# The Arithmetic of Elliptic Curves (Google eBook)

Springer Science & Business Media, Apr 20, 2009 - Mathematics - 534 pages
The theory of elliptic curves is distinguished by its long history and by the diversity of the methods that have been used in its study. This book treats the arithmetic theory of elliptic curves in its modern formulation, through the use of basic algebraic number theory and algebraic geometry. The book begins with a brief discussion of the necessary algebro-geometric results, and proceeds with an exposition of the geometry of elliptic curves, the formal group of an elliptic curve, and elliptic curves over finite fields, the complex numbers, local fields, and global fields. Included are proofs of the Mordell–Weil theorem giving finite generation of the group of rational points and Siegel’s theorem on finiteness of integral points.For this second edition of The Arithmetic of Elliptic Curves, there is a new chapter entitled Algorithmic Aspects of Elliptic Curves, with an emphasis on algorithms over finite fields which have cryptographic applications. These include Lenstra’s factorization algorithm, Schoof’s point counting algorithm, Miller’s algorithm to compute the Tate and Weil pairings, and a description of aspects of elliptic curve cryptography. There is also a new section on Szpiro’s conjecture and ABC, as well as expanded and updated accounts of recent developments and numerous new exercises.The book contains three appendices: Elliptic Curves in Characteristics 2 and 3, Group Cohomology, and a third appendix giving an overview of more advanced topics.

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### Contents

 Algebraic Varieties 1 2 Projective Varieties 6 3 Maps Between Varieties 11 Exercises 14 Algebraic Curves 17 2 Maps Between Curves 19 3 Divisors 27 4 Differentials 30
 7 Torsion Points 240 8 The Minimal Discriminant 243 9 The Canonical Height 247 10 The Rank of an Elliptic Curve 254 11 Szpiros Conjecture and ABC 255 Exercises 261 Integral Points on Elliptic Curves 269 1 Diophantine Approximation 270

 5 The RiemannRoch Theorem 33 Exercises 37 The Geometry of Elliptic Curves 41 1 Weierstrass Equations 42 2 The Group Law 51 3 Elliptic Curves 58 4 Isogenies 66 5 The Invariant Differential 75 6 The Dual Isogeny 80 7 The Tate Module 87 8 The Weil Pairing 92 9 The Endomorphism Ring 99 10 The Automorphism Group 103 Exercises 104 The Formal Group of an Elliptic Curve 115 2 Formal Groups 120 3 Groups Associated to Formal Groups 123 4 The Invariant Differential 125 5 The Formal Logarithm 127 6 Formal Groups over Discrete Valuation Rings 129 7 Formal Groups in Characteristic p 132 Exercises 135 Elliptic Curves over Finite Fields 136 2 The Weil Conjectures 140 3 The Endomorphism Ring 144 4 Calculating the Hasse Invariant 148 Exercises 153 Elliptic Curves over C 157 1 Elliptic Integrals 158 2 Elliptic Functions 161 3 Construction of Elliptic Functions 165 4 Maps Analytic and Maps Algebraic 171 5 Uniformization 173 6 The Lefschetz Principle 177 Exercises 178 Elliptic Curves over Local Fields 185 2 Reduction Modulo π 187 3 Points of Finite Order 192 4 The Action of Inertia 194 5 Good and Bad Reduction 196 6 The Group EE 199 7 The Criterion of NeronOggShafarevich 201 Exercises 203 Elliptic Curves over Global Fields 206 1 The Weak MordellWeil Theorem 208 2 The Kummer Pairing via Cohomology 215 3 The Descent Procedure 218 4 The MordellWeil Theorem over Q 220 5 Heights on Projective Space 224 6 Heights on Elliptic Curves 234
 2 Distance Functions 273 3 Siegels Theorem 276 4 The SUnit Equation 281 5 Effective Methods 286 6 Shafarevichs Theorem 293 7 The Curve Y² X³ + D 296 8 Roths TheoremAn Overview 299 Exercises 302 Computing the MordellWeil Group 308 1 An Example 310 2 TwistingGeneral Theory 318 3 Homogeneous Spaces 321 4 The Selmer and ShafarevichTate Groups 331 5 TwistingElliptic Curves 341 6 The Curve Y² X³ + DX 344 Exercises 355 Algorithmic Aspects of Elliptic Curves 363 1 DoubleandAdd Algorithms 364 2 Lenstras Elliptic Curve Factorization Algorithm 366 3 Counting the Number of Points in EFq 372 4 Elliptic Curve Cryptography 376 The General Case 381 Special Cases 386 7 PairingBased Cryptography 390 8 Computing the Weil Pairing 393 9 The TateLichtenbaum Pairing 397 Exercises 403 Elliptic Curves in Characteristics 2 and 3 409 Exercises 414 Group Cohomology H⁰ and H¹ 415 2 Galois Cohomology 418 3 Nonabelian Cohomology 421 Exercises 422 Further Topics An Overview 424 12 Modular Functions 429 13 Modular Curves 439 14 Tate Curves 443 15 Neron Models and Tates Algorithm 446 16 LSeries 449 17 Duality Theory 453 18 Local Height Functions 454 19 The Image of Galois 455 20 Function Fields and Specialization Theorems 456 21 Variation of ap and the SatoTate Conjecture 458 Notes on Exercises 461 List of Notation 467 References 473 Index 489 Copyright