The Arithmetic of Elliptic Curves

  • Joseph H. Silverman

Part of the Graduate Texts in Mathematics book series (GTM, volume 106)

Table of contents

  1. Front Matter
    Pages i-xx
  2. Joseph H. Silverman
    Pages 1-16
  3. Joseph H. Silverman
    Pages 17-40
  4. Joseph H. Silverman
    Pages 41-114
  5. Joseph H. Silverman
    Pages 115-135
  6. Joseph H. Silverman
    Pages 137-156
  7. Joseph H. Silverman
    Pages 157-183
  8. Joseph H. Silverman
    Pages 185-205
  9. Joseph H. Silverman
    Pages 207-267
  10. Joseph H. Silverman
    Pages 269-307
  11. Joseph H. Silverman
    Pages 309-361
  12. Joseph H. Silverman
    Pages 363-408
  13. Back Matter
    Pages 409-513

About this book


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.


Algebraic Cohomology Geometry Mordell-Weil Number theory algebra cryptography finite field

Authors and affiliations

  • Joseph H. Silverman
    • 1
  1. 1.Department of MathematicsBrown UniversityProvidenceUSA

Bibliographic information