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Elliptic Tales Curves Counting and Number Theory 1st Edition by Avner Ash, Robert Gross ISBN 1400841712 9781400841714

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Elliptic Tales Curves Counting and Number Theory Avner Ash Digital Instant Download

Author(s): Avner Ash; Robert Gross
ISBN(s): 9781400841714, 1400841712
Edition: Course Book
File Details: PDF, 1.28 MB
Year: 2012
Language: english
SKU: EB-51948696 Category: Tags: ,

Elliptic Tales Curves Counting and Number Theory 1st Edition by Avner Ash, Robert Gross – Ebook PDF Instant Download/Delivery: 1400841712, 9781400841714
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ISBN 10: 1400841712 
ISBN 13: 9781400841714
Author: Avner Ash, Robert Gross

A look at one of the most exciting unsolved problems in mathematics today 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.

Elliptic Tales Curves Counting and Number Theory 1st Table of contents:

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

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Tags: Avner Ash, Robert Gross, Elliptic Tales, Curves Counting