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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| Main Author: | |
|---|---|
| Other Authors: | |
| Format: | Electronic eBook |
| Language: | English |
| Published: |
Princeton :
Princeton University Press,
2012.
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| Subjects: | |
| Online Access: | DE-862 DE-863 |
| Summary: | 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. |
| Physical Description: | 1 online resource (276 pages) |
| Bibliography: | Includes bibliographical references and index. |
| ISBN: | 9781400841714 1400841712 6613439754 9786613439758 |
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| 245 | 1 | 0 | |a Elliptic Tales : |b Curves, Counting, and Number Theory. |
| 260 | |a Princeton : |b Princeton University Press, |c 2012. | ||
| 300 | |a 1 online resource (276 pages) | ||
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| 505 | 0 | |a 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. | |
| 505 | 8 | |a 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 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. | |
| 505 | 8 | |a 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. | |
| 505 | 8 | |a 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. | |
| 505 | 8 | |a 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. | |
| 520 | |a 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. | ||
| 504 | |a Includes bibliographical references and index. | ||
| 650 | 0 | |a Elliptic functions. |0 http://id.loc.gov/authorities/subjects/sh85052336 | |
| 650 | 0 | |a Curves, Elliptic. |0 http://id.loc.gov/authorities/subjects/sh85034918 | |
| 650 | 0 | |a Number theory. |0 http://id.loc.gov/authorities/subjects/sh85093222 | |
| 650 | 6 | |a Fonctions elliptiques. | |
| 650 | 6 | |a Courbes elliptiques. | |
| 650 | 6 | |a Théorie des nombres. | |
| 650 | 7 | |a MATHEMATICS |x Complex Analysis. |2 bisacsh | |
| 650 | 7 | |a MATHEMATICS |x Algebra |x Abstract. |2 bisacsh | |
| 650 | 7 | |a Curves, Elliptic |2 fast | |
| 650 | 7 | |a Elliptic functions |2 fast | |
| 650 | 7 | |a Number theory |2 fast | |
| 700 | 1 | |a Gross, Robert. | |
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| contents | 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 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 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. |
| ctrlnum | (OCoLC)773567194 |
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| id | ZDB-4-EBA-ocn773567194 |
| illustrated | Not Illustrated |
| indexdate | 2025-04-11T08:37:33Z |
| institution | BVB |
| isbn | 9781400841714 1400841712 6613439754 9786613439758 |
| language | English |
| lccn | 2011044712 |
| oclc_num | 773567194 |
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| physical | 1 online resource (276 pages) |
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| publishDate | 2012 |
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| publishDateSort | 2012 |
| publisher | Princeton University Press, |
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| spelling | Ash, Avner, 1949- https://id.oclc.org/worldcat/entity/E39PBJvxq7xmXGjt3xFRYTRR8C http://id.loc.gov/authorities/names/n2005048797 Elliptic Tales : Curves, Counting, and Number Theory. Princeton : Princeton University Press, 2012. 1 online resource (276 pages) text txt rdacontent computer c rdamedia online resource cr rdacarrier Print version record. 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 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 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. 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. Includes bibliographical references and index. Elliptic functions. http://id.loc.gov/authorities/subjects/sh85052336 Curves, Elliptic. http://id.loc.gov/authorities/subjects/sh85034918 Number theory. http://id.loc.gov/authorities/subjects/sh85093222 Fonctions elliptiques. Courbes elliptiques. Théorie des nombres. MATHEMATICS Complex Analysis. bisacsh MATHEMATICS Algebra Abstract. bisacsh Curves, Elliptic fast Elliptic functions fast Number theory fast Gross, Robert. has work: Elliptic tales (Text) https://id.oclc.org/worldcat/entity/E39PCGC4KmBvVJfMjfHBdm98fm https://id.oclc.org/worldcat/ontology/hasWork Print version: Ash, Avner. Elliptic Tales : Curves, Counting, and Number Theory. Princeton : Princeton University Press, ©2012 9780691151199 |
| spellingShingle | Ash, Avner, 1949- Elliptic Tales : Curves, Counting, and Number Theory. 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 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 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. Elliptic functions. http://id.loc.gov/authorities/subjects/sh85052336 Curves, Elliptic. http://id.loc.gov/authorities/subjects/sh85034918 Number theory. http://id.loc.gov/authorities/subjects/sh85093222 Fonctions elliptiques. Courbes elliptiques. Théorie des nombres. MATHEMATICS Complex Analysis. bisacsh MATHEMATICS Algebra Abstract. bisacsh Curves, Elliptic fast Elliptic functions fast Number theory fast |
| subject_GND | http://id.loc.gov/authorities/subjects/sh85052336 http://id.loc.gov/authorities/subjects/sh85034918 http://id.loc.gov/authorities/subjects/sh85093222 |
| title | Elliptic Tales : Curves, Counting, and Number Theory. |
| title_auth | Elliptic Tales : Curves, Counting, and Number Theory. |
| title_exact_search | Elliptic Tales : Curves, Counting, and Number Theory. |
| title_full | Elliptic Tales : Curves, Counting, and Number Theory. |
| title_fullStr | Elliptic Tales : Curves, Counting, and Number Theory. |
| title_full_unstemmed | Elliptic Tales : Curves, Counting, and Number Theory. |
| title_short | Elliptic Tales : |
| title_sort | elliptic tales curves counting and number theory |
| title_sub | Curves, Counting, and Number Theory. |
| topic | Elliptic functions. http://id.loc.gov/authorities/subjects/sh85052336 Curves, Elliptic. http://id.loc.gov/authorities/subjects/sh85034918 Number theory. http://id.loc.gov/authorities/subjects/sh85093222 Fonctions elliptiques. Courbes elliptiques. Théorie des nombres. MATHEMATICS Complex Analysis. bisacsh MATHEMATICS Algebra Abstract. bisacsh Curves, Elliptic fast Elliptic functions fast Number theory fast |
| topic_facet | Elliptic functions. Curves, Elliptic. Number theory. Fonctions elliptiques. Courbes elliptiques. Théorie des nombres. MATHEMATICS Complex Analysis. MATHEMATICS Algebra Abstract. Curves, Elliptic Elliptic functions Number theory |
| work_keys_str_mv | AT ashavner elliptictalescurvescountingandnumbertheory AT grossrobert elliptictalescurvescountingandnumbertheory |