Preface Computer Packages Acknowledgments Introduction CHAPTER Ⅰ Elliptic and Modular Functions The Modular Group The Modular Curve X(1) Modular Functions Uniformization and Fields of Moduli Elliptic Functions Revisited q-Expansions of Elliptic Functions q-Expansions of Modular Functions Jacobis Product Formula for A(T) Hecke Operators Hecke Operators Acting on Modular Forms L-Series Attached to Modular Forms Exercises CHAPTER Ⅱ Complex Multiplication Complex Multiplication over C Rationality Questions Class Field Theory —— A Brief Review The Hilbert Class Field The Maximal Abelian Extension Integrality of j Cyclotomic Class Field Theory The Main Theorem of Complex Multiplication The Associated GrSssencharacter The L-Series Attached to a CM Elliptic Curve Exercises CHAPTER Ⅲ Elliptic Surfaces Elliptic Curves over Function Fields The Weak Mordell-Weil Theorem Elliptic Surfaces Heights on Elliptic Curves over Unction Fields Split Elliptic Surfaces and Sets of Bounded Height The Mordell-Weil Theorem for Fhnction Fields The Geometry of Algebraic Surfaces The Geometry of Fibered Surfaces The Geometry of Elliptic Surfaces Heights and Divisors on Varieties Specialization Theorems for Elliptic Surfaces Integral Points on Elliptic Curves over Function Fields Exercises CHAPTER Ⅳ The N6ron Model Group Varieties Schemes and S-Schemes Group Schemes Arithmetic Surfaces N6ron Models Existence of N6ron Models Intersection Theory， Minimal Models， and Blowing-Up The Special Fiber of a N6ron Model Tates Algorithm to Compute the Special Fiber The Conductor of an Elliptic Curve Oggs Formula Exercises CHAPTER Ⅴ Elliptic Curves over Complete Fields Elliptic Curves over C Elliptic Curves over R The Tate Curve The Tate Map Is Surjective Elliptic Curves over p-adic Fields Some Applications of p-adic Uniformization Exercises CHAPTER Ⅵ Local Height Functions Existence of Local Height Functions Local Decomposition of the Canonical Height Archimedean Absolute Values —— Explicit Formulas Non-Archimedean Absolute Values —— Explicit Formulas Exercises APPENDIX A Some Useful Tables Bernoulli Numbers and (2k) Fourier Coefficients of A(T) and j(T) Elliptic Curves over Q with Complex Multiplication Notes on Exercises References List of Notation Index
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