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

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Hlavní autoři:	Ash, Avner, Gross, Robert
Médium:	E-kniha Kniha
Jazyk:	angličtina
Vydáno:	Princeton Princeton University Press 2012
Vydání:	1
Témata:	Abelian group Abstract Addition Additive group Algebra Algebraic Algebraic closure Algebraic curve Algebraic equation Algebraic geometry Analytic continuation Analytic function Birch and Swinnerton-Dyer conjecture Calculation Calculus Coefficient Complex multiplication Complex number Complex plane Computation Conjecture Cubic function Curves, Elliptic Derivative Dirichlet series Elliptic curve Elliptic functions Equation Euler product Exponential function Factorization Finite field Finitely generated abelian group Functional equation Geometry History & Philosophy Identity element Infinite product Integer Line at infinity Line–line intersection Logarithm Mathematician MATHEMATICS MATHEMATICS / Algebra / Abstract MATHEMATICS / Calculus MATHEMATICS / Geometry / Algebraic MATHEMATICS / History & Philosophy MATHEMATICS / Number Theory Modular form Monomial Multiplicative group Natural number Number Theory Originality Parabola Parametric equation Parametrization PBF PBH PBKA PBMW PBX PDZ Point at infinity Polynomial Power series Prime number Princeton University Press Projective line Projective plane Pure mathematics Rational number Rational point Real number Riemann hypothesis Riemann zeta function Science Scientific notation Series (mathematics) Simultaneous equations Solution set Summation Tangent Taylor series The Modern World (novel) Theorem Torsion subgroup Variable (mathematics) Polynomial Parametrization Originality Series (mathematics) Rational number Cubic function Projective line Projective plane Rational point Complex multiplication Line–line intersection Calculation Infinite product Algebraic curve Complex plane Finite field Addition Real number Analytic continuation Conjecture Functional equation Line at infinity Parabola Computation Identity element Derivative Euler product Riemann hypothesis Summation Prime number Natural number Finitely generated abelian group Abelian group Point at infinity Science Mathematics Power series Torsion subgroup Exponential function Pure mathematics Multiplicative group Logarithm Solution set Riemann zeta function Monomial Theorem Coefficient Birch and Swinnerton-Dyer conjecture The Modern World (novel) Equation Taylor series Tangent Analytic function Elliptic curve Mathematician Factorization Additive group Complex number Variable (mathematics) Integer Simultaneous equations Princeton University Press Algebraic equation Dirichlet series Scientific notation Modular form Algebraic closure Number theory Algebraic geometry Parametric equation
ISBN:	0691163502, 9780691163505, 0691151199, 9780691151199, 9781400841714, 1400841712
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Elliptic tales: curves, counting, and number theory -- Contents -- Preface -- Acknowledgments -- Prologue -- Part I: Degree -- Chapter 1: Degree of a Curve -- Chapter 2: Algebraic Closures -- Chapter 3: The Projective Plane -- Chapter 4: Multiplicities and Degree -- Chapter 5: Bézout’s Theorem -- Part II: Elliptic Curves and Algebra -- Chapter 6: Transition to Elliptic Curves -- Chapter 7: Abelian Groups -- Chapter 8: Nonsingular Cubic Equations -- Chapter 9: Singular Cubics -- Chapter 10: Elliptic Curves over Q -- Part III: Elliptic Curves and Analysis -- Chapter 11: Building Functions -- Chapter 12: Analytic Continuation -- Chapter 13: L-Functions -- Chapter 14: Surprising Properties of L-Functions -- Chapter 15: The Conjecture of Birch and Swinnerton-Dyer -- Epilogue -- Bibliography -- Index.
Front matter Table of Contents Preface Acknowledgments Prologue Chapter 1: DEGREE OF A CURVE Chapter 2: ALGEBRAIC CLOSURES Chapter 3: THE PROJECTIVE PLANE Chapter 4: MULTIPLICITIES AND DEGREE Chapter 5: BÉZOUT’S THEOREM Chapter 6: TRANSITION TO ELLIPTIC CURVES Chapter 7: ABELIAN GROUPS Chapter 8: NONSINGULAR CUBIC EQUATIONS Chapter 9: SINGULAR CUBICS Chapter 10: ELLIPTIC CURVES OVER Q Chapter 11: BUILDING FUNCTIONS Chapter 12: ANALYTIC CONTINUATION Chapter 13: L-FUNCTIONS Chapter 14: SURPRISING PROPERTIES OF L-FUNCTIONS Chapter 15: THE CONJECTURE OF BIRCH AND SWINNERTON-DYER EPILOGUE Bibliography Index
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 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 -- 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 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 Do We Go from Here? -- Bibliography -- Index -- A -- B -- C -- D -- E -- F -- G -- H -- I -- J -- K -- L -- M -- N -- O -- P -- R -- S -- T -- V -- W -- Z
Chapter 5. Bézout’s Theorem
Chapter 13. L-functions
Index
Chapter 4. Multiplicities and Degree
Chapter 14. Surprising Properties of L-functions
PART III. ELLIPTIC CURVES AND ANALYSIS --
Chapter 11. Building Functions
Chapter 6. Transition to Elliptic Curves
Prologue
Acknowledgments
Chapter 10. Elliptic Curves over Q
Preface
Chapter 2. Algebraic Closures
PART II. ELLIPTIC CURVES AND ALGEBRA --
Chapter 8. Nonsingular Cubic Equations
Chapter 7. Abelian Groups
Chapter 12. Analytic Continuation
PART I. DEGREE --
Epilogue
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Chapter 1. Degree of a Curve
Chapter 9. Singular Cubics
Contents
Chapter 15. The Conjecture of Birch and Swinnerton-Dyer
Frontmatter --
Chapter 3. The Projective Plane
Bibliography