Elliptic curves椭圆的曲线
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作者： Dale Husemöller著
出 版 社：
出版时间： 2003-12-1字数：版次： 1页数： 487印刷时间： 2003/12/01开本： 16开印次： 1纸张： 胶版纸I S B N ： 9780387954905包装： 精装内容简介
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 Husemoller is a member of the faculty at the Max Planck Institute of Mathematics in Bonn。
目录
Preface to the Second Edition
Preface to the First Edition
Acknowledgments to the Second Edition
Acknowledgments to the First Edition
Introduction to Rational Points on Plane Curves
1 Elementary Properties of the Chord-Tangent Group Law on a Cubic Curve
2 Plane Algebraic Curves
App. to Ch. 2 Factorial Rings and Elimination Theory
3 Elliptic Curves and Their Isomorphisms
4 Families of Elliptic Curves and Geometric Properties of Torsion Points
5 Reduction mod p and Torsion Points
6 Proof of Mordell's Finite Generation Theorem
7 Galois Cohomology and Isomorphism Classification of Elliptic Curves over Arbitrary Fields
8 Descent and Galois Cohomology
9 Elliptic and Hypergeometric Functions
10 Theta Functions
11 Modular Functions
12 Endomorphisms of Elliptic Curves
13 Elliptic Curves over Finite Fields
14 Elliptic Curves over Local Fields
15 Elliptic Curves over Global Fields and l-Adic Representations
16 L-Function of an Elliptic Curve and Its Analytic Continuation
17 Remarks on the Birch and Swinnerton-Dyer Conjecture
18 Remarks on the Modular Elliptic Curves Conjecture and Fermat's Last Theorem
19Higher Dimensional Analogs of Elliptic Curves：Calabi-Yau Varieties
20 Families of Elliptic Curves
Appendix Ⅰ：Calabi-Yau Manifolds and String Theory
Appendix Ⅱ：Eliptic Curves in Algorithmic Number Theroy and Cryptography
Appendix Ⅲ：Elliptic Curves and Topological Modular Forms
Appendix Ⅳ：Guide to the Exercises
References
List of Notation
Index
