Neal Koblitz
Publisher: Springer Verlag (2004)
Details: 268 pages, Hardcover
Edition: 2 Series: Graduate Texts in Mathematics 97
Price: $64.95
ISBN: 0387979662
Category: Monograph
Topics: Number Theory, Modular Forms, Elliptic Curves, Arithmetic Algebraic Geometry
[Reviewed by Allen Stenger, on 09/15/2008]
This book takes the "complex variables" view of elliptic curves. It uses a particular number-theoretic problem to drive the discussion: the problem of characterizing congruent numbers. A congruent number (no relation to congruences modulo a number) is defined as a number that is the area of a right triangle with all sides rational. Thus 6 is a congruent number because it is the area of the familiar 3-4-5 right triangle. There is an equivalent formulation in terms of whether the elliptic curve y2 = x3 - n2x has rational solutions. This enables us to bring the powerful machinery of elliptic curves to bear on the problem, and the book develops that machinery and culminates with a proof of Tunnell's (almost complete) characterization of congruent numbers.
I like the method of using a single difficult program to organize a book. I think it is not completely successful here, because the original problem drops out of view in the middle of the book, with many new concepts being introduced that are not clearly driven by it. The book travels though L and zeta funtions, elliptic functions, and modular functions and forms.
Silverman and Tate's Rational Points on Elliptic Curves is a very different approach to elliptic curves, through abstract algebra and geometry. There is surprisingly little overlap between the two books, considering that they are introductions to the same subject. Koblitz is much faster-paced, and contains a lot of intricate arguments. It covers a much larger amount of material and requires more mathematical maturity (it is correctly placed in Springer's Graduate Texts series, while Silverman and Tate is in the Undergraduate Texts series). I like both books, but I think Silverman and Tate is a better introduction.
Allen Stenger is a math hobbyist, library propagandist, and retired computer programmer. He volunteers in his spare time at MathNerds.com, a math help site that fosters inquiry learning. His mathematical interests are number theory and classical analysis.