Enhanced descriptions from Syndetics:

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.

Table of contents provided by Syndetics

• Preface (p. xiii)
• Acknowledgments (p. xix)
• Prologue (p. 1)
• Part I Degree
• Chapter 1 Degree of a Curve (p. 13)
• 1 Greek Mathematics (p. 13)
• 2 Degree (p. 14)
• 3 Parametric Equations (p. 20)
• 4 Our Two Definitions of Degree Clash (p. 23)
• Chapter 2 Algebraic Closures (p. 26)
• 1 Square Roots of Minus One (p. 26)
• 2 Complex Arithmetic (p. 28)
• 3 Rings and Fields (p. 30)
• 4 Complex Numbers and Solving Equations (p. 32)
• 5 Congruences (p. 34)
• 6 Arithmetic Modulo a Prime (p. 38)
• 7 Algebraic Closure (p. 38)
• Chapter 3 The Projective Plane (p. 42)
• 1 Points at Infinity (p. 42)
• 2 Projective Coordinates on a Line (p. 46)
• 3 Projective Coordinates on a Plane (p. 50)
• 4 Algebraic Curves and Points at Infinity (p. 54)
• 5 Homogenization of Projective Curves (p. 56)
• 6 Coordinate Patches (p. 61)
• Chapter 4 Multiplicities and Degree (p. 67)
• 1 Curves as Varieties (p. 67)
• 2 Multiplicities (p. 69)
• 3 Intersection Multiplicities (p. 72)
• 4 Calculus for Dummies (p. 76)
• Chapter 5 BézoutÆs Theorem (p. 82)
• 1 A Sketch of the Proof (p. 82)
• 2 An Illuminating Example (p. 88)
• Part II Elliptic Curves and Algebra
• Chapter 6 Transition to Elliptic Curves (p. 95)
• Chapter 7 Abelian Groups (p. 100)
• 1 How Big Is Infinity? (p. 100)
• 2 What Is an Abelian Group? (p. 101)
• 3 Generations, (p. 103)
• 4 Torsion (p. 106)
• 5 Pulling Rank (p. 108)
• Appendix: An Interesting Example of Rank and Torsion (p. 110)
• Chapter 8 Nonsingular Cubic Equations (p. 116)
• 1 The Group Law (p. 116)
• 2 Transformations (p. 119)
• 3 The Discriminant (p. 121)
• 4 Algebraic Details of the Group Law (p. 122)
• 5 Numerical Examples (p. 125)
• 6 Topology (p. 127)
• 7 Other Important Facts about Elliptic Curves (p. 131)
• 5 Two Numerical Examples (p. 133)
• Chapter 9 Singular Cubics (p. 135)
• 1 The Singular Point and the Group Law (p. 135)
• 2 The Coordinates of the Singular Point (p. 136)
• 3 Additive Reduction (p. 137)
• 4 Split Multiplicative Reduction (p. 139)
• 5 Nonsplit Multiplicative Reduction (p. 141)
• 6 Counting Points (p. 145)
• 7 Conclusion (p. 146)
• Appendix A Changing the Coordinates of the Singular Point (p. 146)
• Appendix B Additive Reduction in Detail (p. 147)
• Appendix C Split Multiplicative Reduction in Detail (p. 149)
• Appendix D Nonsplit Multiplicative Reduction in Detail (p. 150)
• Chapter 10 Elliptic Curves over Q (p. 152)
• 1 The Basic Structure of the Group (p. 152)
• 2 Torsion Points (p. 153)
• 3 Points of Infinite Order (p. 155)
• 4 Examples (p. 156)
• Part III Elliptic Curves and Analysis
• Chapter 11 Building Functions (p. 161)
• 1 Generating Functions (p. 161)
• 2 Dirichlet Series (p. 167)
• 3 The Riemann Zeta-Function (p. 169)
• 4 Functional Equations (p. 171)
• 5 Euler Products (p. 174)
• 6 Build Your Own Zeta-Function (p. 176)
• Chapter 12 Analytic Continuation (p. 181)
• 1 A Difference that Makes a Difference (p. 181)
• 2 Taylor Made (p. 185)
• 3 Analytic Functions (p. 187)
• 4 Analytic Continuation (p. 192)
• 5 Zeroes, Poles, and the Leading Coefficient (p. 196)
• Chapter 13 L-Functions (p. 199)
• 1 A Fertile Idea (p. 199)
• 2 The Hasse-Weil Zeta-Function (p. 200)
• 3 The L-Function of a Curve (p. 205)
• 4 The L-Function of an Elliptic Curve (p. 207)
• 5 Other L-Functions (p. 212)
• Chapter 14 Surprising Properties of L-Functions (p. 215)
• 1 Compare and Contrast (p. 215)
• 2 Analytic Continuation (p. 220)
• 3 Functional Equation (p. 221)
• Chapter 15 The Conjecture of Birch and Swinnerton-Dyer (p. 225)
• 1 How Big Is Big? (p. 225)
• 2 Influences of the Rank on the N p Æs (p. 228)
• 3 How Small Is Zero? (p. 232)
• 4 The BSD Conjecture (p. 236)
• 5 Computational Evidence for BSD (p. 238)
• 6 The Congruent Number Problem (p. 240)
• Epilogue (p. 245)
• Retrospect (p. 245)
• Where Do We Go from Here? (p. 247)
• Bibliography (p. 249)
• Index (p. 251)

