Elliptic tales : curves, counting, and number theory /

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

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.