i e n t s . o f the fr a c t io n s

FI G,

+00 - 00

with c o e ffi c i e nt s in

K,

c o ns i s t s

q F a nd G a r e Laur e nt s e r i e s

wh e r e

F = �

E

T he function fie ld of

a z n

n -00

c onver ging fo r all val ue s of

z I: 0 ,00 ,

a nd

s u c h that F( q z ) 1 G( q z ) = F ( z ) 1 G( z) . S in c e the modular inva r iant j i s such tha t v(j )

< 0,

of the given e lliptic curve

E

and s inc e the s e r i e s fo r j ( q) ha s inte g r a l

c o effi c i e nt s , o n e c a n c ho o s e

q

s o tha t j = j ( q) .

The e lliptic c urve s

b e come the n i so mo r ph i c o ve r a finite ext e n s io n o f K q (whi c h c an be taken to be of de g r e e 2 ) . Henc e , aft e r p o s s ibly r e -

E and E

plac ing

K

by a fini te e xt e n s io n , w e may a s sume that

E

=

E

q

•

I

E L LI P T I C C U R YES

A. l. 2 .

V- 3 1

An exact s e qu e nc e notation o f

W e c o n s e r v e the

n

':'

in K / r

A. l. l.

Le t

.

be t he ke r n e l o f n gr oup of 1. - t h r o o t s E

n

If Jl. n i s t he of unity i n K s , we have an inj e ction Jl. --7 E . On the o the r hand , n n n 1. if z e E , we have z E r , and hence the r e e xi s ts an inte g e r c n q in s u c h th a t z q C . If we a s s o c i a t e t o z the irn.a ge of c in Z / i nZ , we obtain a horn.orn.o rphi s rn. o f E into Z / 1. n Z , and the r e s ulting n

rn. ul t i p l i c a tion by

1.

s

q

s e qu e n c e

(1 ) i s a n exact s e quenc e of Gal{K / K) -module s , Gal{K / K) a cting s

n

s

trivially on Z / 1. Z . Pa s s ing to the limit, we o btain an exact s e que nc e of Galoi s rn.o dule s o --+

T ( Jl. )

1.

--7

T (E )

1

q

--7

Z --+ 0

(2)

1

whe r e Gal{K / K ) a c t s trivially o n Z l ' T e n s o r ing with Q ' we l o b t a in t h e e xa ct s e q ue n c e o --+ V ( Jl. ) -7 V (E

1

1

q

)

-7 Q

1 -7 0

•

W e now show that thi s s e quenc e of Gal{K / K ) -module s doe s

s

no t spli t .

T o do thi s w e intr o du ce an inva r iant

the gr oup

em H { G , ).I n ) '

1

boundary homomorphi sm:

wher e G

=

x

whi c h belong s to

Gal (K / K ) . Let d be the co

( 3)

A B E LIA N l -A DI C R E P R E S E N T A T lONS

IV- 3 Z

w i th r e s p e c t to the e xa c t s e que n c e ( 1 ) a n d l e t

i s the el eITlent of lj ITl

l

x

n

=

d( l ) .

T he inva r i a n t

define d by the f aITlil y ( x ) , n � l . n n I ':' ::, 1 � H (G, I-l ) o f PR OPOSIT ION ( a ) The i soITlo r p hi sm 5 K / K n n ..' .1 i nto x . KUITlITl e r theo ry transfo r ITl s the cla s s of q mod K " n x

H ( G 'l-l ) n

-

x

( b ) The e l eITl e nt

( R e call that the exa c t sequenc e

5

i s of infinite o r de r .

i s induc e d by the co bounda ry map r e lative to

A s s e r tio n ( a ) i s p r oved by an e a s y cOITlputation. T o prove ( b ) , note that the valuation

v

de f ine s a hOITloITlo rphi sITl

and hence a hOITloITlo r phi s m � Z If

we identify

in

(a) ,

x

we have

COROLLAR Y

-

1

with the c o r r e sponding eleITlent of � K f(x)

=

v( q) , he nce

x

.." .'

/K

.'.". 1

n as

i s of i nfinite o rde r .

The s e quence ( 3 ) doe s not split .

A s sUITle it doe s , i . e . the r e i s a G- subspace X o f V

(E

1 q T {E } n X.

)

whi ch i s ITlapped i s oITlo rphi c ally onto Q l ' Let X = 1 q T . N T he iITlage of X T in Z l 1 S /. Z /. ' fo r S O ITle N > 0 . It i s the n ea sy N to s e e t ha t 1 x = 0 , and thi s c ontr adict s the f a c t that x i s o f

I V- j j

E L LI P T I C C U R Y E S

i nfinite o r d e r .

A. l. 3 .

a nd i po � 1 -- - 1 W e ke e p the no tation of A . l . l a n d A . l .

D e t e r mination o f

u { V1 ) u

e

C X,

E.X

I f X is

V = V ( E) , 1 1

and let !:. X with u{ X ) = O .

b e t h e s uba l g e b r a o f E. X

fo r me d by t h o s e

T HEOREM - ( a ) If k i s a lg e br ai c a ll y c lo s e d and 1 1= p, is

a

one - dime n s ional s ub s pa c e

( c) s u b s pac - e =

X of

then t h e r e

V

s uc h that '& = !:.X · 1 1 ( b) If k i s a l g e br a i c a l l y c l o s e d a nd 1 = p , then the r e

o n e - dim e n s io na l s u b spa c e

£

a one -

l e t E. d e no t e the s u bal g e b r a X c o n s i s ting of tho s e e nd o m o r phi s m s u f o r whi c h

dim e n s i onal s ub s pa c e o f

o f E nd{ V 1 )

Z.

pl ·

If k X

of

i s finite ,

of V

X

the n �l. po --

V ' a nd i I. -- -I.

=

n

-X

1 = r

such that

-X

is

a

.& = E. ' £ X

fo r some one -dim ens ional

( r e sn • �

i

-£

= r

-X

)

if

-

1

-/:. P ( r e s p .

Proof. Note fi r s t that , s i nc e .&£ and 2:.£ are invar iant under finite e xte n s i o n of

(a)

In

K, we may a s sume

tha t

E = E q £n - th r o ot s of unity , hence t hi s ca s e , K c ontains the •

Gal(K / K) acts trivially on T £ ( � ) . Conse quently, the r e i s a ba s i s s e , e Z of T l ( E) s u c h that, fo r all a E Gal(K / K) , w e have 1 a ( e ) = e , a( e Z ) = a (a) e + e Z with a (a) E 2 1 , Mor e over , the l l l ho m o m o r p h i sm a � a(a) cannot be tr ivial since the s e q uence ( 3 ) d o e s not s plit .

