Coates J. - Arithmetic Theory of Elliptic Curves.pdf - Number Theory - edudako

J.Coates R. Greenberg

K. A. Ribet K. Rubin

Arithmetic Theory

of Elliptic ~urves-

Lectures given at the 3rd Session of the

Centro Internazionale Matematico Estivo

(C.I.M.E.) held in Cetraro, Italy,

July 12-19, 1997

Editor: C. Viola

Fonduiione

C.I.M.E.

Springer

Berlin

Heidelberg

New York

Barcelona

Hong Kong

London

Milan

Paris

Singapore

Tokyo

Springer

Authors

John H. Coates

Department of Pure Mathematics

and Mathematical Statistics

University of Cambridge

16 Mill Lane

Cambridge CB2 1 SB, UK

Kenneth A. Ribet

Department of Mathematics

University of California

Berkeley CA 94720, USA

Ralph Greenberg

Department of Mathematics

University of Washington

Seattle, WA 98195, USA

Preface

The C.I.M.E. Session "Arithmetic Theory of Elliptic Curves" was held at

Cetraro (Cosenza, Italy) from July 12 to July 19, 1997.

The arithmetic of elliptic curves is a rapidly developing branch of

mathematics, at the boundary of number theory, algebra, arithmetic alge-

braic geometry and complex analysis. ~fter

Karl Rubin

Department of Mathematics

Stanford University

Stanford CA 94305, USA

the pioneering research in this

field in the early twentieth century, mainly due to H. Poincar6 and B. Levi,

the origin of the modern arithmetic theory of elliptic curves goes back to

L. J. Mordell's theorem (1922) stating that the group of rational points on

an elliptic curve is finitely generated. Many authors obtained in more re-

cent years crucial results on the arithmetic of elliptic curves, with important

connections to the theories of modular forms and L-functions. Among the

main problems in the field one should mention the Taniyama-Shimura con-

jecture, which states that every elliptic curve over Q is modular, and the

Birch and Swinnerton-Dyer conjecture, which, in its simplest form, asserts

that the rank of the Mordell-Weil group of an elliptic curve equals the order of

vanishing of the L-function of the curve at 1. New impetus to the arithmetic

of elliptic curves was recently given by the celebrated theorem of A. Wiles

(1995), which proves the Taniyama-Shimura conjecture for semistable ellip-

tic curves. Wiles' theorem, combined with previous results by K. A. Ribet,

J.-P. Serre and G. Frey, yields a proof of Fermat's Last Theorem. The most

recent results by Wiles, R. Taylor and others represent a crucial progress

towards a complete proof of the Taniyama-Shimura conjecture. In contrast

to this, only partial results have been obtained so far about the Birch and

Swinnerton-Dyer conjecture.

The fine papers by J. Coates, R. Greenberg, K. A. Ribet and K. Rubin

collected in this volume are expanded versions of the courses given by the

authors during the C.I.M.E. session at Cetraro, and are broad and up-to-date

contributions to the research in all the main branches of the arithmetic theory

of elliptic curves. A common feature of these papers is their great clarity and

elegance of exposition.

Much of the recent research in the arithmetic of elliptic curves consists

in the study of modularity properties of elliptic curves over Q, or of the

structure of the Mordell-Weil group E(K) of K-rational points on an elliptic

curve E defined over a number field K. Also, in the general framework of

Iwasawa theory, the study of E(K) and of its rank employs algebraic as well

as analytic approaches.

Various algebraic aspects of Iwasawa theory are deeply treated in

Greenberg's paper. In particular, Greenberg examines the structure of

the pprimary Selmer group of an elliptic curve E over a Z,-extension of

the field K, and gives a new proof of Mazur's control theorem. Rubin gives a

Editor

Carlo Viola

Dipartimento di Matematica

Universiti di Pisa

Via Buonarroti 2

56127 Pisa, Italy

Cataloging-in-Publication Data applied for

Die Deutsche Bibliothek - CIP-Einheitsaufnahme

Arithmetic theory of elliptic curves : held in Cetraro, Italy, July

12 - 19, 1997 / Fondazione CIME. J. Coates ... Ed.: C. Viola. - Berlin

; Heidelberg ; New York ; Barcelona ; Hong Kong ; London ; Milan ;

