Elliptic tales : curves, counting, and number theory /
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| Author / Creator: | Ash, Avner, 1949- |
|---|---|
| Imprint: | Princeton : Princeton University Press, c2012. |
| Description: | 1 online resource. |
| Language: | English |
| Subject: | |
| Format: | E-Resource Book |
| URL for this record: | http://pi.lib.uchicago.edu/1001/cat/bib/8733143 |
Table of 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
- 5. 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 DoWe Go from Here?
- Bibliography
- Index