It

follow s that Im( a) i s an open subgroup of

h e n c e that E. = !:. with X = V /�) . 1 X ( b) Since 1 = p , we mus t have In t h i s

ca s e , the action o f Gal( K/ K ) o n

c ha r a c t e r

x£

( d . c ha p . I,

c ha r (K) = 0 a s

1 1=

Zl '

and

char (K) .

i s by means o f t he 1 . 2 ) whi ch i s of infinite orde r . It follows V (�) 1

IV-34

A B E LIA N

1 - A DI C

R E P R E S E N T A T IO NS

that '&' = !. whe r e X = V /J-I ) ; in fac t , '&' .J .:: s i n c e the s e quence 1 X X' 1 ( 3 ) do e s no t split, and we canno t have '&' 1 = ':: X·

( c ) Since k i s finite , the a ction of Gal(K / K ) on T (J-I) i s 1 not tr ivial no r e ven o f finite o r der . Henc e , the ar gum e nt u s e d i n (b)

show s that '&' = .!.X , whe re X = V / J-I ) . Applying (a) to the c om 1 pletion of the maximal unramifi e d exten sion of K , we s e e that iL

n = _

X

L

if

I:

a n d tha t

p,

i

-1.

=

r

if

-x

1. =

p.

E x e r ci s e

I n c a s e ( a ) , shows that Im( a ) = 1 n Z l ' whe r e 1 n i s the highe st powe r of 1 whi ch divide s v( q ) = - v(j ) . A . I . 4 . Appli cation to i s o g e nie s He r e , we a s sume that k i s finite and K i s of characteri stic o

(i. e . K i s a finite exten s ion o f q , q' e

T HE O R E M - L e t

and E'

=

K

*

wi th

Q ).

p

v( q )

a nd

v(q' )

>

O.

Let

E = E

E , be the c o r r e sponding elliptic cur ve s o ve r K . Then q

q

the following are equival ent : ( 1)

E q i s K - i sogenous to E q' -

(2)

T h e r e a r e inte g e r s

( 3 ) V ( E) and p

Pr oof.

-

( 2 ) =:::3> ( 1 ) .

V

p

'

i. e. , (1)

E

q

==='I> ( 3 ) .

q

A

= q,

B

a s Gal( K / K ) -modul e s . ( E ' ) a r e i somo rphic q

and E

q

A

are

e ve r y me r omorphic function F I G invariant A by q i s i n va r iant und e r mul tip l i c a t i o n by q ;

hence the function field of E A

s u c h that

B ut

unde r m ul tipli c a t io n

q

1

It suffi c e s to show that E

i s ogenous ove r K .

E

A, B �

and E

A

q T r i via l .

q

i s contained in the function field of

a r e i s ogenou s .

I V- 3 5

ELLIPTIC C UR YES C hoo s e an i s omor phi sm

( E) onto p V ( E ' ) . Since V ( JJ ) i s the only o ne - dimen si o nal s ubspa c e of Vp ( E ) p p ( r e sp . V ( E ' » s table by G = Gal( K / K) ,
( 2 ) .

that cp ma p s T ( E ) into T ( E ' ) . p p diag r am :

W e the n have a commutative

o � T (JJ ) � T ( E ) p p

o�

whe r e ( r e sp . E

and

I T

p

p

(JJ)

�1 � T (E') p

(4) �

Z �0 p

0')

i s the mult ipl i cation by a p -adi c inte g e r r l s ) . If X, x ' ar e the element s of Vm H ( G ,JJ ) a s sociated to n E ' ( d. A . I . 2 ) , th e c ommutativi t y of ( 4 ) s how s tha t

p

( r e sp .

rx

=

sx'

.

But the valuation v yields a homomo rphi s m o f n * *p 1 into Z , and we have s e e n that the lJm H ( G, JJ n) = l;im K / K p image of x i s v(q) , and the imag e of x ' i s V( ql ) . Henc e r v ( q)

=

W e will now s how that the element

s v( q ' ) .

IV-3 6 is a

A B E LIA N l -A DIC R E PR E S E N T A TrONS

r o ot o f unity . Fi r st of all , the iIna g e o f p - th r o o t of unity ; in fact, thi s image i s a

a

z

in {f m K / K ��

*p

n is

v( q ' ) x - v(q) x ' , and multiplyi ng by s , we find 0 in vir tue of the above fo rmula e n * *p ( note that � m K / K i s a Z mo d ul e , hence multiplication by s p -

make s s e n s e ) .

K

�

W e the n u s e the fa c t tha t

p VIn K * I K *

n

the ke rnel of

. ( vi ewed, a s usual a s a subg roup of K "" ) . * * 1 de compo s e s K a s a p r odu c t Z X k X U , whe r e is

k

*

To s e e thi s , o ne l* U i s the g r oup o f uni t s c ong r u e nt to 1 . The functo r n * l A � l i m A I A P t r a n s fo r m s Z into Z , kills k and leave s U � p l unc ha ng e d s i n c e U is a finit e ly ge n e rated Z -mo dul e . Henc e , we p * ha ve z E k , and z i s a root o f unity . Thi s implie s ( 1 ) , q. e . d . ,

R e mark T he e quival e nc e ( 1 ) � ( 2 ) wa s r e marked by Tate .

It

i s true

wi thout any hypothe s i s on K .

Exe r ci s e Sho w tha t the hypo the s i s

"k i s A. 1.

of

imag e

v(K )

=

o f transve ctions

/

-

G

in the ine r tia g r oup

( d . A . 1 . I) , l e t G 1 be the -

E

be the e llipti c curve 1 Gal ( K K ) in Aut( T T

of Z.

s ubg r oup *

p

Exi s t e nc e Let

may b e r eplac e d by

ove r F . "

al ge br a i c

5.

" k i s fini t e "

1

.

