Curves, Counting, and Number Theory
Elsewhere $38.92 $28.49 Save $10.43 (27%)
Price includes NZ wide delivery!
Ships from UK supplier
|Format:||Hardback, 280 pages|
|Other Information: ||52 line illus. 16 tables.|
|Published In: ||United States, 21 February 2012|
"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 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. Along the way, they give an informative and entertaining introduction to some of the most profound discoveries of the last three centuries in algebraic geometry, abstract algebra, and number theory. They demonstrate how mathematics grows more abstract to tackle ever more challenging problems, and how each new generation of mathematicians builds on the accomplishments of those who preceded them. Ash and Gross fully explain how the Birch and Swinnerton-Dyer Conjecture sheds light on the number theory of elliptic curves, and how it provides a beautiful and startling connection between two very different objects arising from an elliptic curve, one based on calculus, the other on algebra.
About the Author
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).
Table of 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 Zeta-Function 169 4. Functional Equations 171 5. Euler Products 174 6. Build Your Own Zeta-Function 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. L-functions 199 1. A Fertile Idea 199 2. The Hasse-Weil Zeta-Function 200 3. The L-Function of a Curve 205 4. The L-Function of an Elliptic Curve 207 5. Other L-Functions 212 Chapter 14. Surprising Properties of L-functions 215 1. Compare and Contrast 215 2. Analytic Continuation 220 3. Functional Equation 221 Chapter 15. The Conjecture of Birch and Swinnerton-Dyer 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
The authors present their discussion in an informal, sometimes playful manner and with detail that will appeal to an audience with a basic understanding of calculus. This book will captivate math enthusiasts as well as readers curious about an intriguing and still unanswered question. -- Margaret Dominy Library Journal Minimal prerequisites and its clear writing make this book (which even has a few exercises) a great choice for a seminar for mathematics majors, who at some point should have such an excursion to one of the frontiers of mathematics. Mathematics Magazine The authors of Elliptic Tales do a superb job in demonstrating the approach that mathematicians take when they confront unsolved problems involving elliptic curves. -- Sungkon Chang Times Higher Education One cannot help being impressed, in reading the book and pursuing a few of the references, by the magnitude of the enterprise it chronicles. -- James Case SIAM News Ash and Gross thoroughly explain the statement and significance of the linchpin Birch and Swinnerton-Dyer conjection... [A]sh and Gross deliver ample and current intellectual and technical substance. Choice I would envision this book as an excellent text for an undergraduate 'capstone' course in mathematics; the book lends itself to independent reading, but topics may be explored in much greater depth and rigor in the classroom. Additionally, the book indeed brings together ideas from calculus, complex variables and algebra, showing how a single mathematical research question may require an integrated understanding of the various branches of mathematics. Thus, it encourages students to reinforce their understanding of these various fields, while simultaneously introducing them to an open question in mathematics and a vibrant field of study. -- Lisa A. Berger Mathematical Reviews Clippings
|Publisher: ||Princeton University Press|
|Dimensions: ||22.0 x 15.0 x 2.0 centimeters (0.54 kg)|