Advanced Topics in the Arithmetic of Elliptic Curves

Joseph H. Silverman

Springer-Verlag – Graduate Texts in Mathematics 151
ISBN: 978-0387943282 – 1st ed. – © 1994 – 548 + xiii pages
Math. Subj. Class: 11G05, 14H52
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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 Γ(1), including a discussion of Hecke 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.

Chapter I:Elliptic and Modular Functions
Chapter II:Complex Multiplication
Chapter III:Elliptic Surfaces
Chapter IV:The Néron Model
Chapter V:Elliptic Curves over Complete Fields
Chapter VI:Local Height Functions
Appendix A:Some Useful Tables

     Errata List

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