/

W e a S S UIne that

E

q

a nd let 1 1 be the 1) ' v i s no rmali z ed , i . e .

iner tia that

I Y- 3 7

E LLI P T I C C U R YES

- If

1

do e s not divide v( q) , then 1 1 c o nt a ins a trans 1 1 ve ction , i . e . an element who s e matr ix i s ( ) fo r a s uit a bl e P R O POSIT ION

o

1

P r o o f . Aft e r po s sibly r e pla c ing K by a l a r g e r field, we can suppo s e

that the r e sidue fie ld k i s algebraically clo s e d , and that K c o nta i n s the 1 - th root s of unity . In fac t , if 1 I:. p , thi s l a s t c o ndi ti o n i s i mpl i e d by the fir st; if 1 = p , we mu st adj oin the se roots ; but the degr e e of the exten s io n thus o b t a in e d divide s 1 - 1 ,

henc e

i s p r i me to

1. , a nd the valuation of q r e mains p r i m e to 1 . Thi s being said, the h ypot h e S1· S on v ( q ) s h ows th at qIl l 1. S n o t ln K . T hus t h e r e 1. S a n I/ 1 l/ automo r phi sm s 6 Ga l (K I K ) such that s ( q 1 ) = zq , with s z 1= 1 . Then z i s a pr imitive I - t h r o o t of uni t y , and z , q I I I fo r m .

modulo IT . Since s ( z ) = z , we see that the image 1 1 of s in G 1 = 1 i s the r e quir e d t r a n s v e cti o n .

a ba s i s of T

1

.....

A . 2 . The ca s e

v

(j ) � 0

In thi s s e ction we a s sume that the modular invariant j of the e l li p t ic curve E i s integral , i . e . that v(j ) �

O.

Henc e , afte r

p o s si bly r epla cing K b y a finite exte nsion, we may a s sume that E

ha s goo d r eduction ( d. 1 . 2 ) . W e al s o a s sume that

K i s o f charac -

te ri s tic zero. A . 2 . 1 . The ca s e 1 1= p Suppo s e that 1 1= p . Since E ha s goo d r eduction, the module T 1 can be ide ntified wi th the T at e mo dul e T 1 ( E) of the r e duced --

c urve

E,

cf.

1. 3 .

He nc e t h e i n e r tia a l g e b r a

II

is

O.

I f the

A B E LIA N l - A DI C R E PR E S E N T A T IO NS

I V- 3 8

r e sidue field k i s finite , the g r oup G i s topologically gene rated by 1 the Frobenius F . He nc e , in thi s ca s e , .8. 1 = Lie(G 1 ) i s a one L dime ns iona l s ubalg e br a o f End( V1 ) . = P

A . 2 . 2 . The c a s e 1

with good r e duction of height ""

2

He r e we a s s um e that the r e duced curve E i s of he ight 2 ; r e call that, if A i s an abelian va riety defined over a field of chara c h t e r i s tic p , it s height ca n be defined a s the integ e r h fo r which p i s the inseparable part of the de g r e e of the homothety " multiplication by p . " An elliptic cur ve is of height inva r iant ( c f. Deur ing [ 9] ) is O . S inc e E(p ) o ve r

O

K

E

ha s good r e duction, it

ove r OK hence a p - divi sible v ( cf . T ate [ 3 9 ] , 2 . 1 - see also [2 6 ] , §l, Ex.

define s an abe lian s cheme E �

if and only if its Ha s se

2

'

2).

T he conne cted p component E( p) of E ( p ) cOi � ide s with the fo rmal gr oup ( o ve r O ) K attache d to E ; the he ight o f E i s p r e c i sely the he ight of thi s v o fo rmal g r o up ( in the usual s e ns e ) . In our ca s e , we have E(p) = E(p) since the height i s a s sume d to be 2 . The Tate modul e o f E ( p) can b e i dentified with 0

T HEOREM - One ha s

� = �.

T

•

T hi s Li e algebra is eithe r End( Vp )

o r a no n- split Cartan s ubalgebr a o f End( V ) . p

( R e call that a non - split Ca r ta n s ubalgebra of End( Vp ) i s a co mmutative subalg ebra of r ank 2 w ith re spect to which V i s p i r r e duci bl e . It i s give n by a quadr atic s ubfield of End( V ) . ) p Proof. The Li e algebra no n z e r o element

z

�

of V

ha s the p rope r ty that

�z =

fo r any p In particular,

V

( d . [ 2 7 ] , p . 12 8 , Prop. 8 ) . p Vp i s an i r r e ducible � - mo dul e ; i t s commuting algebr a i s eithe r a fi eld of de g r e e 2 (whi ch i s then ne ce s sarily e qual to �) o r the

ELLIPTIC C UR YES

IV- 3 9

fi eld Q p , in whi ch ca s e � i s a p r i o r i s L Z o r gL Z . But sp � s L Z Z s ince A Vp i s cano ni cally i s omo r phic to V ( J.l ) , and the action o f p Gal(K / K ) on Vp ( J.l ) i s by mean s o f the charac ter X , which i s of p infinite order ( inde ed, no finite exten s ion of K can contain all p n - th roots o f unity , n = 1 , Z , ) . Hence the Lie algebra sp i s e ithe r End( V ) or a no n s plit Cartan s ubalgebr a o f End( V ) . Sinc e the p p above appli e s to the completion of the maximal unramified exten sion •

.

•

o f K, we have the same alte r native for i Mo r eove r , i is t> t> containe d in sp . W e have a priori thre e po s si bilitie s : •

( a) i = g_ = End( V ) . p t> t> ( b) i = g . i s a no n s plit Cartan subalgebra o f End(V ) . p t> t> ( c) � i s a Car tan subalgebra and sp = End( Vp ) . Howe ve r , � i s an ideal of � . He nce , ( c ) i s impo s s ibl e , and thi s

prove s the theorem . Remarks 1)

By a the o r em of Tate ( [3 9 ] , § 4 , cor .

1

to th o 4 ) , the alge bra

sp i s a Ca rtan s ubalg e br a of E nd( Vp ) if and only if E(p) ha s

" fo r mal complex multiplication, " i . e . if and only if the ring of endo

morphi sms of E(p) , o ve r a suita ble extens ion o f K , i s of rank 2 over Z . The r e exi s t elliptic c ur ve s without complex multiplication p (in the algebraic s e n s e ) who s e p - c omple tio n E ( p) have formal complex multipli cation.

