This book is an introduction to the theory of elliptic curves, ranging from elementary topics to current research. The first chapters, which grew out...
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Résumé
This book is an introduction to the theory of elliptic curves, ranging from elementary topics to current research. The first chapters, which grew out of Tate's Haverford Lectures, cover the arithmetic theory of elliptic curves over the field of rational numbers. This theory is then recast into the powerful and more general language of Galois cohomology and descent theory. An analytic section of the book includes such topics as elliptic functions, theta functions, and modular functions. Next, the book discusses the theory of elliptic curves over finite and local fields and provides a survey of results in the global arithmetic theory, especially those related to the conjecture of Birch and Swinnerton-Dyer. This new edition contains three new chapters. The first is an outline of Wiles's proof of Fermat's Last Theorem. The two additional chapters concern higher-dimensional analogues of elliptic curves, including K3 surfaces and Calabi-Yau manifolds. Three new appendices explore recent applications of elliptic curves and their generalizations. The first, written by Stefan Theisen, examines the role of Calabi-Yau manifolds and elliptic curves in string theory, while the second, by Otto Forster, discusses the use of elliptic curves in computing theory and coding theory. The third appendix explains the rote of elliptic curves in homotopy theory.
Sommaire
Introduction to Rational Points on Plane Curves
Elementary Properties of the Chord-Tangent Group Law on a Cubic Curve
Plane Algebraic Curves
Appendix to Chapter 2: Factorial Rings and Elimination Theory
Elliptic Curves and their Isomorphisms
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-fonction of an Elliptic Curve and Its Analytic Continuation
Remarks on the Birch and Swinnerton-Dyer Conjoncture
Remarks on the Modular Elliptic Curves Conjoncture and Fermat's Last Theorem
Higher Dimensional Analogs of Elliptic Curves: Calabi-Yau Varieties