Elliptic Curves Book

Introduction to Rational Points on Plane Curves
• Elementary Properties of the Chord-Tangent Group Law on a Cubic Curve
• Plane Algebraic Curves
• Factorial Rings and Elimination Theory
• Elliptic Curves and Their Isomorphism
• Families of Elliptic Curves and Geometric Properties of Torsion Points
• Reduction mod p and Torsion Points
• Proof of Mordell's Finite Generation Theorem
• Galois Cohomology and Isomorphism Classification of Elliptic Curves over Arbitrary Fields
• Descent and Galois Cohomology
• Elliptic and Hypergeometric Functions
• Theta Functions
• Modular Functions
• Endomorphisms of Elliptic Curves
• Elliptic Curves over Finite Fields
• Elliptic Curves over Local Fields
• Elliptic Curves over Global Fields and l-adic Representations
• L-Functions of an Elliptic Curve and Its Analytic Continuation
• Remarks on the Birch and Swinnerton-Dyer Conjecture
• Remarks on the Modular Curves Conjecture and Fermat's Last Theorem
• Higher Dimensional Analogs of Elliptic Curves: Calabi-Yau Varieties
• Families of Elliptic Curves
• Appendix I: Calabi-Yau Manifolds and String Theory
• Appendix II: Elliptic Curves in Algorithmic Number Theory
• Appendix III: Guide to the Exercises
• Bibliography
• Index