Z ) Suppo s e that sp is a Cartan s ubalge bra o f End( V ) , and p l e t H = � n Aut( Vp ) be the c a r r e sponding Ca r tan subg roup of Aut( V ) . If N is the nor malize r o f H in Aut( V ) , then one know s p p that N / H i s cyclic of order 2 . Sinc e G e N , i t follows that G p p i s commutative if and only if G p C H. T he ca s e Gp C H corre spond s to the ca s e whe r e the for mal co mplex multipli cation of E (p) i s

I V-40

A B E LIA N l -A DlC R E PR E S E N T A T lONS

d e fi n e d o v e r K,

a n d the c a s e

G

rr p I-

H c o r r e s p-o n d s to the ca s e whe r e

thi s fo r ma l multipl i c a t i o n i s d e f in e d o ve r a qua d r a t i c e xte n s i o n of K . 3)

S up p o s e that G

fi e l d k i s fin it e .

L et

m ul t i p l i c a t i o n ( i . e .

o f E nd ( V ) ) .

p Ga l ( K / K ) o n

�

p

F

i s c o m mu ta t i ve , a nd t ha t t h e r e s i due

be

the

it se lf,

p

vi

e w e d a s a n a s s o c i a t i v e s u ba l g e b r a

d e n o t e s t h e g r o up o f uni t s o f F , F i s g i ve n by a ho mo m o r p hi s m

If U

V

qu a d r a t i c fie l d o f fo r m a l c o m p l ex

q>

G a l ( K I K ) --7 U

t h e a c ti o n o f

F

B y l o c a l c la s s fi e l d the o r y , w e ma y i d e n tify the ine r tia g r oup o f

ab Ga l ( K / K ) with t h e g r oup

U

K

o f uni t s of K .

Henc e the r e s t r i ction

CP I of

into U o F K we fi r s t r e m a r k that the a c ti o n o f End ( E ( p ) ) o n

T o d e t e r mi ne

t h e ta ng ent s p a c e t o E ( p ) d e fi n e s a n e m be d di n g o f F i nto K . t h a t e m b e dding , o n e ha s ( c o m p a r e with c ha p . Ill ,

qJ I

( x)

fo r a ll

x

EO

U

Inde e d , by a r e s ul t of Lubin ( A n n . of Math.

85 ,

fo r m a l g r oup

E(p) ,

E'

w hi c h i s K - i s o g e n o u s to

e n domo r phi s m s the ring o f inte g e r s of F . Lubi n - T at e g r oup o ve r K E'

For

A . 4)

K

1 9 6 7 ) , the r e i s a and ha s fo r r i ng of

But the n , i f E "

is a

( d. Lubin - T a t e [ 1 7 ] ) , the fo r m a l g r o up s

and E " a r e i so mo r p h i c ove r the c o m p l e ti o n o f the maximal

r a m i fi e d e xte n s i o n o f K ( d . Lubin [ 1 6 ] , th o 4 . 3 . 2 ) . the fo r mula ( ':' ) , we may a s s um e that E ( p )

un

H e n c e t o p r o ve

i s a Lubi n - T a t e g r oup , ln

whi c h c a s e the fo r m ul a ( ,:' ) foll o w s fr o m the m a i n r e s u l t o f [ 1 7 ] .

E L LI P T I C C U R Y E S

A. 2 .

3.

I V- 41

A uxi l i a r y r e s u l t s o n a b e l ia n va r i e t i e s

Let A

and

B

be two a be l i a n va r i e ti e s o v e r

r e du c t i o n , s o t h a t t h e a s s o c ia t e d p - divi s i b l e g r o up s

a r e d e f i ne d ( th e s e a r e

Tate [3 9]) .

o f A a nd

.....

and l e t

--

f( p )

( re sp.

B

( r e sp . of A(p)

T HEOR E M 1 - L e t

--

A{p)

--

f : A � B

and

.....

B{ p »

and

B( p)

' d. K be the r e du c ti o n s O

B(p» .

a nd

be a m o r p h i s m o f a be l i a n va r i e tie s ,

b e t h e co r r e s po nd i ng mo rphi s m o f

--

A( p )

f( p ) : A ( p ) � B ( p )

A s sume the r e i s a mo r phi sm

f( p ) .

--

with good

A(p)

p - di vi s i b l e g r o up s o ve r t h e r ing

L e t A and

B

K,

T h e n , the r e i s a mo rphi s m

f :

A --7

B

i nto

B( p ) .

who s e r e d u c t i o n i s

who s e r e du c ti o n i s

f.

A p r o o f o f t h i s " l ifting " the o r e m ha s b e e n g i v e n by Tate i n a

S e mi na r ( W o o d s H o l e , 1 9 6 4 ) , but ha s no t y e t b e e n publi s he d ; a d i ffe r

ent p ro o f h as b e e n given b y

W.

M essi n g ( L .

N.

264, 1972 ).

T ( A ) i s a d i r e c t s um o f Z - m odul e s of p p inva r i ant unde r the a c ti o n of Ga l ( K / K ) . T h e n e ve r y e ndomo r -

T HEOR E M 2 - A s s um e r a nk 1

--

phi s m o f A lift s to an e n domo r phi s m o f A , homomo r p hi s m

--

E n d ( A ) � End ( A )

i . e . , t h e r e duction

i s s u rj e c ti ve ( a n d h e n c e bij e ctive ,

s i n c e i t i s known to be inj e c tive ) .

U s i ng the o r e m 1 , one s e e s tha t it i s e n o u g h t o s how tha t any e ndomo rphi sm o f

A ( p ) can b e l ift e d t o an e nd o m o r p hi s m o f A( p ) .

But the a s s um p t i o n made on A ( p)

T

i M p l i e s ( c f. T a t e [ 3 9] , 4 . 2 ) tha t p i s a p r o du c t o f p - di vi s ibl e g r oup s of he i g ht 1 . H e n c e w e a r e

r e du c e d t o p r oving t h e fo llowing e l e m e nta r y r e s u l t : b e t w o p - di vi s i b l e g r o up s , o ve r H ' H Z I T h e n the r e du c t i o n map :

L E MMA - L e t b o th of h e i g ht o n e .

H o m ( H 1 , H Z ) � H o m( H 1 , H Z )

i s bij e c ti ve .

OK '

A B E LIA N 1. - A DI C R E PR E S E N TA T IONS

I V- 4Z

P r o o f . Thi s i s c le a r if both H

a r e e ta l e . I f both a r e no t Z e tal e , the i r dual s a r e e t a l e a nd w e a r e r e duc e d to the p r e vio u s c a s e .

----

and H

1

If one o f them i s e ta l e , a n d the othe r i s n o t , o n e c he ck s r e adily tha t Hom ( H , H ) = Ho m ( H , H ) Z l Z l -

=

O.

