The theme of the 2009 session of the Park City Mathematics Institute was “The Arithmetic of L-functions.” All sorts of number theorists, young and old, were in attendance. Lunch was served every day in a tent outside the conference center. Every so often the PCMI staff assigned us randomly to tables. On one of those days, I was at a table with a collection of young mathematicians, graduate students, and undergraduates. There were three older mathematicians as well: John Tate, David Grant, and me.

As we were having lunch, the conversation turned to elliptic curves, and one of the younger persons asked, “So how did one learn about elliptic curves before Joe Silverman’s book was published?” That question tells you most of what you need to know about *The Arithmetic of Elliptic Curves*.

When it was published in 1986, Silverman’s book was the only available book-length systematic account of the number-theoretical take on theory of elliptic curves. Many others have been written since, but Silverman’s book remains the gold standard.

This new edition preserves most of the structure of the original. It has one new chapter, on “Algorithmic Aspects of Elliptic Curves,” a subject that has become vastly more important than it was in the mid-80s. The new chapter was added at the end, preserving the original chapter numbers. Similarly, while new exercises have been added, they have been placed at the end of each problem set, so that the original numbering is unchanged (and references to it remain valid). Other additions include new material on the Sato-Tate Conjecture (in Appendix C) and a section on Szpiro’s conjecture and the ABC conjecture (in Chapter VIII).

The new edition also allowed extensive rewriting. Proofs have been corrected and/or simplified, exercises have been rewritten, graphics have been redrawn, and the whole book has been re-typeset. Overall, these changes have been positive. The graphics, alas, mostly look horrible; for example, check the pictures in section VI.1. These are way below the standard of the original edition, whose graphics were already nothing to write home about. (The publisher has told me that these are being corrected.)

The new typesetting is not as elegant as the original: the pages are overfull, and the new font makes for less contrast and a grayer page. Maybe if I keep complaining authors and publishers will worry more about typesetting and stop overfilling pages… but probably not.

These formal criticisms should not, however, obscure the main point. Silverman writes well and clearly. He has thought through the mathematics, orginized it well, and produced a book that will continue to be the standard reference in the field.

Fernando Q. Gouvêa first learned about elliptic curves by reading the function-theoretical account in

*Vorlesungen über allgemeine Funktionentheorie und elliptische Funktionen, *by Courant and Hurwitz and the

*Springer Lecture Notes* volume by Alain Robert. When

*The Arithmetic of Elliptic Curves* came out, he bought a copy (despite being on a graduate student’s budget) and read the whole thing.

**BLL* — The Basic Library List Committee recommends this book for acquisition by undergraduate mathematics libraries.**