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"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.

Avner Ash is professor of mathematics at Boston College. Robert Gross is associate professor of mathematics at Boston College. They are the coauthors of Fearless Symmetry: Exposing the Hidden Patterns of Numbers (Princeton).

Cover
Half title
Title
Copyright
Dedication
Contents
Preface
Acknowledgments
Prologue
Part I. Degree
Chapter 1. Degree of a Curve
1. Greek Mathematics
2. Degree
3. Parametric Equations
4. Our Two Definitions of Degree Clash
Chapter 2. Algebraic Closures
1. Square Roots of Minus One
2. Complex Arithmetic
3. Rings and Fields
4. Complex Numbers and Solving Equations
5. Congruences
6. Arithmetic Modulo a Prime
7. Algebraic Closure
Chapter 3. The Projective Plane
1. Points at Infinity
2. Projective Coordinates on a Line
3. Projective Coordinates on a Plane
4. Algebraic Curves and Points at Infinity
5. Homogenization of Projective Curves
6. Coordinate Patches
Chapter 4. Multiplicities and Degree
1. Curves as Varieties
2. Multiplicities
3. Intersection Multiplicities
4. Calculus for Dummies
Chapter 5. Bézout’s Theorem
1. A Sketch of the Proof
2. An Illuminating Example
Part II. Elliptic Curves and Algebra
Chapter 6. Transition to Elliptic Curves
Chapter 7. Abelian Groups
1. How Big Is Infinity?
2. What Is an Abelian Group?
3. Generations
4. Torsion
5. Pulling Rank
Appendix: An Interesting Example of Rank and Torsion
Chapter 8. Nonsingular Cubic Equations
1. The Group Law
2. Transformations
3. The Discriminant
4. Algebraic Details of the Group Law
5. Numerical Examples
6. Topology
7. Other Important Facts about Elliptic Curves
8. Two Numerical Examples
Chapter 9. Singular Cubics
1. The Singular Point and the Group Law
2. The Coordinates of the Singular Point
3. Additive Reduction
4. Split Multiplicative Reduction
5. Nonsplit Multiplicative Reduction
6. Counting Points
7. Conclusion
Appendix A: Changing the Coordinates of the Singular Point
Appendix B: Additive Reduction in Detail
Appendix C: Split Multiplicative Reduction in Detail
Appendix D: Nonsplit Multiplicative Reduction in Detail
Chapter 10. Elliptic Curves Over Q
1. The Basic Structure of the Group
2. Torsion Points
3. Points of Infinite Order
4. Examples
Part III. Elliptic Curves and Analysis
Chapter 11. Building Functions
1. Generating Functions
2. Dirichlet Series
3. The Riemann Zeta-Function
4. Functional Equations
5. Euler Products
6. Build Your Own Zeta-Function
Chapter 12. Analytic Continuation
1. A Difference that Makes a Difference
2. Taylor Made
3. Analytic Functions
4. Analytic Continuation
5. Zeroes, Poles, and the Leading Coefficient
Chapter 13. L-Functions
1. A Fertile Idea
2. The Hasse-Weil Zeta-Function
3. The L-Function of a Curve
4. The L-Function of an Elliptic Curve
5. Other L-Functions
Chapter 14. Surprising Properties of L-Functions
1. Compare and Contrast
2. Analytic Continuation
3. Functional Equation
Chapter 15. The Conjecture of Birch and Swinnerton-Dyer
1. How Big Is Big?
2. Influences of the Rank on the Np’s
3. How Small Is Zero?
4. The BSD Conjecture
5. Computational Evidence for BSD
6. The Congruent Number Problem
Epilogue
Retrospect
Where Do We Go from Here?
Bibliography
Index
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