Silverman J.H. The arithmetic of elliptic curves. - 2nd ed. - New York: Springer-Verlag, 2009. - xx, 513 p.: ill. - (Graduate texts in mathematics; 106). - Bibliogr.: p.473-487. - Ind.: p.489-513. - ISBN 978-0-387-09493-9  Место хранения:   050 | Филиал Института математики CO РАН | Омск | Библиотека

Оглавление / 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. Isogenics .................................................. 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 ρ ......................... 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 ........................................ 157 §1. Elliptic Integrals ........................................ 158 § 2. Elliptic Functi on s ..................................... 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 Criterion of Neron-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 ........................... 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(q) .................... 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 В Group Cohomology (H0 and H1) .................................. 415 §1. Cohomology of Finite Groups ............................... 415 §2. Galois Cohomology ......................................... 418 §3. Nonabelian Cohomology ..................................... 421 Exercises ..................................................... 422 APPENDIX С 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 αp and the Sato-Tate Conjecture .............. 458 Notes on Exercises ............................................ 461 List of Notation .............................................. 467 References .................................................... 473 Index ......................................................... 489