Abelian l-Adic Representations and Elliptic

This classic book contains an introduction to systems of l-adic representations, a topic of great importance in number theory and algebraic geometry, as reflected by the spectacular recent developments on the Taniyama-Weil conjecture and Fermat's Last Theorem. The initial chapters are devoted to the Abelian case (complex multiplication), where one finds a nice correspondence between the l-adic representations and the linear representations of some algebraic groups (now called Taniyama groups). The last chapter handles the case of elliptic curves with no complex multiplication, the main result of which is that the image of the Galois group (in the corresponding l-adic representation) is "large."

INTRODUCTION

xiii

(4)

NOTATIONS

xvii

Chapter I l-adic Representations

1 The notion of an l-adic representation

I-1

(4)

1.1 Definition

I-1

(2)

1.2 Examples

I-3

(2)

2 l-adic representations of number fields

I-5

(13)

2.1 Preliminaries

I-5

(2)

2.2 Cebotarev's density theorem

I-7

(2)

2.3 Rational l-adic representations

I-9

(5)

2.4 Representations with values in a linear algebraic group

I-14

(2)

2.5 L-functions attached to rational representations

I-16

(2)

Appendix Equipartition and L-functions

I-18

A.1 Equipartition

I-18

(3)

A.2 The connection with L-functions

I-21

(5)

A.3 Proof of theorem 1

I-26

Chapter II The Groups S(m)

1 Preliminaries

II-1

(5)

1.1 The torus T

II-1

(1)

1.2 Cutting down T

II-2

(1)

1.3 Enlarging groups

II-3

(3)

2 Construction of T(m) and S(m)

II-6

(23)

2.1 Ideles and idele-classes

II-6

(2)

2.2 The groups T(m) and S(m)

II-8

(2)

2.3 The canonical l-adic representation with values in S(m)

II-10

(3)

2.4 Linear representations of S(m)

II-13

(5)

2.5 l-adic representations associated to a linear representation of S(m)

II-18

(3)

2.6 Alternative construction

II-21

(2)

2.7 The real case

II-23

(2)

2.8 An example: complex multiplication of abelian varieties

II-25

(4)

3 Structure of T(m) and applications

II-29

(9)

3.1 Structure of X(Tm)

II-29

(2)

3.2 The morphism j*: G(m) --> T(m)

II-31

(1)

3.3 Structure of T(m)

II-32

(3)

3.4 How to compute Frobeniuses

II-35

(3)

Appendix Killing arithmetic groups in tori

II-38

A.1 Arithmetic groups in tori

II-38

(2)

A.2 Killing arithmetic subgroups

II-40

Chapter III Locally Algebraic Abelian Representations

1 The local case

III-1

(6)

1.1 Definitions

III-1

(4)

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

III-5

(2)

2 The global case

III-7

(13)

2.1 Definitions

III-7

(2)

2.2 Modulus of a locally algebraic abelian representation

III-9

(3)

2.3 Back to S(m)

III-12

(4)

2.4 A mild generalization

III-16

(1)

2.5 The function field case

III-16

(4)

3 The case of a composite of quadratic fields

III-20

(10)

3.1 Statement of the result

III-20

(1)

3.2 A criterion for local algebraicity

III-20

(4)

3.3 An auxiliary result on tori

III-24

(4)

3.4 Proof of the theorem

III-28

(2)

Appendix Hodge-Tate decompositions and locally algebraic representations

III-30

A.1 Invariance of Hodge-Tate decompositions

III-31

(3)

A.2 Admissible characters

III-34

(4)

A.3 A criterion for local triviality

III-38

(2)

A.4 The character (XXX)E

III-40

(2)

A.5 Characters associated with Hodge-Tate decompositions

III-42

(5)

A.6 Locally compact case

III-47

(5)

A.7 Tate's theorem

III-52

Chapter IV l-adic Representations Attached to Elliptic Curves

1 Preliminaries

IV-2

(7)

1.1 Elliptic curves

IV-2

(1)

1.2 Good reduction

IV-3

(1)

1.3 Properties of V(l) related to good reduction

IV-4

(3)

1.4 Safarevic's theorem

IV-7

(2)

2 The Galois modules attached to E

IV-9

(9)

2.1 The irreducibility theorem

IV-9

(2)

2.2 Determination of the Lie algebra of G(l)

IV-11

(3)

2.3 The isogeny theorem

IV-14

(4)

3 Variation of G(l) and G(l) with l

IV-18

(11)

3.1 Preliminaries

IV-18

(2)

3.2 The case of a non integral j

IV-20

(1)

3.3 Numerical example

IV-21

(2)

3.4 Proof of the main lemma of 3.1

IV-23

(6)

Appendix Local results

IV-29

A.1 The case v(j) is less than 0

IV-29

(8)

A.1.1 The elliptic curves of Tate

IV-29

(2)

A.1.2 An exact sequence

IV-31

(2)

A.1.3 Determination of g(l) and i(l)

IV-33

(1)

A.1.4 Application to isogenies

IV-34

(2)

A.1.5 Existence of transvections in the inertia group

IV-36

(1)

A.2 The case v(j) is greater than or equal to 0

IV-37

A.2.1 The case l not equal to p

IV-37

(1)

A.2.2 The case l = p with good reduction of height 2

IV-38

(3)

A.2.3 Auxiliary results on abelian varieties

IV-41

(1)

A.2.4 The case l = p with good reduction of height 1

IV-42

BIBLIOGRAPHY

B-1

INDEX

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Abelian l-Adic Representations and Elliptic Curves

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