The Arithmetic of Elliptic Curves / Edition 2

The Arithmetic of Elliptic Curves / Edition 2

by Joseph H. Silverman
     
 

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

See more details below

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.

Read More

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:
514
Product dimensions:
6.30(w) x 9.30(h) x 1.20(d)

Meet the Author

Dr. Joseph Silverman is a professor at Brown University and has been an instructor or professors since 1982. He was the Chair of the Brown Mathematics department from 2001-2004. He has received numerous fellowships, grants and awards, as well as being a frequentlyinvited lecturer. He is currently a member of the Council of the American Mathematical Society. His research areas of interest are number theory, arithmetic geometry, elliptic curves, dynamical systems and cryptography. He has co-authored over 120 publications and has had over 20 doctoral students under his tutelage. He has published9 highly successful books with Springer,including the recently released, An Introduction to Mathematical Cryptography, for Undergraduate Texts in Mathematics.

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.

Customer Reviews

Average Review:

Write a Review

and post it to your social network

     

Most Helpful Customer Reviews

See all customer reviews >