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More About This Textbook
Overview
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 algorthims 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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Meet the Author
Dr. Joseph Silverman is a professor at Brown University and has been an instructor or professors since 1982. He was the Chair of the Brown Mathematics department from 2001-2004. He has received numerous fellowships, grants and awards, as well as being a frequentlyinvited lecturer. He is currently a member of the Council of the American Mathematical Society. His research areas of interest are number theory, arithmetic geometry, elliptic curves, dynamical systems and cryptography. He has co-authored over 120 publications and has had over 20 doctoral students under his tutelage. He has published9 highly successful books with Springer,including the recently released, An Introduction to Mathematical Cryptography, for Undergraduate Texts in Mathematics.
Table of Contents
Preface to the Second Edition v
Preface to the First Edition vii
Introduction xvii
Chapter I Algebraic Varieties 1
1 Affine Varieties 1
2 Projective Varieties 6
3 Maps Between Varieties 11
Exercises 14
Chapter II Algebraic Curves 17
1 Curves 17
2 Maps Between Curves 19
3 Divisors 27
4 Differentials 30
5 The Riemann-Roch Theorem 33
Exercises 37
Chapter III 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
Chapter IV The Formal Group of an Elliptic Curve 115
1 Expansion Around O 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
Chapter V Elliptic Curves over Finite Fields 137
1 Number of Rational Points 137
2 The Weil Conjectures 140
3 The Endomorphism Ring 144
4 Calculating the Hasse Invariant 148
Exercises 153
Chapter VI 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
Chapter VII Elliptic Curves over Local Fields 185
1 Minimal Weierstrass Equations 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 E/E0 199
7 The Criterionof Néron-Ogg-Shafarevich 201
Exercises 203
Chapter VIII Elliptic Curves over Global Fields 207
1 The Weak Mordell-Weil Theorem 208
2 The Kummer Pairing via Cohomology 215
3 The Descent Procedure 218
4 The Mordell-Weil Theorem over Q 220
5 Heights on Projective Space 224
6 Heights on Elliptic Curves 234
7 Torsion Points 240
8 The Minimal Discriminant 243
9 The Canonical Height 247
10 The Rank of an Elliptic Curve 254
11 Szpiro's Conjecture and ABC 255
Exercises 261
Chapter IX Integral Points on Elliptic Curves 269
1 Diophantine Approximation 270
2 Distance Functions 273
3 Siegel's Theorem 276
4 The S-Unit Equation 281
5 Effective Methods 286
6 Shafarevich's Theorem 293
7 The Curve Y2 = X3+ D 296
8 Roth's Theorem-An Overview 299
Exercises 302
Chapter X Computing the Mordell-Weil Group 309
1 An Example 310
2 Twisting-General Theory 318
3 Homogeneous Spaces 321
4 The Selmer and Shafarevich-Tate Groups 331
5 Twisting-Elliptic Curves 341
6 The Curve Y2 = X3 + DX 344
Exercises 355
Chapter XI Algorithmic Aspects of Elliptic Curves 363
1 Double-and-Add Algorithms 364
2 Lenstra's Elliptic Curve Factorization Algorithm 366
3 Counting the Number of Points in E(Fq) 372
4 Elliptic Curve Cryptography 376
5 Solving the ECDLP: The General Case 381
6 Solving the ECDLP: Special Cases 386
7 Pairing-Based Cryptography 390
8 Computing the Weil Pairing 393
9 The Tate-Lichtenbaum Pairing 397
Exercises 403
Appendix A Elliptic Curves in Characteristics 2 and 3 409
Exercises 414
Appendix B Group Cohomology (H0 and H1) 415
1 Cohomology of Finite Groups 415
2 Galois Cohomology 418
3 Nonabelian Cohomology 421
Exercises 422
Appendix C Further Topics: An Overview 425
11 Complex Multiplication 425
12 Modular Functions 429
13 Modular Curves 439
14 Tate Curves 443
15 Néron Models and Tate's Algorithm 446
16 L-Series 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 Sato-Tate Conjecture 458
Notes on Exercises 461
List of Notation 467
References 473
Index 489