C OR O L LA R Y - A s s um e :

( i) V ( A ) i s a dir e c t s um o f o ne - dim e n s io nal s u b s p a c e s s ta b l e p u nde r Gal( K I K ) . Then

( U ) T he r e s idue fi e ld

A.

k of K i s finite .

i s i s o g e no u s t o a p r od u c t of a b e lian va r i e ti e s o f ( C M) .!y.p!:.

( i n t h e s e n s e o f Shimur a - T a niyama

[ 3 4] ,

d.

a l s o cha p . II , Z 8 ) . .

A s s umption ( i ) impli e s tha t T ( A ) c onta i n s a latti c e T ' p whi c h i s a di r e c t sum o f fr e e Z - mo d ul e s of r ank 1 stable unde r p Gal ( K / K ) . One can find a n i s o g e ny A ---,;.. A s u c h that T ( A ) i s l p l m a p p e d onto T ' . T hi s m e a n s tha t , a ft e r r e pl a c ing A b y a n P r oo f .

A,

i s o g e no u s var i e t y , w e ma y a pply T h . Z t o -

E nd( A ) ---,;.. E nd(A) i s a n i s o mo r phi s m .

B u t , s i n c e k i s finite , i t

fo llow s fr o m a r e s ul t o f T ate [ 3 8] that Q s i m p l e c o m m u ta t i ve Q - s uba l g e br a

A

i. e .

�

-

End( A ) c ontains a s e mi -

o f r a nk Z dim ( A ) ( thi s i s not

e xpl i c i tl y s ta t e d i n [ 3 8] . but fol l ow s e a s il y from its "Main The o r em " ) . Henc e , the same i s t r u e fo r Q p r odu c t of c omm uta tive fi e l d s a p r oduct A , a

q.

nA

e . d.

A. Z . 4.

a

,

The c a s e

whe r e A

1.

a

�

E nd(A ) .

A , a

If w e now w r i t e

A

as a

one s e e s that A i s i s o ge no u s to

ha s c om p l e x multip l i c a ti o n o f type

= P with good r e du c ti o n of he ight

1

In thi s s e ction, w e a s sume that the r e duc e d cur ve E i s o f i . e . that i t s Ha s s e inva r iant i s f 0 ( d . D e u r ing [ 9] ) . o c o nne c t e d c omponent -£ 1 = E ( p) o f the p - divi s i b l e g r o up E ( p)

height

1

The

I V- 43

E L LI P T I C C UR YES

atta ched to E Sinc e E ( p )

( d.

Tate [ 3 9 ] ) i s the n a fo rrnal g r oup of he i ght

1.

an exten sion o f E by an etale gr oup , w e o btain an l exact s e quenc e of Gal ( K/ K) - mo dul e s is

whe r e X cor r e spond s t o the T at : modul e o f E l , and Y to the points of orde r a powe r of p of E . T HEOREM - Suppo s e that the r e sidue fie ld k i s finite . T hen the following s tatement s a r e e quivalent: ( a) The elliptic cur ve E ha s c omplex mul tiplication ove r K . ( a l ) The e lli p tic cur ve E ha s complex multi p lication ove r an extension of K. ( b) The r e exi s t s a one - dimens ional sub spa of V , which - ce D P i s a sup plementa r y sub s p ace of X , and i s stable unde r the a ction of

--

Gp .

( b ' ) The r e exi s t s a one -dimens ional sub spa of V which - ce D p i s a sup ple mentar y subsp ace of X , and is sta ble unde r the a ction of

--

.£

p

=

Li e ( G p ) .

Pr oof. If D i s sta ble under the a ctio n o f Gp , it i s al s o stable under -the a ction of its Lie algebra � ' hence ( b) =='l> ( b ' ) . Conver sely, if D is stable unde r �' i t s transfo rm s by G a r e in finite numbe r ; p a standa rd me an value ar gum ent then show s that the seque nc e ( * ) split s , hence (b ' ) ='l>(b) . The impli cation ( b) =='l> ( a ) ( the only non

trivial o ne ) follow s fro m the cor ollary to theo r em 2 of A . 2 . 3 . C on ve r s ely , if E ha s complex multiplication by an imaginar y quadratic field F , the group Gal( K / K ) a c t s on Vp thr ough F � Q ( s e e p chap . I I , 2 . 8 ) and thi s action i s thu s s em i - simple . Co n s e que ntly, the

I V- 44

A B E LIA N l -A DIC R E PR E S E N T A T IO NS

exact s e quenc e (a' )

=:l>

( 1,, )

split s ; thi s s hows tha t ( a ) =:l>

( b ' ) . Since (a)

=:l>

( b) , hence al s o that

( a l ) i s trivia l , the the o r e m is pr ove d .

If E ha s no c omplex mul tiplication, � is the Bor e l subalge b of End( V ) fo r me d b -y tho s e u e E nd( V ) - bra X P P s uch that u( X) C X ; the ine r tia al gebra � i s the subalgebra !.X o f C OR O L LAR y

1

-

b formed b y tho s e u X Le t Xx and

e

E nd( V ) s u c h that u ( V ) C P

p

x.

Xy

be the characte r s o f Gal( K / K ) defined by the one - dimensional module s X and y . Since k i s finite , Xy i s

o f infinite o rder . I f X i s the cha r a c t e r de fine d by the ac tion of Gal( K/ K ) on V ( IJ ) , the i somorphi s m s p X

�

y

- A2 V - V __

__

P

P

( IJ )

-1 s how that X X y = X · Hence the r e striction of X x and X X y to X X the ine r tia subgr oup of Gal(K/ K) a r e of infinite orde r . T hi s show s

fi r st that � i s either � o r a C artan subalgebra of � ; s inc e X X the s e c ond ca s e would imply ( b ' ) , i t i s impo s s ibl e , henc e � = �X .

Similarly, one s e e s fi r st that i

i s c ontained in r ' the n that its X action on X i s non tri vial ; sinc e it i s an ideal in � = � ' the s e X p r operti e s imply i = r ' X -p -p

Remark

T he above r e sult is g ive n in [ 2 5 ] . p. 2 4 5 , Th . 1 , but mi s stated: the algebra !. has be en wrongly defined a s fo rme d of X tho s e u such that u( X) = 0 (in stead of u( V ) C X) . p C OROLLAR Y 2 - If E ha s c omplex multiplication,

�

is a split

Cartan s ubalg ebra of End( Vp l . If D is a sup plementary sub s p ace

I V-45

E L LI P T I C C UR YE S

t o X s table unde r Gal( K / K) , the n X and D a r e the characte r i s ti c s ub spa c e s o f � and the inertia a lgebra � is the s ubalgebra of End( V p ) fo rm e d b y tho se u e End( V ) s uch that u( D) p The p r oof i s analo gou s to the one of C o r . s imple r ) .