Paris ; Singapore ; Tokyo : Springer, 1999

(Lectures given at the ... session of the Centro Internazionale

Matematico Estivo (CIME) ... ; 1997,3) (Ixcture notes in mathematics

; Vol. 1716 : Subseries: Fondazione CIME)

ISBN 3-540-66546-3

Mathematics Subject Classification (1991):

l 1605, 11607, 31615, 11618, 11640, 11R18, llR23, 11R34, 14G10, 14635

ISSN 0075-8434

ISBN 3-540-66546-3 Springer-Verlag Berlin Heidelberg New York

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is

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detailed and thorough description of recent results related to the Birch and

Swinnerton-Dyer conjecture for an elliptic curve defined over an imaginary

quadratic field K. with complex multiplication by K . Coates' contribution is

mainly concerned with the construction of an analogue of Iwasawa theory for

elliptic curves without complex multiplication. and several new results are

included in his paper . Ribet's article focuses on modularity properties. and

contains new results concerning the points on a modular curve whose images

in the Jacobian of the curve have finite order .

The great success of the C.I.M.E. session on the arithmetic of elliptic

curves was very rewarding to me . I am pleased to express my warmest thanks

to Coates. Greenberg. Ribet and Rubin for their enthusiasm in giving their

fine lectures and for agreeing to write the beautiful papers presented here .

Special thanks are also due to all the participants. who contributed. with

their knowledge and variety of mathematical interests. to the success of the

session in a very co-operative and friendly atmosphere .

Table of Contents

Fragments of the GL2 Iwasawa Theory of Elliptic Curves

without Complex Multiplication

John Coates ............................... 1

Statement of results ......................... 2

Basic properties of the Selmer group ................ 14

Local cohomology calculations .................... 23

Global calculations .......................... 39

Carlo Viola

Iwasawa Theory for Elliptic Curves

Ralph Greenberg ............................. 51

Introduction .............................. 51

Kummer theory for E ........................ 62

Control theorems ........................... 72

Calculation of an Euler characteristic ................ 85

Conclusion .............................. 105

Torsion Points on Jo(N) and Galois Representations

Kenneth A Ribet ............................ 145

Introduction .............................. 145

A local study at N .......................... 148

The kernel of the Eisenstein ideal .................. 151

Lenstra's input ............................ 154

Proof of Theorem 1.7 ......................... 156

Adelic representations ........................ 157

Proof of Theorem 1.6 ......................... 163

Elliptic Curves with Complex Multiplication

and the Conjecture of Birch and Swinnerton-Dyer

Karl Rubin ................................ 167

Quick review of elliptic curves .................... 168

........................

Fragments of the GL2 Iwasawa Theory

of Elliptic Curves

without Complex Multiplication

Elliptic curves over C

170

....................

172

..................

Elliptic curves over local fields

178

............

Elliptic curves over number fields

181

................................

Elliptic curves with complex multiplication

Descent

188

.............................

Elliptic units

193

.............................

John Coates

Euler systems

203

.....................

Bounding ideal class groups

209

..................

213

.............

The theorem of Coates and Wiles

216

....................

Iwasawa theory and the "main conjecture"

"Fearing the blast

Of the wind of impermanence,

I have gathered together

The leaflike words of former mathematicians

And set them down for you."

227

Computing the Selmer gmup

Thanks to the work of many past and present mathematicians, we now know

a very complete and beautiful Iwasawa theory for the field obtained by ad-

joining all ppower roots of unity to Q, where p is any prime number. Granted

the ubiquitous nature of elliptic curves, it seems natural to expect a precise

analogue of this theory to exist for the field obtained by adjoining to Q all

the ppower division points on an elliptic curve E defined over Q. When E

admits complex multiplication, this is known to be true, and Rubin's lectures

in this volume provide an introduction to a fairly complete theory. However,

when E does not admit complex multiplication, all is shrouded in mystery

and very little is known. These lecture notes are aimed at providing some

fragmentary evidence that a beautiful and precise Iwasawa theory also exists

in the non complex multiplication case. The bulk of the lectures only touch

on one initial question, namely the study of the cohomology of the Selmer

group of E over the field of all ppower division points, and the calculation

of its Euler characteristic when these cohomology groups are finite. But a

host of other questions arise immediately, about which we know essentially

nothing at present.

