| Preface to the Second Edition Preface to the First Edition Acknowledgments to the Second Edition Acknowledgments to the First Edition Introduction to Rational Points on Plane Curves 1 Elementary Properties of the Chord-Tangent Group Law on a Cubic Curve 2 Plane Algebraic Curves App. to Ch. 2 Factorial Rings and Elimination Theory 3 Elliptic Curves and Their Isomorphisms 4 Families of Elliptic Curves and Geometric Properties of Torsion Points 5 Reduction mod p and Torsion Points 6 Proof of Mordell's Finite Generation Theorem 7 Galois Cohomology and Isomorphism Classification of Elliptic Curves over Arbitrary Fields 8 Descent and Galois Cohomology 9 Elliptic and Hypergeometric Functions 10 Theta Functions 11 Modular Functions 12 Endomorphisms of Elliptic Curves 13 Elliptic Curves over Finite Fields 14 Elliptic Curves over Local Fields 15 Elliptic Curves over Global Fields and l-Adic Representations 16 L-Function of an Elliptic Curve and Its Analytic Continuation 17 Remarks on the Birch and Swinnerton-Dyer Conjecture 18 Remarks on the Modular Elliptic Curves Conjecture and Fermat's Last Theorem 19 Higher Dimensional Analogs of Elliptic Curves：Calabi-Yau Varieties 20 Families of Elliptic Curves Appendix Ⅰ：Calabi-Yau Manifolds and String Theory Appendix Ⅱ：Eliptic Curves in Algorithmic Number Theroy and Cryptography Appendix Ⅲ：Elliptic Curves and Topological Modular Forms Appendix Ⅳ：Guide to the Exercises References List of Notation Index |
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