1

=

0, u(X) C X.

( and in fac t

B IB LIOGRAPH Y

[1]

[2 ] [3 ]

E . ARTIN - Coll e c te d Pap e r s (e dited b y S . Lang and J . T ate ) , Addi s on - W e sley, 19 6 5 .

E . ARTIN and J . TATE - Cla s s field the o ry . Ha rvard, 1961 . M. ARTIN e t A. GROTHENDIECK - Cohomo10gie e ta1e d e s s ch e m a s . S e m . G e om . a1 g . ,

sur Yvette. [4 ] [5] [6 ] (7]

[8 ] [9]

[1 0 ]

W.

1. H.

E. S . , 196 3 / 6 4, Bure s

BURNSIDE - The Theo ry of Group s (Sec ond Edit. ) . Camb r idg e Uill v . P r e s s . 1911 .

J . CASSELS - Diophantine equations with special r efe re nc e to elliptic curve s . J . London Math. Soc 41. 19 6 6 . p . 193 - 2 9 1 . .. J . CASSELS and A. FROHLICH - Algeb raic Numbe r Theory . Ac ademic Pr e s s . 196 7 . N. CEBOTARE V - Die B e s timmung de r Dichtigkeit eine r Menge von P rimzah1en. we1che zu e ine r geg ebenen Substitutionskla s s e g ehl:>ren. Math. Annalen. 9 5 . 19 2 5 . p . 1 51 - 228 . C . CHEVALLE Y - Deux th e or�m e s d a r i thm e tiqu e . J . Math. Soc . Japan. 3. 19 51, p. 3 6 - 4 4 . M. DEURING - Die Typen d e r Multiplikato renring e ellipti sche r Funktione nk�rp e r . Abh. Math . Sem. Hamburg. 14. 1941, p. 197 -27 2 . J . IGUSA - Fibre sy stems of Jac obian va rietie s . 1 . Ame r . • •

'

J.

Ill.

o f Ma th s . , 7 8 , 1 9 5 6 , p . 1 7 1 - 1 9 9 ; II, i d . , p . 7 4 5 - 7 6 0 ; id . ,

81,

19 5 9 .

p.

453 -476 .

B -2

[ 11 ] [1 2 ] [1 3 ]

AB ELIA N l -ADI C R E PRES ENTATIONS

S . KOI Z U MI and G . SHIMU RA - On Sp e c ia l i z a t i o n s of Abe lian Var ietie s . Sc . Pape r s C ol I . G e n . E d . , U niv . T okyo . 9 . 19 5 9 . p . 18 7 - 211 .

o S . LANG - Abe lian va r i e tie s . Inte r s c . T ra c t s n 7 . New York. 19 5 7 . S . LANG - Alg e braic numbe r s . Addi s o n - W e sley. Ne w Yo rk.

[ 14 ]

1964 . o S . LANG - Diophantine Geometry . Inte r s c . T ract s n 11 .

[15 ]

S . LANG - Intr oduc tion to tran s c endental numbe r s . Addi s on -

[ 16 ] [1 7 ] [ 18 ]

New York. 1 96 2 . W e sley. New Y o rk, 1 9 6 6 . J . LUBIN - One parame ter formal Lie g r oup s ove r p -adic inte g e r ring s . Ann. of Maths . • 8 0. 19 64 . p. 46 4 -484 . J . LU BIN and J . TAT E - F o rmal c omplex multipl ication in lo cal field s . Ann. of Math s

.

•

8 1 , 1 9 6 5 . p . 3 8 0 - 38 7 .

G . D . MOST OW and T . TAMAGAWA - On the c ompac tne s s of a r ithmetic ally define d homo geneous spac e s . Ann. of Math s .

•

7 6 , 19 6 2 . p . 446 - 4 6 3 .

[1 9 ]

D. MUMF OR D - Geometric Inva r iant Theory . Erg ebni s s e

[ 2 0j

A. P . OGG - Abe lian curv e s of s mall c onduc tor . J ourn . fUr

[21]

T . ONO - A rithme tic of alg e b raic tori . Ann . of Math s

[22] [23]

der Ma th •

•

Bd. 3 4 . Sp r ing e r - Ve rlag . 196 5 .

die reine und ang . Math .

•

2 26 . 1 9 6 7 . p. 2 04 - 21 5 . .

•

74.

19 61 . p . 1 01 - 1 3 9 . ,

-

G . POL YA und G. S Z EGO - Aufgaben und Lehr sa.tze aus de r te Analy s i s . Band 1 . Sp r ing e r -V e rlag . 2 Aufl . . 19 5 4 .

1.

�AFAREVI � - Alg ebraic Number Field s . Proc . Int .

Cong r e s s . Stockholm . 1 9 6 2 , p . 16 3 -1 7 6 (A . M . S . T rans l . , S e r . 2 , vol . 31, p . 2 5 - 3 9 ) .

B-3

B I B L IOGRAP H Y

[24]

J . - P o SERRE - Sur l e s g r oup e s de c ongruenc e de s va riete s abelienne s . I z v . Aka d . Nauk . S . S . S . R . , 28 , 1964 , p. 3 -20.

[25]

J.

-Po

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Tate ) .

[26]

J. -Po

[27]

J . - P o SERRE - Sur l e s g roup e s de Galois attac he s aux

J.

g roup e s p - div i sibl e s . P r o c e e d . Conf . on Local F iel d s , Sp ring e r -Ve rlag, 1 9 6 7 , p . 113 -13l . [28]

J . - P o SERRE - L i e alg e b r a s and Lie g roup s . B enjamin, New Y ork, 196 5 .

[ 2 9]

J . - Po SERR E - C o rp s Locaux. He rmann, Paris , 196 2 .

[ 3 0]

J . - P o SERRE - Dependanc e d' exponentiell e s p -adique s . S e m . De l an g e - P i s o t - Poi t o u , 7 e anne e, 196 5/66 , e x p o s e

15.

[31]

J . - P o SERRE - R e sume de s c ou r s 1 96 5 /6 6 . Annuai r e du College de F ranc e , 1 9 6 6 -6 7 , p .

[ 3 2] [33]

49 -58 .

