Elliptic Curves

Dale Husemöller (auth.)0387954902, 9780387954905

This book is an introduction to the theory of elliptic curves, ranging from its most elementary aspects to current research. The first part, which grew out of Tate’s Haverford lectures, covers the elementary arithmetic theory of elliptic curves over the rationals. The next two chapters recast the arguments used in the proof of the Mordell theorem into the context of Galois cohomology and descent theory. This is followed by three chapters on the analytic theory of elliptic curves, including such topics as elliptic functions, theta functions, and modular functions. Next, the theory of endomorphisms and elliptic curves over infinite and local fields are discussed. The book then continues by providing a survey of results in the arithmetic theory, especially those related to the conjecture of the Birch and Swinnerton-Dyer.
This new edition contains three new chapters which explore recent directions and extensions of the theory of elliptic curves and the addition of two new appendices. The first appendix, written by Stefan Theisan, examines the role of Calabi-Yau manifolds in string theory, while the second, by Otto Forster, discusses the use of elliptic curves in computing theory and coding theory.
Dale Husemöller is a member of the faculty at the Max Planck Institute of Mathematics in Bonn.

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Table of contents :
Front Matter….Pages I-XV
Introduction to Rational Points on Plane Curves….Pages 1-21
Elementary Properties of the Chord-Tangent Group Law on a Cubic Curve….Pages 22-42
Plane Algebraic Curves….Pages 43-61
Elliptic Curves and Their Isomorphisms….Pages 62-80
Families of Elliptic Curves and Geometric Properties of Torsion Points….Pages 81-98
Reduction mod p and Torsion Points….Pages 99-119
Proof of Mordell’s Finite Generation Theorem….Pages 120-137
Galois Cohomology and Isomorphism Classification of Elliptic Curves over Arbitrary Fields….Pages 138-151
Descent and Galois Cohomology….Pages 152-161
Elliptic and Hypergeometric Functions….Pages 162-182
Theta Functions….Pages 183-201
Modular Functions….Pages 202-221
Endomorphisms of Elliptic Curves….Pages 222-241
Elliptic Curves over Finite Fields….Pages 242-261
Elliptic Curves over Local Fields….Pages 262-271
Elliptic Curves over Global Fields and ℓ -Adic Representations….Pages 272-289
L -Function of an Elliptic Curve and Its Analytic Continuation….Pages 290-305
Remarks on the Birch and Swinnerton-Dyer Conjecture….Pages 306-314
Back Matter….Pages 315-350