Advanced Topics in the Arithmetic of Elliptic Curves (Graduate Texts in Mathematics (151))

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 book: 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.

In The Arithmetic of Elliptic Curves, the author presented the basic theory culminating in two fundamental global results, the Mordell-Weil theorem on the finite generation of the group of rational points and Siegel's theorem on the finiteness of the set of integral points. This book continues the study of elliptic curves by presenting six important, but somewhat more specialized topics: I. Elliptic and modular functions for the full modular group. II. Elliptic curves with complex multiplication. III. Elliptic surfaces and specialization theorems. IV. Néron models, Kodaira-N ron classification of special fibres, Tate's algorithm, and Ogg's conductor-discriminant formula. V. Tate's theory of q-curves over p-adic fields. VI. Néron's theory of canonical local height functions.

This book is meant to be an introductory text, albeit at an upper graduate level. The main prerequisite for reading this book is some familiarity with the basic theory of elliptic curves as described, for example, in the first volume. Numerous exercises have been included at the end of each chapter. A list of comments and citations for the exercises will be found at the end of the book. In the first volume of The Arithmetic of Elliptic Curves, we presented the basic theory culminating in two fundamental global results, the Mordell-Weil theorem on the finite generation of the group of rational points and Siegel's theorem on the finiteness of the set of integral points. Joseph H. Silverman. Includes Bibliographical References (p. [488]-497) And Index.