J . - P o SERRE and J . TATE - Good r e duc ti on of abelian v a r i etie s , A n n . o f M a t h . 8 8 ( 1 9 6 8 ) , p . 4 9 2 -5 1 7

G.

SHIMURA - A r ec ipr o c ity law in nons o1vab1 e extensions .

J ournal fUr die r e ine und ang . Ma th . , 2 21 , 1 9 6 6 , p . 2 0 9 220. [ 3 4]

G.

SHI MURA a n d Y .

T A NI Y AMA - C om p l e x

m ult i p l i c a t i o n of

abelian va rietie s and it s applic ation s to numbe r the ory . Publ . Math. So c . Japan, 6 , 1 9 6 1 .

[ 3 5]

Y . TANIYAMA

-

L fun c tions

of

fun c t i o n s of a b e l i a n v a r i e t i e s .

numbe r fields and z e ta J ou r n .

M a th .

Soc .

Japan,

B -4

[ 3 6]

A B E LIAN l -A DI C R E PR E S E N T A TIONS

J.

9, 19 5 7 , p. 3 3 0 - 3 6 6 . TATE Alg ebraic cycl e s and pol e s of zeta func tion s . _

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[38]

J . TAT E

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A.

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W ElL

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g roup of an alg ebraic numbe r fiel d . P r oc . Int . Symp . T okyo -Nikko , 1 9 5 5 , [4 2 ]

A . W ElL

[4 3 ]

A.

p.

1 -7 .

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S u p p l e m e n tary B i b l i o g r a p h y

B-6

[60] 1.-M. FONTAINE-Formes differentiel les e t mod u les d e Tate d es varictcs abeliennes sur les corps locaux , Invent. M ath . 65 ( 1 982) , p. 379-409 . [6 1 ] 1.-M . FONTAINE-Representations p ad i q ues , Proc. Int. Congress 1 9 8 3 , vol . I , p. 475-486. -

[62] 1.-M. FONTAINE and W. :MESS ING-p-adic periods and p-a dic etale coho mology, Contemp. Math. 67 ( 1987), p. 1 79-207. [63] G . HENNIART-Representations /-adiques abeliennes, S emin ai re de Th cori e des Nombres 1 980/8 1 , Birkhau ser- Verlag 1 982, p. 1 07- 1 26. [64] N. KATZ-Galois properties of torsion points on abelian varieties, I n v en t . Math. 62 ( 1 98 1 ), p. 48 1 -502. [65] . R. P. LANGLANDS-Modular forms and /-adic representations, Lecture Notes in Math. 349, p. 36 1 -500, Springer-Verlag, 1 973 . [66] R. P. LANGLANDS-Automorphic representations, Shimura varieti es, and motives. Ein Mru-chen, Proc. Symp. Pure Math. 33, A.M.S . ( 1 979), vol . 2, p. 205-246. [67] D. MUMFORD-Families of abelian varieties, Proc. Symp. Pure Math . IX, A.M.S. 1 966, p. 347-35 1 . ' [68] M. OHTA-On /-adic rep rese n ta ti on s attached to automorphic forms, lap. 1 . Math. 8 ( 1 982) p . 1 -47. ' [69] K. RIBET-On /-adic representations attached to m od ul ar forms, Invent. M a Lil . 28 ( 1 975), p. 245-275 ; II, Glasgow Math. 1 . 27 ( 1 985), p. 1 85- 1 94. (70] K. RIBET-Galois action on division points of abelian v arieti es w i th many rea l multiplications, Amer. 1. Math. 98 ( 1 976), p. 75 1 -804. [7 1 ] K . RIBET-Galois representations attached to eigenforms with Neben typus, Lecture Notes in Math. 60 1 , p. 1 8-52, Springer-Verlag, 1977. [72] S. SEN-Lie algebras of Galois groups arising from Hodge-Tate m odules , A n n . of Math. 97 ( 1 973), p. 1 60- 1 70. (73] loP. SERRE-Une interpretation des congruences relatives a la fonction t de Ramanujan, Seminaire D.P.P. 1 967/68, n0 1 4 (=Oe.80). (74] 1.-P. SERRE-Facteurs locaux des fonctions zeta des varietes al gebri q u e s (definitions et conjectures), Sem inaire D.P.P. 1 969nO, n0 1 9 (=Oe. 8 7 ) .

[75] 1 . -P. SERRE-Sur les groupes de congruence des varietes abel iennes II, Izv. Akad. Nauk S . S . S .R. 3 5 ( 1 97 1), p . 73 1 -735 (=Oe.89) .

B-7

Supplementary Bibliography

[76] J.-P. SERRE-Proprietes galoisiennes des points d ' ordre fini des courbes elliptiques, Invent. Math. 15 ( 1 972), p. 259-33 1 (=Oe.94). [77] J.-P. SERRE-Congruences et formes modulaires (d'apres H.P.F. S w i n nerto n D y er) , Seminaire Bourbaki 1 97 1n2, n04 1 6 (=Oe.95). [78] J.-P. SERRE-Representations l-adiques, Kyoto Symp. on Algebraic Num ber Theory, 1 977, p. 1 77- 1 93 (=0e. 1 1 2).

[79] J.-P. SERRE-Groupes algebriques associes au x modules de Hodge-Ta te , Asterisque 65 ( 1 979), p . 1 55- 188 (=0e. 1 1 9). [80] J.-P. SERRE-Resume des cours de 1985-86, Annuaire du C o l l ege de France, 1 986, p. 95- 1 00. [8 1 ] H. P. F. SWINNE RTON-D YER-On l-adic rep rese n tati on s and c o n g r u en c e s for coefficients of modular forms, Lecture Notes in Math . 350, p. 1 -5 5 , S prin ger- Verlag, 1973; II. ibid. 60 1 , p . 63-90, Springer-Verlag. 1 977. [82] L. SZPIRO-Seminaire sur les pinceaux arithmetiques: la conjec ture de Mordell, Asterisque 1 27, S .M.F., 1 985. [83] M. WALDSCHMIDT Transcendance et exponentielles en plusieurs vari a b le s Invent Math. 63 (198 1), p. 97- 1 27. -

,

[84] A. WILES-On ordinary }..-adic representations associated to modular forms, preprint, Princeton, 1 987. [85] J.-P. WINTENBERGER-Groupes algebriques associes a certaines represenla tions p-adiques, Amer. J. Math. 108 ( 1986), p. 1425- 1466. [86] Y. G. ZARHIN-Abelian varieties, l-adic representations and SLl, MaLh . US S R lzv. 1 4 ( 1 9 80), p . 275-288.

