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.

http://www.amazon.com/Arithmetic-Elliptic-Curves-Graduate-Mathematics/dp/0387094938

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