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Elliptic Tales

Book Description

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. 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 may appear simple, but 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.

Table of Contents

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