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
