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.

Unfortunately, it turned out that even those ten topics would not fit into a single book, so I was forced to make some choices.

Preface

Computer Packages

Acknowledgments

Introduction

CHAPTER Ⅰ

Elliptic and Modular Functions

1.The Modular Gronp

2.The Modular Curve X(1)

3.Modular Functions

4.Uniformization and Fields of Moduli

5.Elliptic Functions Revisited

6.q-Expansions of Elliptic Functions

7.q-Expansions of Modular Functions

8.Jacobi's Product Formula for A(r)

9.Hecke Operators

10.Hecke Operators Acting on Modular Forms

11.L-Series Attached to Modular Forms

Exercises

CHAPTER Ⅱ

Complex Multiplication

.1.Complex Multiplication over C

2.Rationality Questions

3.Class Field Theory -- A Brief Review

4.The Hilbert Class Field

5.The Maximal Abelian Extension

6.Integrality of j

7.Cyclotomic Class Field Theory

8.The Main Theorem of Complex Multiplication

9.The Associated GrSssencharacter

10. The L-Series Attached to a CM Elliptic Curve

Exercises

CHAPTER Ⅲ

Elliptic Surfaces

1.Elliptic Curves over Function Fields

2.The Weak Mordell-Weil Theorem

3.Elliptic Surfaces

4.Heights on Elliptic Curves over Fhnction Fields

5.Split Elliptic Surfaces and Sets of Bounded Height

6.The Mordell-Weil Theorem for Fhnction Fields

7.The Geometry of Algebraic Surfaces

8.The Geometry of Fibered Surfaces

9.The Geometry of Elliptic Surfaces

10.Heights and Divisors on Varieties

11.Specialization Theorems for Elliptic Surfaces

12.Integral Points on Elliptic Curves over Function Fields

Exercises

CHAPTER Ⅳ

The Neron Model

1.Group Varieties

2.Schemes and S-Schemes

3.Group Schemes

4.Arithmetic Surfaces

5.Neron Models

6.Existence of Neron Models

7.Intersection Theory, Minimal Models, and Blowing-Up

8.The Special Fiber of a Neron Model

9.Tate's Algorithm to Compute the Special Fiber

10. The Conductor of an Elliptic Curve

11. Ogg's Formula

Exercises

CHAPTER Ⅴ

Elliptic Curves over Complete Fields

1.Elliptic Curves over C

2.Elliptic Curves over R

3. The Tate Curve

4.The Tate Map Is Surjective

5.Elliptic Curves over p-adic Fields

6.Some Applications of p-adic Uniformization

Exercises

CHAPTER Ⅵ

Local Height Functions

1.Existence of Local Height Functions

2.Local Decomposition of the Canonical Height

3.Archimedean Absolute Values -- Explicit Formulas

4.Non-Archimedean Absolute Values -- Explicit Formulas

Exercises

APPENDIX A

Some Useful Tables

1.Bernoulli Numbers and (2k)

2.Fourier Coefficients of A(T)and j(T)

3.Elliptic Curves over Q with Complex Multiplication

Notes oil Exercises

References

List of Notation

Index