The Arithmetic of Elliptic Curves

Preface to the Second Edition.- Preface to the First Edition.- Introduction.- Algebraic Varieties.- Algebraic Curves.- The Geometry of Elliptic Curves.- The Formal Group of an Elliptic Curve.- Elliptic Curves over Finite Fields.- Elliptic Curves over C.- Elliptic Curves over Local Fields.- Elliptic Curves over Global Fields.- Integral Points on Elliptic Curves.- Computing the Mordell Weil Group.- Algorithmic Aspects of Elliptic Curves.- Appendix A: Elliptic Curves in Characteristics 2 and 3.- Appendix B: Group Cohomology (H 0 and H 1 ).- Appendix C: Further Topics: An Overview.- Notes on Exercises.- Bibliography.- List of Notation.- References.- Index.

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.

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. Andri chuk, 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 textand a standard reference.