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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 solut...
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| Formato: | Electrónico eBook |
| Idioma: | Inglés |
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Princeton :
Princeton University Press,
2012.
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| Colección: | Book collections on Project MUSE.
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| Acceso en línea: | Texto completo |
Tabla de Contenidos:
- Cover; Title; Copyright; 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 Infinity5. 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 Bezout'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 Equations1. 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 DetailAppendix 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 Continuation5. 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.