The Arithmetic of Elliptic Curves (2nd Edition)

Joseph H. Silverman

Springer-Verlag – Graduate Texts in Mathematics 106
ISBN: 978-0-387-09493-9 – 2nd ed. – © 2009 – 513 + xviii pages
Math. Subj. Class: Primary 11G05; Secondary 11D25, 11G07, 14G05, 14H25, 14H52

The Arithmetic of Elliptic Curves is a graduate-level textbook designed to introduce the reader to an important topic in modern mathematics. 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.

  1. Algebraic Varieties
  2. Algebraic Curves
  3. The Geometry of Elliptic Curves
  4. The Formal Group of an Elliptic Curve
  5. Elliptic Curves over Finite Fields
  6. Elliptic Curves over C
  7. Elliptic Curves over Local Fields
  8. Elliptic Curves over Global Fields
  9. Integral Points on Elliptic Curves
  10. Computing the Mordell--Weil Group
  11. Algorithmic Aspects of Elliptic Curves (new to the 2nd edition)
  1. Elliptic Curves in Characteristics 2 and 3
  2. Group Cohomology: H0 and H1
  3. Further Topics: An Overview
The Arithmetic of Elliptic Curves book jacket

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