Elliptic Curves (Hardcover, 2, 2004)

From the Contents: 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.