[87] Y. G. ZARHIN-Abelian varieties, I-adic representations and Lie algebras. Rank independence on I, Invent Math. 55 ( 1 979), p. 165- 1 76. [88] Y. G. ZARHIN-Weights of simple Lie algebras in the c oho m ol og y of alge brai c varieties, Math. USSR Izv. 24 ( 1 985) , p. 245-28 1 .

INDEX

Admis s ib l e ( cha ra c t e r ) : A lmo s t loca l ly a lgebra i c Ani s o t ro pic ( to rus )

:

I I I .A . 2 . :

111 . 3 . 3 .

I I .A . 1 .

Arithmet i c ( subgroup ) :

I I .A . 1 .

As s oc i a t ed ( a l gebra i c morph i s m

•

a lgebra i c repres enta tion ) :

•

with a loca l ly

•

111 . 1 . 1 ,

111. 2. 1 .

Aut ( V ) : No ta tions . C, C

:

m

"

11 . 2 . 1 .

Cebo ta rev ' s theo rem

1 . 2. 2 .

Character g rou p ( o f a torus )

ll1 .A . 2

c ( ep ) : C

..

11 . 2. 1 .

•

K : 111 . 1 . 2 .

=

c

: l I I . A . 6 . Exer . 1 . K/ E Compa t ib le ( repres ent a t ions ) : Comp l ex mu l t i p li ca t ion :

1 . 2.3, 1 . 2. 4.

11. 2.8,

IV . 2 . 1 .

Conducto r ( o f a loca l ly a lgebra i c repres enta t ion ) : C oo , c l1l D :

:

11 . 3 . 1 .

ll . 2 . 1 .

Decompo s i t ion group : De f ined over

1. 2. 1 .

k ( repres enta tion

Dens ity ( o f a s et o f p la c es ) :

E

et

: 11 . 2 . 2 . :

II . 2 . 3 .

E l l i pt i c curve E

IP

:

IV . 1 . 3 .

IV . 1 .

1.

•

•

•

) :

1 . 2. 2.

11 . 2 . 4.

111. 2 . 2 .

I nd ex - 2

Em : I I . 2 . 1

•

: IV . A . 1 . 1 . q E q uid is tribu t ion : E

'"

I .A . 1 .

E : IV . 1 . 2 . v Ex c e p t i o na l s e t ( o f a s tr i c t ly compa t ib l e s y s t em ) � "...J � '

�t

: I11 .A . 2 .

: 11 . 2 . 5 .

F ' f : I!.2.3 v v Fro b enius e l ement : •

1. 2. 1 .

Fro benius endomo rph i s m :

IV . A . 3 .

rE

Gt

Gt

IV . 1 . 2 .

IV . 2 . 2 .

...

�t GLy

11. 2.8,

IV . 3 . 1 . ..

IV . 2 . 2 ,

IV . App

•

: No ta t i ons .

, Om :

II. 1. 1.

Good reduct ion ( o f an e l liptic curve ) :

Gro s s encha rakter o f type ( A ) : o He i ght : 1V . A . 2 . 2 . Hod ge- Ta t e d ecompo s i t ion Hod ge- Tate modu le :

1,

1

m

:

Id e le : :

111 . 1 . 2.

111 . 1 . 2 .

11 . 2 . 1 . II . 2 . 1 .

Id e l e c las s es

1t

:

IV . 1 . 2 .

11. 2. 7.

11. 2. 1 .

1V . A pp .

Inert ia group :

1.2. 1 .

Inte gra l ( repres enta t ion ) :

1. 2.3.

Is o geny , i s o genous curves

1V . 1 . 3 .

j : IV. 1 . 1

•

K, K s : No ta t io ns .

t-ad ic repres enta t ion ( o f a fi eld )

1.1 . 1 .

1.2.3.

Ind ex - 3

1.2.3.

A - ad ic repres enta t io n ( o f a fie ld ) La t t i c e :

1. 1 . 1 -

L- func t ion :

1. 2.5.

Loca l ly a l gebra ic ( repres entat ion ) :

111. 1 . 1 , 111 . 2 . 1 ,

111 . 2 . 4, 111 . 3 . 3 .

Modu lar invariant ( o f a n e l l i pt ic curve ) a

Modu lus ( o f

•

Ra tiona l ( repres enta t ion )

•

•

•

•

) :

1V. 1 . 3 .

: 1. 2 . 3 , 1 . 2 . 4. 1V . 1 . 2 .

: 11 . 2 . 4.

S a farevi c ( theo rem o f

•

•

•

) : 1V. 1 . 4 .

: 11. 2 . 2.

Stric t ly compa t i b l e

( sys t em o f repres entations )

1. 2. 4. Supp( m ) : 1 1 . 2 . 1 . Tate ' s e l l i pt ic curv es : 1V . A.1 1 1 . Ta t e ' s theorem : 1 1 1 . 1 . 2 , 1 1 1 . A . 7 .

g

: 1 1 . 2 . 4.

ql

T .r, ( \.l) : 1 . 1 . 2 . T : 11.2 . 2 . m Torus : 1 1 . 1 . 1 . Transve c t ion :

T

Um

111. 2 . 2 .

: 11 . 1 . 3 .

•

Reduct ion ( o f an e l l i p t i c curve ) :

Sm

IV . 1 . 1 .

loca l ly a lgebra ic repres enta t i o n ) :

Mu l t i p l i ca t ive type ( grou p o f ) S Neron- Ogg- a fa revi c ( c r i t erion o f

Re � ( H )

:

=

1V . 3 . 2 . 11. 1 . 1 .

� / Q ( Gm/ K ) :

, U

: 1 1. 2 . 1 . Uni formly d i s t ribu ted ( s equenc e ) : I . A . 1 . Unrami f i ed ( repres enta tion ) : 1 . 2 . 1 . V .r, ( 1-1) : I 1 2 . Weiers tra s s form ( o f an e l l i pt i c curve ) : 1V . 1 . 1 . v ,m

•

•

1.2.3,

1nd ex - 4

'XE

:

1II.A. 4.

'XL

:

X( T ) , X ( T ) m

11. 3. 1 .

Y,

yO , Y - , y+

:

I: K

1. 1 2. •

1. 2. 1 .

11. 3 . 1 , II.A . 2 .

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