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E-grāmata: Abelian l-Adic Representations and Elliptic Curves

  • Formāts: 202 pages
  • Sērija : Research Notes in Mathematics
  • Izdošanas datums: 15-Nov-1997
  • Izdevniecība: A K Peters
  • Valoda: eng
  • ISBN-13: 9781040187043
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Abelian l-Adic Representations and Elliptic CurvesPalielināt
  • Formāts: 202 pages
  • Sērija : Research Notes in Mathematics
  • Izdošanas datums: 15-Nov-1997
  • Izdevniecība: A K Peters
  • Valoda: eng
  • ISBN-13: 9781040187043
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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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