Rather than tempt uncertain fate by making premature conjectures, let

me illustrate two key questions by one concrete example. Let E be the elliptic

curve XI (1 I), given by the equation

Take p to be the prime 5, let K be the field obtained by adjoining the

5-division points on E to Q, and let F, be the field obtained by adjoin-

ing all 5-power division points to Q. We write R for the Galois group of F,

over K. The action of R on the group of all 5-power division points allows

us to identify R with a subgroup of GL2(iZ5),and a celebrated theorem of

Serre tells us that R is an open subgroup. Now it is known that the Iwasawa

Elliptic curves without complex multiplication

John Coates

Write

algebra A(R) (see (14)) is left and right Noetherian and has no divisors of

zero. Let C(E/F,) denote the compact dual of the Selmer group of E over

F, (see (12)), endowed with its natural structure as a left A(R)-module. We

prove in these lectures that C(E/F,)

is large in the sense that

for the Galois groups of F, over Fn, and F,

over F, respectively. Now the

action of C on E,-

defines a canonical injection

But we also prove that every element of C(E/Fw) has a non-zero annihi-

lator in A(R). We strongly suspect that C(E/F,) has a deep and interest-

ing arithmetic structure as a representation of A(R). For example, can one

say anything about the irreducible representations of A(R) which occur in

C(E/F,)? Is there some analogue of Iwasawa's celebrated main conjecture

on cyclotomic fields, which, in this case, should relate the A(R)-structure of

C(E/F,) to a 5-adic L-function formed by interpolating the values at s = 1

of the twists of the complex L-function of E by all Artin characters of R?

I would be delighted if these lectures could stimulate others to work on these

fascinating non-abelian problems.

In conclusion, I want to warmly thank R. Greenberg, S. Howson and

Sujatha for their constant help and advice throughout the time that these

lectures were being prepared and written. Most of the material in Chapters

3 and 4 is joint work with S. Howson. I also want to thank Y. Hachimori,

K. Matsuno, Y. Ochi, J.-P. Serre, R. Taylor, and B. Totaro for making im-

portant observations to us while this work was evolving. Finally, it is a great

pleasure to thank Carlo Viola and C.I.M.E. for arranging for these lectures

to take place at an incomparably beautiful site in Cetraro, Italy.

When there is no danger of confusion, we shall drop the homomorphism i

from the notation, and identify C with a subgroup of GL2(Zp). Note that i

maps En into the subgroup of GL2(Zp)consisting of all matrices which are

congruent to the identity modulo pn+'. In particular, it follows that & is

always a pro-pgroup. However, it is not in general true that C is a pro-p

group. The following fundamental result about the size of C is due to Serre

[261.

Theorem 1.1.

(i) C is open in GL2(Zp) for all primes p, and

(ii) C = GL2(Zp) for all but a finite number of primes p.

Serre's method of proof in [26] of Theorem 1.1 is effective, and he gives

many beautiful examples of the calculations of C for specific elliptic curves

and specific primes p. We shall use some of these examples to illustrate the

theory developed in these lectures. For convenience, we shall always give the

name of the relevant curves in Cremona's tables [9].

Statement of Results

Example. Consider the curves of conductor 11

1.1

Serre's theorem

Throughout these notes, F will denote a finite extension of the rational field

Q, and E will denote an elliptic curve defined over F, which will always be

assumed to satisfy the hypothesis:

The first curve corresponds to the modular group ro(ll) and is often de-

noted by Xo(ll), and the second curve corresponds to the group (ll), and

is often denoted by X1(ll). Neither curve admits complex multiplication (for

example, their j-invariants are non-integral). Both curves have a Q-rational

point of order 5, and they are linked by a Q-isogeny of degree 5. For both

curves, Serre [26] has shown that C = GL2(Zp)for all primes p 2 7. Subse-

quently, Lang and Trotter [21] determined C for the curve ll(A1) and the

primes p = 2,3,5.

Hypothesis. The endomorphism ring of E overQ is equal to Z, i.e. E does

not admit complex multiplication.

Let p be a prime number. For all integers n 2 0, we define

We now briefly discuss C-Euler characteristics, since this will play an

important role in our subsequent work. By virtue of Theorem 1.1, C is a

padic Lie group of dimension 4. By results of Serre [28] and Lazard [22], C

will have pcohomological dimension equal to 4 provided C has no ptorsion.

We define the corresponding Galois extensions of F

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