Joseph H. Silverman (auth.)0387094938, 9780387094939
The theory of elliptic curves is distinguished by its long history and by the diversity of the methods that have been used in its study. This book treats the arithmetic theory of elliptic curves in its modern formulation, through the use of basic algebraic number theory and algebraic geometry. The book begins with a brief discussion of the necessary algebro-geometric results, and proceeds with an exposition of the geometry of elliptic curves, the formal group of an elliptic curve, and elliptic curves over finite fields, the complex numbers, local fields, and global fields. Included are proofs of the Mordell–Weil theorem giving finite generation of the group of rational points and Siegel’s theorem on finiteness of integral points.
For this second edition of The Arithmetic of Elliptic Curves, there is a new chapter entitled Algorithmic Aspects of Elliptic Curves, with an emphasis on algorithms over finite fields which have cryptographic applications. These include Lenstra’s factorization algorithm, Schoof’s point counting algorithm, Miller’s algorithm to compute the Tate and Weil pairings, and a description of aspects of elliptic curve cryptography. There is also a new section on Szpiro’s conjecture and ABC, as well as expanded and updated accounts of recent developments and numerous new exercises.
The book contains three appendices: Elliptic Curves in Characteristics 2 and 3, Group Cohomology, and a third appendix giving an overview of more advanced topics.
Table of contents :
Front Matter….Pages 1-17
Algebraic Varieties….Pages 1-16
Algebraic Curves….Pages 17-40
The Geometry of Elliptic Curves….Pages 41-114
The Formal Group of an Elliptic Curve….Pages 115-135
Elliptic Curves over Finite Fields….Pages 137-156
Elliptic Curves over C….Pages 157-183
Elliptic Curves over Local Fields….Pages 185-205
Elliptic Curves over Global Fields….Pages 207-267
Integral Points on Elliptic Curves….Pages 269-307
Computing the Mordell–Weil Group….Pages 309-361
Algorithmic Aspects of Elliptic Curves….Pages 363-408
Back Matter….Pages 1-102
Reviews
There are no reviews yet.