Reviews provided by Syndetics

Library Journal Review

Ash and Gross (mathematics, Boston Coll.; Fearless Symmetry: Exposing the Hidden Patterns of Number) offer readers insight into a venerable problem in number theory known as the Birch and Swinnerton-Dyer (BSD) Conjecture, involving cubic equations in the form of elliptic curves. The proof of the BSD Conjecture is one of the Millennium Prize Problems chosen by the Clay Mathematics Institute in 2000 and whose winners receive $1 million. As of today, six of the original seven problems remain unsolved, including the proof of the BSD Conjecture. Ash and Gross aim to illuminate this particular problem, which concerns the number of solutions to cubic equations. The authors associate their approach to the discussion of elliptic curves with Chaucer's Canterbury Tales, multiple tactics that are loosely connected but make a fascinating whole. Verdict 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, Drexel Univ., Philadelphia (c) Copyright 2012. Library Journals LLC, a wholly owned subsidiary of Media Source, Inc. No redistribution permitted.

CHOICE Review

Diophantine analysis, roughly the business of finding all integral or rational solutions of a given polynomial equation, has its roots in ancient Greece yet remains a frontier of current mathematical research. As the topic of a semipopular book, Diophantine analysis, specifically of elliptic curves (two-variable polynomials of degree 3), might seem at first rather narrow. But intermediate complexity makes elliptic curves the critical case, quite mysterious yet tractable pending the solution of certain precisely formulated and apparently very difficult conjectures. The vast literature on elliptic curves includes monographs of unlimited sophistication, but for undergraduates, J. Silverman and J. Tate's basic Rational Points on Elliptic Curves (1992) offers an excellent portal. Elliptic Tales, though much less demanding, still reaches beyond Silverman and Tate in one crucial respect. Ash and Gross (both, Boston College; coauthors of Fearless Symmetry, CH, Mar'07, 44-3913) thoroughly explain the statement and significance of the linchpin Birch and Swinnerton-Dyer conjecture, one of the seven $1 million Millennium Prize Problems; the Silverman-Tate book does not mention it. Even without the proofs and exercises a textbook would offer, Ash and Gross deliver ample and current intellectual and technical substance. One odd omission: Manin's conditional algorithm deserves a place in this story! Summing Up: Highly recommended. All levels/libraries. D. V. Feldman University of New Hampshire

Author notes provided by Syndetics

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).