**Author**: Joseph H. Silverman

**Publisher:** Springer Science & Business

**ISBN:**

**Category:** Mathematics

**Page:** 525

**View:** 133

In "The Arithmetic of Elliptic Curves," the author presented the basic theory culminating in two fundamental global results, the Mordell-Weil theorem on the finite generation of the group of rational points and Siegel's theorem on the finiteness of the set of integral points. This book continues the study of elliptic curves by presenting six important, but somewhat more specialized topics: I. Elliptic and modular functions for the full modular group. II. Elliptic curves with complex multiplication. III. Elliptic surfaces and specialization theorems. IV. NA(c)ron models, Kodaira-N ron classification of special fibres, Tate's algorithm, and Ogg's conductor-discriminant formula. V. Tate's theory of q-curves over p-adic fields. VI. NA(c)ron's theory of canonical local height functions.

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 approach in its modern formulation, through the use of basic algebraic number theory and algebraic geometry. Following a brief discussion of the necessary algebro-geometric results, the book proceeds with an exposition of the geometry and the formal group of elliptic curves, elliptic curves over finite fields, the complex numbers, local fields, and global fields. Final chapters deal with integral and rational points, including Siegels theorem and explicit computations for the curve Y = X + DX, while three appendices conclude the whole: Elliptic Curves in Characteristics 2 and 3, Group Cohomology, and an overview of more advanced topics.

Written by an authority with great practical and teaching experience in the field, this book addresses a number of topics in computational number theory. Chapters one through five form a homogenous subject matter suitable for a six-month or year-long course in computational number theory. The subsequent chapters deal with more miscellaneous subjects.

This collection of articles grew out of an expository and tutorial conference on public-key cryptography held at the Joint Mathematics Meetings (Baltimore). The book provides an introduction and survey on public-key cryptography for those with considerable mathematical maturity and general mathematical knowledge. Its goal is to bring visibility to the cryptographic issues that fall outside the scope of standard mathematics. These mathematical expositions are intended for experienced mathematicians who are not well acquainted with the subject. The book is suitable for graduate students, researchers, and engineers interested in mathematical aspects and applications of public-key cryptography.

This book constitutes the refereed proceedings of the International Conference on the Theory and Applications of Cryptology and Information Security, ASIACRYPT'98, held in Beijing, China, in October 1998. The 32 revised full papers presented were carefully reviewed and selected from a total of 118 submissions. The book is divided in topical sections on public-key cryptosystems, elliptic-curve cryptosystems, cryptanalysis, digital signature schemes, finite automata, authentication codes, electronic cash, stream ciphers, cryptographic protocols, key escrow, new cryptography, and information theory.

These proceedings are based on a conference at the Chinese University of Hong Kong, held in response to Andrew Wile's conjecture that every elliptic curve over Q is modular. The survey article describing Wile's work is included as the first article in the present edition.