Elliptic curves /

Elliptic curves (not ellipses!), as defined by cubic equations, rank in complexity just above conic sections in the hierarchy of algebraic curves. In many respects this means that elliptic curves live on the narrow margin separating the well understood from the utterly intractable, and what lives on this margin necessarily makes itself the focus of intense research. Any unexpected, salient appearance of elliptic curves in a mathematical or scientific investigation constitutes evidence, prima facie, that somebody has done something right. The book by Husemoller (Max-Planck-Institut fur Mathematik, Germany) has held pride of place as the preferred middlebrow introduction to the subject since its first publication in 1987--and in a sense even before that, seeing as how it grew from unpublished but influential notes for Tate's 1961 Haverford Philips Lectures. (But for readers with a strength in algebraic geometry, J. Silverman's two volumes seem best, and those interested in automorphic forms and the Langlands program should see the book by A. Knapp.) Besides many modest improvements, this new edition documents some recent, unexpected appearances as mentioned above: the role of elliptic curves in the proof of Fermat's last theorem; the rise of elliptic cohomology theory in algebraic topology; the use physicists make of elliptic curves (and algebraic curves more generally) in string theory; the crucial role of elliptic curves in cryptography and computational number theory. These pithy new additions make a valuable book even more valuable; perhaps a future edition could address the role of elliptic curves and their generalizations in connection with integrable systems and solitons. That research continues to make elliptic curves more tractable means, in part, that tools continually emerge that provide for the computational determination of their structure. An algorithm, even in principle, that determines any elliptic curve's rational solutions (the so-called Mordell-Weil group) still remains elusive, but unproved conjectures already predict the shape of one such algorithm. Moreover, existing tools can handle completely certain special elliptic curves and also give important partial information in a wide range of cases. By dint of its currency and focus on computation, the book by Schmitt (Max-Planck-Institut fur Informatik, Germany) and Zimmer (Universitat des Saarlandes, Germany) makes an important contribution to the literature. Students, even those interested chiefly in theory, will find this book illuminating for the range and depth of examples. The elementary character, especially of the initial considerations, make this book especially appropriate for undergraduates. As Schmitt and Zimmer work over arbitrary number fields, not just the rationals, Alaca and Williams (both, Carleton Univ., Canada) provide a well-matched introduction to the fundamentals. Most treatments of algebraic number theory rely on so-called local methods--one completes the fields by taking limits in various ways. Though much larger (uncountable), these "local" fields turn out to have a simpler logical structure that the smaller (countable) "global" number fields they contain. Although experts often find it easier to answer questions about local fields, arguably, local methods constitute a conceptual hurdle for beginners. So Alaca and Williams simply avoid them and push elementary methods as far as they will go. This constitutes their book's special niche. The book's climax treats applications to special elliptic curves, particularly Bachet's equation, which asks, essentially, what numbers occur as the difference between a cube and square. ^BSumming Up: All three books: Recommended. Lower- and upper-division undergraduates. D. V. Feldman University of New Hampshire