Modular forms and Fermat's last theorem /

The statement known generally as "Fermat's last theorem" (but more accurately described as Fermat's conjecture) stood as mathematics' most celebrated challenge until its proof in 1994 by Andrew Wiles (assisted by Richard Taylor), building on previous work of Shimura, Langlands, Tunnell, Frey, Serre, Ribet, Mazur, and others. Intense interest surrounding this event spawned a flurry of publication, tempered only by the ferocious complexity of the ideas involved. Popular but shallow treatments have come from A. Aczel and S. Singh (but avoid M. Vos Savant's wrongheaded and worthless diatribe!); however, A.J. Van Der Poorten's semitechnical account well suits undergraduates. The serious reader pursuing the necessary background for penetrating Wiles's work absolutely must turn to the volume reviewed here edited by Cornell, Silverman, and Stevens (based on a 1995 Boston University conference), with contributions by many of the best respected number theorists of our day. Though one should expect no comparable volume anytime soon, libraries should also acquire Current Developments in Mathematics for 1995 for the worthy 154-page article by H. Darmon, F. Diamond, and R. Taylor, and also the premature publications (assembled prior to full access to Wiles's manuscript) Seminar on Fermat's Last Theorem: 1993-1994, ed. by V. Kumar Murty (1995), and Elliptic Curves, Modular Forms, & Fermat's Last Theorem, ed. by John Coates and S.T. Yau (CH, Feb'96). Although most undergraduates can only hope to admire from afar the details in these books, the great importance and even glamour of the work therein make these volumes essential for college libraries. Elliptic curves in various guises have received close scrutiny from mathematicians for more than 200 years, but the vital if auxiliary role they play in Wiles's proof of Fermat's Last Theorem only intensifies their centrality in modern mathematics. Worthy expositions abound, especially those of Cassels, Husemuller, Knapp, Koblitz, Silverman, and Tate-Silverman. A shade more elementary than most of these, undergraduates will find that McKean and Moll's book also offers more diverse viewpoints, describing connections to dynamical systems, solitons, elliptic integrals, and theta functions, as well as Diophantine problems. Chapter 5, "Ikosaeder and the Quintic," by itself deserves comparison with two recent books, Geometry of the Quintic, by J. Shurman (CH, Oct'97) and Beyond the Quartic Equation, by R.B. King (1996). Chapter 6 describes the almost magical properties of the j-function as it relates to extensions of imaginary quadratic fields, a theory with famous elementary consequences but rarely exposited. Highly recommended. Mollin advertises his book as structured for undergraduate number theory courses at any level. To this end he places two optional sections at the end of each chapter, one pitched to computer science students, usually emphasizing cryptography, and another to upper-division undergraduate students concerning the algebraic theory of quadratic fields. A look in chapter 5 at his treatment of continued fractions illustrates the difficulty of serving upper- and lower-division undergraduates simultaneously, as well as some of the book's other weaknesses. The lower-division student finds very formal prose, few applications, proofs that depend on exercises, and no diagrams such as those that enliven, say, the classic by C. Olds (Continued Fractions, 1963); the upper-division students will not see how linear algebra clarifies the basic continued fraction formalism. Moreover, "mandatory" material in this chapter concerning periodic continued fractions depends on the difficult definition of discriminant tucked in an optional section of chapter 2. Finally, the proof here of Lagrange's key theorem on the periodicity of continued fraction expansions of quadratic irrationalities seems to stop short of the desired conclusion. The book's strengths lie in its currency, its many worked examples, the historical footnotes, and the references to the literature. D. V. Feldman University of New Hampshire