Advanced Topics in the Arithmetic of Elliptic Curves

Graduate Texts in Mathematics #151

by Joseph H. Silverman

In the introduction to the first volume of The Arithmetic of Elliptic Curves (Springer-Verlag, 1986), I observed that "the theory of elliptic curves is rich, varied, and amazingly vast," and as a consequence, "many important topics had to be omitted." I included a brief introduction to ten additional topics as an appendix to the first volume, with the tacit understanding that eventually there might be a second volume containing the details. You are now holding that second volume. it turned out that even those ten topics would not fit Unfortunately, into a single book, so I was forced to make some choices. The following material is covered in this I. Elliptic and modular functions for the full modular group. II. Elliptic curves with complex multiplication. III. Elliptic surfaces and specialization theorems. IV. Neron models, Kodaira-Neron classification of special fibers, Tate's algorithm, and Ogg's conductor-discriminant formula. V. Tate's theory of q-curves over p-adic fields. VI. Neron's theory of canonical local height functions.

Genres: Mathematics

541 pages, Paperback

First published September 24, 1994

Reviews

[Will · Rating 5 out of 5]

A little less focused than the prequel, and a more difficult read, but the ending on "Tate curves" came at a complete surprise.

[Wissam Raji · Rating 4 out of 5]

This is a graduate book on the theory of elliptic curves with an algebraic geometry approach to varieties. The book deals with the advanced topics of elliptic curves and the cohomological aspect of rational points on finite fields. Difficult to read if one does not have the basic knowledge of the theory of elliptic curves.