Elliptic Curves

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Springer Science & Business Media, 29 ביוני 2013 - 350 עמודים
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The book divides naturally into several parts according to the level of the material, the background required of the reader, and the style of presentation with respect to details of proofs. For example, the first part, to Chapter 6, is undergraduate in level, the second part requires a background in Galois theory and the third some complex analysis, while the last parts, from Chapter 12 on, are mostly at graduate level. A general outline ofmuch ofthe material can be found in Tate's colloquium lectures reproduced as an article in Inven tiones [1974]. The first part grew out of Tate's 1961 Haverford Philips Lectures as an attempt to write something for publication c10sely related to the original Tate notes which were more or less taken from the tape recording of the lectures themselves. This inc1udes parts of the Introduction and the first six chapters The aim ofthis part is to prove, by elementary methods, the Mordell theorem on the finite generation of the rational points on elliptic curves defined over the rational numbers. In 1970 Tate teturned to Haverford to give again, in revised form, the originallectures of 1961 and to extend the material so that it would be suitable for publication. This led to a broader plan forthe book.
 

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Introduction to Rational Points on Plane Curves
1
CHAPTER
7
4 Rational Cubics and Mordells Theorem
10
6 Rational Points on Rational Curves Faltings and the Mordell
17
2 Illustrations of the Elliptic Curve Group Law
27
3 The Curves with Equations y x + ax and y x + a
33
5 Remarks on the Group Law on Singular Cubics
39
2 Irreducible Plane Algebraic Curves and Hypersurfaces
45
2 Generalities on Elliptic Functions
164
5 Preliminaries on Hypergeometric Functions
174
CHAPTER 10
183
3 Embeddings of a Torus by Theta Functions
189
6 Introduction to Tates Theory of pAdic Theta Functions
197
2 Families of Elliptic Curves with Additional Structures
204
4 Modular Functions
213
7 Hecke Operators
220

Appendix to Chapter 2
55
2 Factorial Properties of Polynomial Rings
57
3 Remarks on Valuations and Algebraic Curves
58
4 Resultant of Two Polynomials
59
CHAPTER 3
62
2 Elliptic Curves in Normal Form
64
3 The Discriminant and the Invariant j
67
4 Isomorphism Classification in Characteristics 2 3
70
5 Isomorphism Classification in Characteristic 3
72
6 Isomorphism Classification in Characteristic 2
74
7 Singular Cubic Curves
77
CHAPTER 4
81
The Hessian Family
84
3 The Jacobi Family
87
4 Tates Normal Form for a Cubic with a Torsion Point
88
5 An Explicit 2Isogeny
91
6 Examples of Noncyclic Subgroups of Torsion Points
96
CHAPTER 5
99
2 Minimal Normal Forms for an Elliptic Curve
102
3 Good Reduction of Elliptic Curves
105
4 The Kernel of Reduction mod p and the pAdic Filtration
107
NagellLutz Theorem
112
6 Computability of Torsion Points on Elliptic Curves from Integrality and Divisibility Properties of Coordinates
114
7 Bad Reduction and Potentially Good Reduction
116
8 Tates Theorem on Good Reduction over the Rational Numbers
118
CHAPTOR 6
120
2 Fermat Descent and x + y 1
122
2EQ for E Eab
123
2Ek
124
5 Quasilinear and Quasiquadratic Maps
127
6 The General Notion of Height on Projective Space
130
7 The Canonical Height and Norm on an Elliptic Curve
133
8 The Canonical Height on Projective Spaces over Global Fields
136
Galois Cohomology and Isomorphism Classification of Elliptic
138
4 Long Exact Sequence in GCohomology
146
CHAPTER 8
152
3 Basic Descent Formalism
158
2 Symplectic Pairings on Lattices and Division Points
224
3 Isogenies in the General Case
226
4 Endomorphisms and Complex Multiplication
230
5 The Tate Module of an Elliptic Curve
234
6 Endomorphisms and the Tate Module
236
7 Expansions Near the Origin and the Formal Group
237
CHAPTER 13
242
1 The Riemann Hypothesis for Elliptic Curves over a Finite Field
243
2 Generalities on Zeta Functions of Curves over a Finite Field
245
3 Definition of Supersingular Elliptic Curves
248
4 Number of Supersingular Elliptic Curves
252
5 Points of Order p and Supersingular Curves
253
6 The Endomorphism Algebra and Supersingular Curves
255
7 Summary of Criteria for a Curve To Be Supersingular
257
8 Tates Description of Homomorphisms
259
CHAPTER 14
262
2 The Néron Minimal Model
264
3 Galois Criterion for Good Reduction of NéronOggSafarević
267
CHAPTER 15
272
2 Generalities on 4Adic Represeñtations
275
3 Galois Representations and the NéronOggSafarević Criterion in the Global Case
277
Čebotarevs Density Theorem
279
Variation of 4
282
Faltings Finiteness Theorem
284
7 Tates Conjecture Safarevićs Theorem and Faltings Proof
286
Serres Open Image Theorem
288
CHAPTER 16
290
2 Zeta Functions of Curves over Q
291
3 HasseWeil LFunction and the Functional Equation
293
4 Classical Abelian LFunctions and Their Functional Equations
296
5 Grössencharacters and Hecke LFunctions
299
8 The TaniyamaWeil Conjecture
305
6 The Conjecture of Birch and SwinnertonDyer on the Leading Term
311
Bibliography
333
Index 345
344
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