The Arithmetic of Elliptic Curves

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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.