Elliptic Tales: Curves, Counting, and Number Theory

Elliptic Tales: Curves, Counting, and Number Theory

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Princeton University Press
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Elliptic Tales: Curves, Counting, and Number Theory

Elliptic Tales describes the latest developments in number theory by looking at one of the most exciting unsolved problems in contemporary mathematics—the Birch and Swinnerton-Dyer Conjecture. The Clay Mathematics Institute is offering a prize of $1 million to anyone who can discover a general solution to the problem. In this book, Avner Ash and Robert Gross guide readers through the mathematics they need to understand this captivating problem.

The key to the conjecture lies in elliptic curves, which are cubic equations in two variables. These equations may appear simple, yet they arise from some very deep—and often very mystifying—mathematical ideas. Using only basic algebra and calculus while presenting numerous eye-opening examples, Ash and Gross make these ideas accessible to general readers, and in the process venture to the very frontiers of modern mathematics. Along the way, they give an informative and entertaining introduction to some of the most profound discoveries of the last three centuries in algebraic geometry, abstract algebra, and number theory. They demonstrate how mathematics grows more abstract to tackle ever more challenging problems, and how each new generation of mathematicians builds on the accomplishments of those who preceded them. Ash and Gross fully explain how the Birch and Swinnerton-Dyer Conjecture sheds light on the number theory of elliptic curves, and how it provides a beautiful and startling connection between two very different objects arising from an elliptic curve, one based on calculus, the other on algebra.

Product Details

ISBN-13: 9780691151199
Publisher: Princeton University Press
Publication date: 03/12/2012
Pages: 280
Sales rank: 1,113,539
Product dimensions: 6.10(w) x 9.20(h) x 1.00(d)

Table of Contents

Preface xiii
Acknowledgments xix
Prologue 1


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. Bézout’s Theorem 82
1.A Sketch of the Proof 82
2.An Illuminating Example 88


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 10. Elliptic Curves over Q 152
1.The Basic Structure of the Group 152
2.Torsion Points 153
3.Points of Infinite Order 155
4.Examples 156


Chapter 11. Building Functions 161
1.Generating Functions 161
2.Dirichlet Series 167
3.The Riemann Zeta-Function 169
4.Functional Equations 171
5.Euler Products 174
6.Build Your Own Zeta-Function 176

Chapter 12. Analytic Continuation 181
1.A Difference that Makes a Difference 181
2.Taylor Made 185
3.Analytic Functions 187
4.Analytic Continuation 192
5.Zeroes, Poles, and the Leading Coefficient 196

Chapter 13. L-functions 199
1.A Fertile Idea 199
2.The Hasse-Weil Zeta-Function 200
3.The L-Function of a Curve 205
4.The L-Function of an Elliptic Curve 207
5.Other L-Functions 212

Chapter 14. Surprising Properties of L-functions 215
1.Compare and Contrast 215
2.Analytic Continuation 220
3.Functional Equation 221

Chapter 15. The Conjecture of Birch and
Swinnerton-Dyer 225
1.How Big Is Big? 225
2.Influences of the Rank on the Np’s 228
3.How Small Is Zero? 232
4.The BSD Conjecture 236
5.Computational Evidence for BSD 238
6.The Congruent Number Problem 240
Epilogue 245
Retrospect 245
Where DoWe Go from Here? 247

Bibliography 249
Index 251

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