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Introduction 1

CHAPTER Ⅰ Elliptic and Modular Functions 5

1．The Modular Group 6

2．The Modular Curve X(1) 14

3．Modular Functions 23

4．Uniformization and Fields of Moduli 34

5．Elliptic Functions Revisited 38

6．q-Expansions of Elliptic Functions 47

7．q-Expansions of Modular Functions 55

8．Jacobi's Product Formula for △(τ) 62

9．Hecke Operators 67

10．Hecke Operators Acting on Modular Forms 74

11．L-Series Attached to Modular Forms 80

Exercises 85

CHAPTER Ⅱ Complex Multiplication 95

1．Complex Multiplication over C 96

2．Rationality Questions 104

3．Class Field Theory—A Brief Review 115

4．The Hilbert Class Field 121

5．The Maximal Abelian Extension 128

6．Integrality of j 140

7．Cyclotomic Class Field Theory 151

8．The Main Theorem of Complex Multiplication 157

9．The Associated Gr？ssencharacter 165

10．The L-Series Attached to a CM Elliptic Curve 171

Exercises 178

CHAPTER Ⅲ Elliptic Surfaces 187

1．Elliptic Curves over Function Fields 188

2．The Weak Mordell-Weil Theorem 191

3．Elliptic Surfaces 200

4．Heights on Elliptic Curves over Function Fields 212

5．Split Elliptic Surfaces and Sets of Bounded Height 220

6．The Mordell-Weil Theorem for Function Fields 230

7．The Geometry of Algebraic Surfaces 231

8．The Geometry of Fibered Surfaces 236

9．The Geometry of Elliptic Surfaces 245

10．Heights and Divisors on Varieties 255

11．Specialization Theorems for Elliptic Surfaces 265

12．Integral Points on Elliptic Curves over Function Fields 274

Exercises 278

CHAPTER Ⅳ The Néron Model 289

1．Group Varieties 290

2．Schemes and S-Schemes 297

3．Group Schemes 306

4．Arithmetic Surfaces 311

5．Néron Models 318

6．Existence of Néron Models 325

7．Intersection Theory,Minimal Models,and Blowing-Up 338

8．The Special Fiber of a Néron Model 350

9．Tate's Algorithm to Compute the Special Fiber 361

10．The Conductor of an Elliptic Curve 379

11．Ogg's Formula 389

Exercises 396

CHAPTER Ⅴ Elliptic Curves over Complete Fields 408

1．Elliptic Curves over C 408

2．Elliptic Curves over R 413

3．The Tate Curve 422

4．The Tate Map Is Surjective 429

5．Elliptic Curves over p-adic Fields 438

6．Some Applications of p-adic Uniformization 445

Exercises 448

CHAPTER Ⅵ Local Height Functions 454

1．Existence of Local Height Functions 455

2．Local Decomposition of the Canonical Height 461

3．Archimedean Absolute Values—Explicit Formulas 463

4．Non-Archimedean Absolute Values—Explicit Formulas 469

Exercises 476

APPENDIX A Some Useful Tables 481

1．Bernoulli Numbers and ζ(2k) 481

2．Fourier Coefficients of △(τ) and j(τ) 482

3．Elliptic Curves over Q with Complex Multiplication 483

Notes on Exercises 484

References 488

List of Notation 498

Index 504