Arithmetic of Elliptic Curves 2E : Revised Edition - Silverman, Joseph H.

Arithmetic of Elliptic Curves 2E : Revised Edition

Joseph H. Silverman

Yayınevi: Springer

Yayın tarihi: 07/2009

ISBN: 9780387094939

Ciltli | İngilizce | 514 Sayfa | 15,75 cm x 23,62 cm; 14 illustrations

Tür: Matematik-İstatistik

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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.

Chapters:
I Algebraic Varieties
II Algebraic Curves
III The Geometry of Elliptic Curves
IV The Formal Group of an Elliptic Curve
V Elliptic Curves over Finite Fields
VI Elliptic Curves over C
VII Elliptic Curves over Local Fields
VIII Elliptic Curves over Global Fields
IX Integral Points on Elliptic Curves
X Computing the Mordell–Weil Group
XI Algorithmic Aspects of Elliptic Curves
Exercises
Apendix A Elliptic Curves in Characteristics 2 and 3
Exercises
Apendix B Group Cohomology (H0 and H1)
Exercises
Apendix C Further Topics: An Overview
Notes on Exercises

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