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# Rational Points on Elliptic Curves

Joseph H. Silverman and John Tate

Preface

Computer Packages

Acknowledgments

Introduction

CHAPTER I — Geometry and Arithmetic
1. Rational Points on Conics
2. The Geometry of Cubic Curves
3. Weierstrass Normal Form
4. Explicit Formulas for the Group Law
Exercises

CHAPTER II — Points of Finite Order
1. Points of Order Two and Three
2. Real and Complex Points on Cubic Curves
3. The Discriminant
4. Points of Finite Order Have Integer Coordinates
5. The Nagell-Lutz Theorem and Further Developments
Exercises

CHAPTER III — The Group of Rational Points
1. Heights and Descent
2. The Height of P + P_0
3. The Height of 2P
4. A Useful Homomorphism
5. Mordell's Theorem
6. Examples and Further Developments
7. Singular Cubic Curves
Exercises

CHAPTER IV — Cubic Curves over Finite Fields
1. Rational Points over Finite Fields
2. A Theorem of Gauss
3. Points of Finite Order Revisited
4. A Factorization Algorithm Using Elliptic Curves
Exercises

CHAPTER V — Integer Points on Cubic Curves
1. How Many Integer Points?
2. Taxicabs and Sums of Two Cubes
3. Thue's Theorem and Diophantine Approximation
4. Construction of an Auxiliary Polynomial
5. The Auxiliary Polynomial Is Small
6. The Auxiliary Polynomial Does Not Vanish
7. Proof of the Diophantine Approximation Theorem
8. Further Developments
Exercises

CHAPTER VI — Complex Multiplication
1. Abelian Extensions of Q
2. Algebraic Points on Cubic Curves
3. A Galois Representation
4. Complex Multiplication
5. Abelian Extensions of Q(i)
Exercises

APPENDIX A — Projective Geometry
1. Homogeneous Coordinates and the Projective Plane
2. Curves in the Projective Plane
3. Intersections of Projective Curves
4. Intersection Multiplicities and a Proof of Bezout's Theorem
5. Reduction Modulo p
Exercises

Bibliography

List of Notation

Index