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Contents

Preface to the Second Edition ...................................... vii

Preface to the First Edition ........................................ ix

Acknowledgments to the Second Edition ............................. xi
Acknowledgments to the First Edition ............................... xiii

Introduction to Rational Points on Plane Curves ...................... 1
1 Rational Lines in the Projective Plane.......................... 2
2 Rational Points on Conics ................................... 4
3 Pythagoras, Diophantus, and Fermat ........................... 7
4 Rational Cubics and Mordell’s Theorem . ...................... 10
5 The Group Law on Cubic Curves and Elliptic Curves . . .......... 13
6 Rational Points on Rational Curves. Faltings and the Mordell
Conjecture . . .............................................. 17
7 Real and Complex Points on Elliptic Curves .................... 19
8 The Elliptic Curve Group Law on the Intersection of Two Quadrics
in Projective Three Space . . .................................. 20

1 Elementary Properties of the Chord-Tangent Group Law
on a Cubic Curve ............................................ 23
1 Chord-Tangent Computational Methods on a
Normal Cubic Curve ........................................ 23
2 Illustrations of the Elliptic Curve Group Law . . . ................ 28
2
3
2
3
3 The Curves with Equations y = x + ax and y = x + a ....... 34
4 Multiplication by 2 on an Elliptic Curve. . ...................... 38
5 Remarks on the Group Law on Singular Cubics . ................ 41
2 Plane Algebraic Curves ....................................... 45
1 Projective Spaces . . . ........................................ 45
2 Irreducible Plane Algebraic Curves and Hypersurfaces . .......... 47