The Arithmetic of Elliptic Curves / Edition 2

The Arithmetic of Elliptic Curves / Edition 2

by Joseph H. Silverman
     
 

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ISBN-10: 0387094938

ISBN-13: 9780387094939

Pub. Date: 06/12/2009

Publisher: Springer New York

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

Overview

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.

Product Details

ISBN-13:
9780387094939
Publisher:
Springer New York
Publication date:
06/12/2009
Series:
Graduate Texts in Mathematics Series, #106
Edition description:
2nd ed. 2009
Pages:
513
Product dimensions:
6.30(w) x 9.30(h) x 1.20(d)

Table of Contents

Preface to the Second Edition.- Preface to the First Edition.- Introduction.- Algebraic Varieties.- Algebraic Curves.- The Geometry of Elliptic Curves.- The Formal Group of an Elliptic Curve.- Elliptic Curves over Finite Fields.- Elliptic Curves over C.- Elliptic Curves over Local Fields.- Elliptic Curves over Global Fields.- Integral Points on Elliptic Curves.- Computing the Mordell–Weil Group.- Algorithmic Aspects of Elliptic Curves.- Appendix A: Elliptic Curves in Characteristics 2 and 3.- Appendix B: Group Cohomology (H0 and H1).- Appendix C: Further Topics: An Overview.- Notes on Exercises.- Bibliography.- List of Notation.- References.- Index.

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