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Tables of Contents for Abelian L-Adic Representations and Elliptic Curves
Chapter/Section Title
Page #
Page Count
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