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Introduction 1

CHAPTER ⅠGeometry and Arithmetic 9

1．Rational Points on Conics 9

2．The Geometry of Cubic Curves 15

3．Weierstrass Normal Form 22

4．Explicit Formulas for the Group Law 28

Exercises 32

CHAPTER Ⅱ Points of Finite Order 38

1．Points of Order Two and Three 38

2．Real and Complex Points on Cubic Curves 41

3．The Discriminant 47

4．Points of Finite Order Have Integer Coordinates 49

5．The Nagell-Lutz Theorem and Further Developments 56

Exercises 58

CHAPTER Ⅲ The Group of Rational Points 63

1．Heights and Descent 63

2．The Height of P+P0 68

3．The Height of 2P 71

4．A Useful Homomorphism 76

5．Mordell's Theorem 83

6．Examples and Further Developments 89

7．Singular Cubic Curves 99

Exercises 102

CHAPTER Ⅳ Cubic Curves over Finite Fields 107

1．Rational Points over Finite Fields 107

2．A Theorem of Gauss 110

3．Points of Finite Order Revisited 121

4．A Factorization Algorithm Using Elliptic Curves 125

Exercises 138

CHAPTER ⅤInteger Points on Cubic Curves 145

1．How Many Integer Points? 145

2．Taxicabs and Sums of Two Cubes 147

3．Thue's Theorem and Diophantine Approximation 152

4．Construction of an Auxiliary Polynomial 157

5．The Auxiliary Polynomial Is Small 165

6．The Auxiliary Polynomial Does Not Vanish 168

7．Proof of the Diophantine Approximation Theorem 171

8．Further Developments 174

Exercises 177

CHAPTER Ⅵ Complex Multiplication 180

1．Abelian Extensions of Q 180

2．Algebraic Points on Cubic Curves 185

3．A Galois Representation 193

4．Complex Multiplication 199

5．Abelian Extensions of Q(i) 205

Exercises 213

APPENDIX A Projective Geometry 220

1．Homogeneous Coordinates and the Projective Plane 220

2．Curves in the Projective Plane 225

3．Intersections of Projective Curves 233

4．Intersection Multiplicities and a Proof of Bezout's Theorem 242

5．Reduction Modulo p 251

Exercises 254

Bibliography 259

List of Notation 263

Index 267