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# Elliptic Tales : Curves, Counting, and Number Theory

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## Description

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.show more

## Product details

- Hardback | 280 pages
- 152.4 x 238.76 x 27.94mm | 521.63g
- 24 May 2012
- Princeton University Press
- New Jersey, United States
- English
- 52 line illus. 16 tables.
- 0691151199
- 9780691151199
- 404,444

## Review quote

"This book has many nice aspects. Ash and Gross give a truly stimulating introduction to elliptic curves and the BSD conjecture for undergraduate students. The main achievement is to make a relative easy exposition of these so technical topics."--Jonathan Sanchez-Hernandez, Mathematical Society "[T]his book is a wonderful introduction to what is arguably one of the most important mathematical problems of our time and for that reason alone it deserves to be widely read. Another reason to recommend this book is the opportunity to share in the readily apparent joy the authors have for their subject and the beauty they see in it, not least because . . . joy and beauty are the most important reasons for doing mathematics, irrespective of its dollar value."--Rob Ashmore, Mathematics Today "The book's most important contributions . . . are the sense of discovery, invention, and insight into the habits of mind used by mathematicians on this journey. I would recommend this book to anyone who wants to be challenged mathematically or who wants to experience mathematics as creative and exciting."--Jacqueline Coomes, Mathematics Teacher "The book is very pleasantly written, and in my opinion, the authors have done an admirable job in giving an idea to non-experts what the Birch-Swinnerton Dyer conjecture is about."--Jan-Hendrik Evertse, Zentralblatt MATH "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 "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 "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 "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 "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 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 Journalshow more

## Back cover copy

"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 Universityshow more

## About Avner Ash

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).show more

## Table of contents

Preface xiiiAcknowledgments xixPrologue 1PART I. DEGREEChapter 1. Degree of a Curve 131.Greek Mathematics 132.Degree 143.Parametric Equations 204.Our Two Definitions of Degree Clash 23Chapter 2. Algebraic Closures 261.Square Roots of Minus One 262.Complex Arithmetic 283.Rings and Fields 304.Complex Numbers and Solving Equations 325.Congruences 346.Arithmetic Modulo a Prime 387.Algebraic Closure 38Chapter 3. The Projective Plane 421.Points at Infinity 422.Projective Coordinates on a Line 463.Projective Coordinates on a Plane 504.Algebraic Curves and Points at Infinity 545.Homogenization of Projective Curves 566.Coordinate Patches 61Chapter 4. Multiplicities and Degree 671.Curves as Varieties 672.Multiplicities 693.Intersection Multiplicities 724.Calculus for Dummies 76Chapter 5. B'ezout's Theorem 821.A Sketch of the Proof 822.An Illuminating Example 88PART II. ELLIPTIC CURVES AND ALGEBRAChapter 6. Transition to Elliptic Curves 95Chapter 7. Abelian Groups 1001.How Big Is Infinity? 1002.What Is an Abelian Group? 1013.Generations 1034.Torsion 1065.Pulling Rank 108Appendix: An Interesting Example of Rank and Torsion 110Chapter 8. Nonsingular Cubic Equations 1161.The Group Law 1162.Transformations 1193.The Discriminant 1214.Algebraic Details of the Group Law 1225.Numerical Examples 1256.Topology 1277.Other Important Facts about Elliptic Curves 1315.Two Numerical Examples 133Chapter 9. Singular Cubics 1351.The Singular Point and the Group Law 1352.The Coordinates of the Singular Point 1363.Additive Reduction 1374.Split Multiplicative Reduction 1395.Nonsplit Multiplicative Reduction 1416.Counting Points 1457.Conclusion 146Appendix A: Changing the Coordinates of the Singular Point 146Appendix B: Additive Reduction in Detail 147Appendix C: Split Multiplicative Reduction in Detail 149Appendix D: Nonsplit Multiplicative Reduction in Detail 150Chapter 10. Elliptic Curves over Q 1521.The Basic Structure of the Group 1522.Torsion Points 1533.Points of Infinite Order 1554.Examples 156PART III. ELLIPTIC CURVES AND ANALYSISChapter 11. Building Functions 1611.Generating Functions 1612.Dirichlet Series 1673.The Riemann Zeta-Function 1694.Functional Equations 1715.Euler Products 1746.Build Your Own Zeta-Function 176Chapter 12. Analytic Continuation 1811.A Difference that Makes a Difference 1812.Taylor Made 1853.Analytic Functions 1874.Analytic Continuation 1925.Zeroes, Poles, and the Leading Coefficient 196Chapter 13. L-functions 1991.A Fertile Idea 1992.The Hasse-Weil Zeta-Function 2003.The L-Function of a Curve 2054.The L-Function of an Elliptic Curve 2075.Other L-Functions 212Chapter 14. Surprising Properties of L-functions 2151.Compare and Contrast 2152.Analytic Continuation 2203.Functional Equation 221Chapter 15. The Conjecture of Birch andSwinnerton-Dyer 2251.How Big Is Big? 2252.Influences of the Rank on the Np's 2283.How Small Is Zero? 2324.The BSD Conjecture 2365.Computational Evidence for BSD 2386.The Congruent Number Problem 240Epilogue 245Retrospect 245Where DoWe Go from Here? 247Bibliography 249Index 251show more