Elliptic Tales: Curves, Counting, and Number Theory
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Elektronisch E-Book |
| Sprache: | English |
| Veröffentlicht: |
Princeton
Princeton University Press
2012
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| Schlagworte: | |
| Online-Zugang: | FAW01 FAW02 Volltext |
| Beschreibung: | Epilogue 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 |
| Beschreibung: | 1 Online-Ressource (276 pages) |
| ISBN: | 0691151199 1400841712 9780691151199 9781400841714 |
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| 500 | |a Epilogue | ||
| 500 | |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 | ||
| 500 | |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 | ||
| 500 | |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 | ||
| 500 | |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 | ||
| 500 | |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 | ||
| 500 | |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 | ||
| 500 | |a Includes bibliographical references and index | ||
| 650 | 4 | |a Counting | |
| 650 | 4 | |a Elliptic functions | |
| 650 | 4 | |a Curves, Elliptic | |
| 650 | 4 | |a Number theory | |
| 650 | 7 | |a MATHEMATICS / Complex Analysis |2 bisacsh | |
| 650 | 7 | |a MATHEMATICS / Algebra / Abstract |2 bisacsh | |
| 650 | 7 | |a Curves, Elliptic |2 fast | |
| 650 | 7 | |a Elliptic functions |2 fast | |
| 650 | 7 | |a Number theory |2 fast | |
| 650 | 4 | |a Elliptic functions | |
| 650 | 4 | |a Curves, Elliptic | |
| 650 | 4 | |a Number theory | |
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| 700 | 1 | |a Gross, Robert |e Sonstige |4 oth | |
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Datensatz im Suchindex
| _version_ | 1804175505653497856 |
|---|---|
| any_adam_object | |
| author | Ash, Avner |
| author_facet | Ash, Avner |
| author_role | aut |
| author_sort | Ash, Avner |
| author_variant | a a aa |
| building | Verbundindex |
| bvnumber | BV043098294 |
| collection | ZDB-4-EBA |
| ctrlnum | (OCoLC)773567194 (DE-599)BVBBV043098294 |
| dewey-full | 515.983 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515.983 |
| dewey-search | 515.983 |
| dewey-sort | 3515.983 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Electronic eBook |
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| id | DE-604.BV043098294 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-10T07:17:21Z |
| institution | BVB |
| isbn | 0691151199 1400841712 9780691151199 9781400841714 |
| language | English |
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| physical | 1 Online-Ressource (276 pages) |
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| publisher | Princeton University Press |
| record_format | marc |
| spelling | Ash, Avner Verfasser aut Elliptic Tales Curves, Counting, and Number Theory Princeton Princeton University Press 2012 1 Online-Ressource (276 pages) txt rdacontent c rdamedia cr rdacarrier Epilogue 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 Counting Elliptic functions Curves, Elliptic Number theory MATHEMATICS / Complex Analysis bisacsh MATHEMATICS / Algebra / Abstract bisacsh Curves, Elliptic fast Elliptic functions fast Number theory fast Elliptische Funktion (DE-588)4134665-8 gnd rswk-swf Zahlentheorie (DE-588)4067277-3 gnd rswk-swf Elliptische Funktion (DE-588)4134665-8 s Zahlentheorie (DE-588)4067277-3 s 1\p DE-604 Gross, Robert Sonstige oth http://search.ebscohost.com/login.aspx?direct=true&scope=site&db=nlebk&db=nlabk&AN=444098 Aggregator Volltext 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Ash, Avner Elliptic Tales Curves, Counting, and Number Theory Counting Elliptic functions Curves, Elliptic Number theory MATHEMATICS / Complex Analysis bisacsh MATHEMATICS / Algebra / Abstract bisacsh Curves, Elliptic fast Elliptic functions fast Number theory fast Elliptische Funktion (DE-588)4134665-8 gnd Zahlentheorie (DE-588)4067277-3 gnd |
| subject_GND | (DE-588)4134665-8 (DE-588)4067277-3 |
| 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 | Counting Elliptic functions Curves, Elliptic Number theory MATHEMATICS / Complex Analysis bisacsh MATHEMATICS / Algebra / Abstract bisacsh Curves, Elliptic fast Elliptic functions fast Number theory fast Elliptische Funktion (DE-588)4134665-8 gnd Zahlentheorie (DE-588)4067277-3 gnd |
| topic_facet | Counting Elliptic functions Curves, Elliptic Number theory MATHEMATICS / Complex Analysis MATHEMATICS / Algebra / Abstract Elliptische Funktion Zahlentheorie |
| url | http://search.ebscohost.com/login.aspx?direct=true&scope=site&db=nlebk&db=nlabk&AN=444098 |
| work_keys_str_mv | AT ashavner elliptictalescurvescountingandnumbertheory AT grossrobert elliptictalescurvescountingandnumbertheory |