Advanced Topics in the Arithmetic of Elliptic Curves

1. Front Matter

   Pages i-xiii

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2. Introduction

   • Joseph H. Silverman

   Pages 1-4

3. Elliptic and Modular Functions

   • Joseph H. Silverman

   Pages 5-94

4. Complex Multiplication

   • Joseph H. Silverman

   Pages 95-186

5. Elliptic Surfaces

   • Joseph H. Silverman

   Pages 187-288

6. The Néron Model

   • Joseph H. Silverman

   Pages 289-407

7. Elliptic Curves over Complete Fields

   • Joseph H. Silverman

   Pages 408-453

8. Local Height Functions

   • Joseph H. Silverman

   Pages 454-480

9. Back Matter

   Pages 481-528

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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.

• Elliptic Curve
• algebraic surface
• arithmetic
• Divisor
• elliptic curve
• modular curve

• Mathematics Department, Brown University, Providence, USA

  Joseph H. Silverman