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Elliptic Tales : Curves, Counting, and Number Theory [electronic resource]
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| 자료유형 | E-Book |
|---|---|
| 개인저자 | Ash, Avner, 1949- Gross, Robert. |
| 서명/저자사항 | Elliptic Tales[electronic resource] :Curves, Counting, and Number Theory. |
| 발행사항 | Princeton : Princeton University Press, 2012. |
| 형태사항 | 1 online resource (276 p.) |
| ISBN | 9781400841714 (electronic bk.) 1400841712 (electronic bk.) |
| 기타표준부호 | 9786613439758 |
| 일반주기 |
Description based upon print version of record.
Epilogue |
| 서지주기 | Includes bibliographical references and index. |
| 내용주기 | 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 Be?zout'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 |
| 요약 | 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 so. |
| 일반주제명 | Counting. Elliptic functions. Curves, Elliptic. Number theory. MATHEMATICS / Complex Analysis. Elliptic functions. Curves, Elliptic. Number theory. MATHEMATICS / Algebra / Abstract. |
| 언어 | 영어 |
| 기타형태 저록 | Print version:Ash, AvnerElliptic Tales : Curves, Counting, and Number TheoryPrinceton : Princeton University Press,c20129780691151199 |
| 대출바로가기 | http://search.ebscohost.com/login.aspx?direct=true&scope=site&db=nlebk&db=nlabk&AN=444098 |
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