This book is an introduction to the theory of elliptic curves, ranging from its most elementary aspects to current research. This manuscript grew out of Tate's Haverford Lectures. For the second edition, the author has written three new chapters and there are also two new appendices which were written by S. Theisen and O. Forster.

Klappentext

First Edition sold over 2500 copies in the Americas; New Edition contains three new chapters and two new appendices

Inhalt

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