© 2009

The Arithmetic of Elliptic Curves


  • Second Edition of highly successful introductory textbook, with new content, from acclaimed author

  • Thorough introduction to arithmetic theory of elliptic curves

  • Many exercises to hone the reader's knowledge

  • Text enlightens proofs through general principles, rather than line-by-line algebraic proof

  • Ideal for students to learn the basics of the subject and as a reference for researchers


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

  1. 1.Department of MathematicsBrown UniversityProvidenceUSA

About the authors

Dr. Joseph Silverman is a professor at Brown University and has been an instructor or professors since 1982. He was the Chair of the Brown Mathematics department from 2001-2004. He has received numerous fellowships, grants and awards, as well as being a frequently invited lecturer. He is currently a member of the Council of the American Mathematical Society. His research areas of interest are number theory, arithmetic geometry, elliptic curves, dynamical systems and cryptography. He has co-authored over 120 publications and has had over 20 doctoral students under his tutelage. He has published 9 highly successful books with Springer, including the recently released, An Introduction to Mathematical Cryptography, for Undergraduate Texts in Mathematics.

Bibliographic information

  • Book Title The Arithmetic of Elliptic Curves
  • Authors Joseph H. Silverman
  • Series Title Graduate Texts in Mathematics
  • DOI
  • Copyright Information Springer-Verlag New York 2009
  • Publisher Name Springer, New York, NY
  • eBook Packages Mathematics and Statistics Mathematics and Statistics (R0)
  • Hardcover ISBN 978-0-387-09493-9
  • Softcover ISBN 978-1-4419-1858-1
  • eBook ISBN 978-0-387-09494-6
  • Series ISSN 0072-5285
  • Series E-ISSN 2197-5612
  • Edition Number 2
  • Number of Pages XX, 513
  • Number of Illustrations 14 b/w illustrations, 0 illustrations in colour
  • Topics Algebraic Geometry
    Number Theory
  • Buy this book on publisher's site
Industry Sectors
Finance, Business & Banking


From the reviews of the second edition:

"This well-written book covers the basic facts about the geometry and arithmetic of elliptic curves, and is sure to become the standard reference in the subject. It meets the needs of at least three groups of people: students interested in doing research in Diophantine geometry, mathematicians needing a reference for standard facts about elliptic curves, and computer scientists interested in algorithms and needing an introduction to elliptic curves..."-- MATHEMATICAL REVIEWS

“The book under review is the second, revised, enlarged, and updated edition of J. Silverman’s meanwhile classical primer of the arithmetic of elliptic curves. … All together, this enlarged and updated version of J. Silverman’s classic ‘The Arithmetic of Elliptic Curves’ significantly increases the unchallenged value of this modern primer as a standard textbook in the field. … This makes the entire text a perfect source for teachers and students, for courses and self-study, and for further studies in the arithmetic of elliptic curves likewise.” (Werner Kleinert, Zentralblatt MATH, Vol. 1194, 2010)

“For the second edition of his masterly book, the author considerably updated and improved several results and proofs. … book contains a great many exercises, many of which develop or complement the results from the main body of the book. … The reference list contains 317 items and reflects both classical and recent achievements on the topic. Notes on the exercises are an aid to the reader. … Summarizing, this is an excellent book … . useful both for experienced mathematicians and for graduate students.”­­­ (Vasil' I. Andriĭchuk, Mathematical Reviews, Issue 2010 i)

“This is the second edition of an excellent textbook on the arithmetical theory of elliptic curves … . Although there are now a number of good books on this topic it has stood the test of time and become a popular introductory text and a standard reference. … The author has added remarks which point out their significance and connection to those parts of the theory he presents. These will give readers a good start if they want to study one of them.” (Ch. Baxa, Monatshefte für Mathematik, Vol. 164 (3), November, 2011)

“The book is written for graduate students … and for researchers interested in standard facts about elliptic curves. … A wonderful textbook on the arithmetic theory of elliptic curves and it is a very popular introduction to the subject. … I recommend this book for anyone interested in the mathematical study of elliptic curves. It is an excellent introduction, elegant and very well written. It is one of the best textbooks to graduate level studies I have ever had contact yet.” (Book Inspections Blog, 2012)