Elliptic Tales

Preface xiii Acknowledgments xix Prologue 1 PART I. DEGREE Chapter 1. Degree of a Curve 13 1. Greek Mathematics 13 2. Degree 14 3. Parametric Equations 20 4. Our Two Definitions of Degree Clash 23 Chapter 2. Algebraic Closures 26 1. Square Roots of Minus One 26 2. Complex Arithmetic 28 3. Rings and Fields 30 4. Complex Numbers and Solving Equations 32 5. Congruences 34 6. Arithmetic Modulo a Prime 38 7. Algebraic Closure 38 Chapter 3. The Projective Plane 42 1. Points at Infinity 42 2. Projective Coordinates on a Line 46 3. Projective Coordinates on a Plane 50 4. Algebraic Curves and Points at Infinity 54 5. Homogenization of Projective Curves 56 6. Coordinate Patches 61 Chapter 4. Multiplicities and Degree 67 1. Curves as Varieties 67 2. Multiplicities 69 3. Intersection Multiplicities 72 4. Calculus for Dummies 76 Chapter 5. B'ezout's Theorem 82 1. A Sketch of the Proof 82 2. An Illuminating Example 88 PART II. ELLIPTIC CURVES AND ALGEBRA Chapter 6. Transition to Elliptic Curves 95 Chapter 7. Abelian Groups 100 1. How Big Is Infinity? 100 2. What Is an Abelian Group? 101 3. Generations 103 4. Torsion 106 5. Pulling Rank 108 Appendix: An Interesting Example of Rank and Torsion 110 Chapter 8. Nonsingular Cubic Equations 116 1. The Group Law 116 2. Transformations 119 3. The Discriminant 121 4. Algebraic Details of the Group Law 122 5. Numerical Examples 125 6. Topology 127 7. Other Important Facts about Elliptic Curves 131 5. Two Numerical Examples 133 Chapter 9. Singular Cubics 135 1. The Singular Point and the Group Law 135 2. The Coordinates of the Singular Point 136 3. Additive Reduction 137 4. Split Multiplicative Reduction 139 5. Nonsplit Multiplicative Reduction 141 6. Counting Points 145 7. Conclusion 146 Appendix A: Changing the Coordinates of the Singular Point 146 Appendix B: Additive Reduction in Detail 147 Appendix C: Split Multiplicative Reduction in Detail 149 Appendix D: Nonsplit Multiplicative Reduction in Detail 150 Chapter

Avner Ash is Professor in the Department of Mathematics at Boston College, Massachusetts.