Elliptic CurvesSpringer Science & Business Media, 29 ביוני 2013 - 350 עמודים 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. |
תוכן
| 1 | |
| 2 | |
| 7 | |
5 The Group Law on Cubic Curves and Elliptic Curves | 15 |
CHAPTER 1 | 21 |
2 Illustrations of the Elliptic Curve Group | 27 |
3 The Curves with Equations y² x² + ax and y² x³ + | 36 |
CHAPTER 2 | 43 |
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 | 238 |
Elliptic Curves over Finite Fields | 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 Multiple or Singular Points | 50 |
CHAPTER 3 | 64 |
4 Isomorphism Classification in Characteristics 2 3 | 70 |
7 Singular Cubic Curves | 77 |
The Hessian Family | 84 |
5 An Explicit 2Isogeny | 91 |
Reduction mod p and Torsion Points | 99 |
3 Good Reduction of Elliptic Curves | 105 |
NagellLutz Theorem | 112 |
8 Tates Theorem on Good Reduction over the Rational Numbers | 118 |
2Ek | 124 |
6 The General Notion of Height on Projective Space | 130 |
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 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 |
2 Symplectic Pairings on Lattices and Division Points | 224 |
3 Isogenies in the General Case | 226 |
4 Endomorphisms and Complex Multiplication | 230 |
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 NéronOggŠafarevič | 267 |
CHAPTER 15 | 272 |
2 Generalities on Adic Representations | 274 |
3 Galois Representations and the NéronOggŠafarevič Criterion in the Global Case | 277 |
Čebotarevs Density Theorem | 279 |
Variation of | 282 |
Faltings Finiteness Theorem | 284 |
7 Tates Conjecture Šafarevičs 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 |
| 334 | |
| 344 | |
| 348 | |
מהדורות אחרות - הצג הכל
מונחים וביטויים נפוצים
a₁ abelian group abelian varieties algebraically closed assertion automorphism ax² b₂ bad reduction change of variable Chapter characteristic coefficients cohomology complex multiplication complex numbers conic conjecture cubic curve cubic equation cyclic Definition denote discriminant divisors E(Q)tors elements elliptic curve elliptic curve defined elliptic function equation y² equivalent Euler product extension factor Fermat field of fractions finite formula Gal(K/K Galois Galois extension given group law H¹(G Hence homomorphism integer intersection points irreducible isogeny isomorphism j-invariant L-function Math modular modulo morphism nonsingular nonzero normal form notations number field p-adic P₁ plane curve point of order polynomial prime projective space PROOF proves the proposition proves the theorem quadratic quotient rational numbers rational points relation Remark ring roots Serre singular subgroup supersingular tangent line Tate valuation Weierstrass equation zero zeta function