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Ⅰ．Overview 3

Ⅱ．Curves in Projective Space 19

1．Projective Space 19

2．Curves and Tangents 24

3．Flexes 32

4．Application to Cubics 40

5．Bezout's Theorem and Resultants 44

Ⅲ．Cubic Curves in Weierstrass Form 50

1．Examples 50

2．Weierstrass Form,Discriminant,j-invariant 56

3．Group Law 67

4．Computations with the Group Law 74

5．Singular Points 77

Ⅳ．Mordell's Theorem 80

1．Descent 80

2．Condition for Divisibility by 2 85

3．E(Q)/2E(Q),Special Case 88

4．E(Q)/2E(Q),General Case 92

5．Height and Mordell's Theorem 95

6．Geometric Formula for Rank 102

7．Upper Bound on the Rank 107

8．Construction of Points in E(Q) 115

9．Appendix on Algebraic Number Theory 122

Ⅴ．Torsion Subgroup of E(Q) 130

1．Overview 130

2．Reduction Modulo p 134

3．p-adic Filtration 137

4．Lutz-Nagell Theorem 144

5．Construction of Curves with Prescribed Torsion 145

6．Torsion Groups for Special Curves 148

Ⅵ．Complex Points 151

1．Overview 151

2．Elliptic Functions 152

3．Weierstrass ？ Function 153

4．Effect on Addition 162

5．Overview of Inversion Problem 165

6．Analytic Continuation 166

7．Riemann Surface of the Integrand 169

8．An Elliptic Integral 174

9．Computability of the Correspondence 183

Ⅶ．Dirichlet's Theorem 189

1．Motivation 189

2．Dirichlet Series and Euler Products 192

3．Fourier Analysis on Finite Abelian Groups 199

4．Proof of Dirichlet's Theorem 201

5．Analytic Properties of Dirichlet L Functions 207

Ⅷ．Modular Forms for SL(2,Z) 221

1．Overview 221

2．Definitions and Examples 222

3．Geometry of the q Expansion 227

4．Dimensions of Spaces of Modular Forms 231

5．L Function of a Cusp Form 238

6．Petersson Inner Product 241

7．Hecke Operators 242

8．Interaction with Petersson Inner Product 250

Ⅸ．Modular Forms for Hecke Subgroups 256

1．Hecke Subgroups 256

2．Modular and Cusp Forms 261

3．Examples of Modular Forms 265

4．L Function of a Cusp Form 267

5．Dimensions of Spaces of Cusp Forms 271

6．Hecke Operators 273

7．Oldforms and Newforms 283

Ⅹ．L Function of an Elliptic Curve 290

1．Global Minimal Weierstrass Equations 290

2．Zeta Functions and L Functions 294

3．Hasse's Theorem 296

Ⅺ．Eichler-Shimura Theory 302

1．Overview 302

2．Riemann surface X0(N) 311

3．Meromorphic DifFerentials 312

4．Properties of Compact Riemann Surfaces 316

5．Hecke Operators on Integral Homology 320

6．Modular Function j(T) 333

7．Varieties and Curves 341

8．Canonical Model of X0(N) 349

9．Abstract Elliptic Curves and Isogenies 359

10．Abelian Varieties and J acobian Variety 367

11．Elliptic Curves Constructed from S2(г0(N) 374

12．Match of L Functions 383

Ⅻ．Taniyama-Weil Conjecture 386

1．Relationships among Conjectures 386

2．Strong Weil Curves and Twists 392

3．Computations of Equations of Weil Curves 394

4．Connection with Fermat's Last Theorem 397

Notes 401

References 409

Index of Notation 419

Index 423