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Graduate Texts in Mathematics #106
The Arithmetic of Elliptic Curves
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 the 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 parings, 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.
For the 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 parings, 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.
533 pages, Hardcover
First published October 14, 1985
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Displaying 1 - 5 of 5 reviews
December 30, 2016
The character development was excellent, but the plot was a little threadbare.
August 31, 2009
I used this book for a term paper I was writing.
Joseph Silverman is a good author to learn from. His books are easier to read and more user friendly than Serge Lang's.
Joseph Silverman is a good author to learn from. His books are easier to read and more user friendly than Serge Lang's.
October 31, 2019
A classic and one of the most famous introductory books to the theory of elliptic curves. The treatment of the topics in elliptic curves is very clear. The book is self-contained and well-written. There is another book that deals with advanced topics. This book can be read by advanced undergraduate students who are interested in the theory of elliptic curves and the geometric aspects of the theory.
June 2, 2023
Great little reference book, perfect for a beginner who is studying cryptography, for the mathematicians, not so much, as many proofs are omitted.
Displaying 1 - 5 of 5 reviews