Elliptic tales : curves, counting, and number theory
 Author/Creator
 Ash, Avner, 1949
 Language
 English.
 Imprint
 Princeton [N.J.] : Princeton University Press, c2012.
 Physical description
 xvii, 253 p. : ill ; 24 cm.
Access
Available online
 proquest.safaribooksonline.com Safari Books Online

Stacks

Unknown
QA343 .A97 2012

Unknown
QA343 .A97 2012
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Contributors
 Contributor
 Gross, Robert, 1959
Contents/Summary
 Bibliography
 Includes bibliographical references (p. [249]250) and index.
 Contents

 Preface xiii Acknowledgments xix Prologue 1 PART I. DEGREE Chapter 1. Degree of a Curve 13 1. Greek Mathematics 13 2. Degree 14 3. Parametric Equations 20 4. Our Two Definitions of Degree Clash 23 Chapter 2. Algebraic Closures 26 1. Square Roots of Minus One 26 2. Complex Arithmetic 28 3. Rings and Fields 30 4. Complex Numbers and Solving Equations 32 5. Congruences 34 6. Arithmetic Modulo a Prime 38 7. Algebraic Closure 38 Chapter 3. The Projective Plane 42 1. Points at Infinity 42 2. Projective Coordinates on a Line 46 3. Projective Coordinates on a Plane 50 4. Algebraic Curves and Points at Infinity 54 5. Homogenization of Projective Curves 56 6. Coordinate Patches 61 Chapter 4. Multiplicities and Degree 67 1. Curves as Varieties 67 2. Multiplicities 69 3. Intersection Multiplicities 72 4. Calculus for Dummies 76 Chapter 5. B'ezout's Theorem 82 1. A Sketch of the Proof 82 2. An Illuminating Example 88 PART II. ELLIPTIC CURVES AND ALGEBRA Chapter 6. Transition to Elliptic Curves 95 Chapter 7. Abelian Groups 100 1. How Big Is Infinity? 100 2. What Is an Abelian Group? 101 3. Generations 103 4. Torsion 106 5. Pulling Rank 108 Appendix: An Interesting Example of Rank and Torsion 110 Chapter 8. Nonsingular Cubic Equations 116 1. The Group Law 116 2. Transformations 119 3. The Discriminant 121 4. Algebraic Details of the Group Law 122 5. Numerical Examples 125 6. Topology 127 7. Other Important Facts about Elliptic Curves 131 5. Two Numerical Examples 133 Chapter 9. Singular Cubics 135 1. The Singular Point and the Group Law 135 2. The Coordinates of the Singular Point 136 3. Additive Reduction 137 4. Split Multiplicative Reduction 139 5. Nonsplit Multiplicative Reduction 141 6. Counting Points 145 7. Conclusion 146 Appendix A: Changing the Coordinates of the Singular Point 146 Appendix B: Additive Reduction in Detail 147 Appendix C: Split Multiplicative Reduction in Detail 149 Appendix D: Nonsplit Multiplicative Reduction in Detail 150 Chapter 10. Elliptic Curves over Q 152 1. The Basic Structure of the Group 152 2. Torsion Points 153 3. Points of Infinite Order 155 4. Examples 156 PART III. ELLIPTIC CURVES AND ANALYSIS Chapter 11. Building Functions 161 1. Generating Functions 161 2. Dirichlet Series 167 3. The Riemann ZetaFunction 169 4. Functional Equations 171 5. Euler Products 174 6. Build Your Own ZetaFunction 176 Chapter 12. Analytic Continuation 181 1. A Difference that Makes a Difference 181 2. Taylor Made 185 3. Analytic Functions 187 4. Analytic Continuation 192 5. Zeroes, Poles, and the Leading Coefficient 196 Chapter 13. Lfunctions 199 1. A Fertile Idea 199 2. The HasseWeil ZetaFunction 200 3. The LFunction of a Curve 205 4. The LFunction of an Elliptic Curve 207 5. Other LFunctions 212 Chapter 14. Surprising Properties of Lfunctions 215 1. Compare and Contrast 215 2. Analytic Continuation 220 3. Functional Equation 221 Chapter 15. The Conjecture of Birch and SwinnertonDyer 225 1. How Big Is Big? 225 2. Influences of the Rank on the Np's 228 3. How Small Is Zero? 232 4. The BSD Conjecture 236 5. Computational Evidence for BSD 238 6. The Congruent Number Problem 240 Epilogue 245 Retrospect 245 Where DoWe Go from Here? 247 Bibliography 249 Index 251.
 (source: Nielsen Book Data)
 Publisher's Summary
 Elliptic Tales describes the latest developments in number theory by looking at one of the most exciting unsolved problems in contemporary mathematicsthe Birch and SwinnertonDyer Conjecture. 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 may appear simple, but arise from some very deepand often very mystifyingmathematical ideas. Using only basic algebra and calculus while presenting numerous eyeopening examples, Ash and Gross make these ideas accessible to general readers, and, in the process, venture to the very frontiers of modern mathematics.
(source: Nielsen Book Data)
Subjects
Bibliographic information
 Publication date
 2012
 Responsibility
 Avner Ash, Robert Gross.
 ISBN
 9780691151199
 0691151199