Advanced Topics in the Arithmetic of Elliptic Curves

Graduate Texts in Mathematics #151

by Joseph H. Silverman

This book continues the treatment of the arithmetic theory of elliptic curves begun in the first volume. The book begins with the theory of elliptic and modular functions for the full modular group r(1), including a discussion of Hekcke operators and the L -series associated to cusp forms. This is followed by a detailed study of elliptic curves with complex multiplication, their associated Grössencharacters and L -series, and applications to the construction of abelian extensions of quadratic imaginary fields. Next comes a treatment of elliptic curves over function fields and elliptic surfaces, including specialization theorems for heights and sections. This material serves as a prelude to the theory of minimal models and Néron models of elliptic curves, with a discussion of special fibers, conductors, and Ogg's formula. Next comes a brief description of q -models for elliptic curves over C and R, followed by Tate's theory of q -models for elliptic curves with non-integral j -invariant over p -adic fields. The book concludes with the construction of canonical local height functions on elliptic curves, including explicit formulas for both archimedean and non-archimedean fields.

Genres: Mathematics, Geometry, Algebra

538 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.