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Zusatztext "A carefully thought out presentation." ---Danny Yee, Danny Reviews Informationen zum Autor Avner Ash is professor of mathematics at Boston College. Robert Gross is associate professor of mathematics at Boston College. They are the coauthors of Fearless Symmetry: Exposing the Hidden Patterns of Numbers (Princeton). Klappentext "Assuming only what every mathematically inclined freshman should know, this book leads the reader to an understanding of one of the most important conjectures in current number theory--whose proof is one of the Clay Mathematics Institute's million-dollar prize problems. The book is carefully and clearly written, and can be recommended without hesitation." --Peter Swinnerton-Dyer, University of Cambridge "The Birch and Swinnerton-Dyer Conjecture is one of the great insights in number theory from the twentieth century, and Ash and Gross write with care and a clear love of the subject. Elliptic Tales will have wide appeal." --Peter Sarnak, Princeton University Zusammenfassung A look at one of the most exciting unsolved problems in mathematics today 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. 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 deep—and often very mystifying—mathematical ideas. Using only basic algebra and calculus while presenting numerous eye-opening examples, Ash and Gross make these ideas accessible to general readers, and, in the process, venture to the very frontiers of modern mathematics. Inhaltsverzeichnis 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 Multip...
