Invitation to the Mathematics of Fermat-Wiles

Rating:
Summary: An excellent introduction
Review: Modulo some sections that require more mathematical maturity, this book gives a straightforward introduction to the mathematics behind Fermat's Last Theorem that is accessible to the first or second year graduate student in mathematics. This is due not only to the excellence of the presentation, but also the many problems at the end of each chapter, making this book qualify more as a textbook than a monograph. Its perusal will give the reader an appreciation of the role of elliptic curves in the proof of Fermat's Last Theorem. Readers familiar with the applications of elliptic curves will find another impressive one in this context. It is a sizeable book filled with many definitions and theorems, so only a few features that make the book stand out will be mentioned.

The first of these is the chapter on elliptic curves, which the author keeps at a level that does not presuppose a heavy background in algebraic geometry. Instead, he develops them using an approach that one might find in elementary analytic or projective geometry. Mathematical rigor however is not sacrificed, and the author does not hesitate to use diagrams when appropriate. Readers therefore will find the presentation fairly easy to follow, and will not be stymied by the complicated constructions that can easily accompany discussions on elliptic curves in the context of Fermat's Last Theorem. The necessary algebra, such as Galois theory, is given in another chapter.

There are two "million-dollar" problems mentioned in this book, such as the Riemann hypothesis and the Birch-Swinnerton-Dyer conjecture. The Riemann hypothesis arises in the discussion of zeta functions for elliptic curves. In this context, the author characterizes the zeta function in a way that makes its role in number theory very transparent, namely in the role it plays for expressing an integer as a product of primes, and the fact that it can be associated with the valuations of non-zero ideals in the integers. Groups that are "simpler" than the integers, such as the p-adic integers, also have zeta functions and similar product representations. The need for zeta functions in the book comes in the context of elliptic curves E over the rational numbers Q. The fields "simpler" than Q are the finite fields F[p] modulo a prime p also result in a representation of the zeta functions as a product, but now the product is taken over the prime ideals of quadratic extensions of the polynomial ring F[p;X] generated by an elliptic curve over F[p]. By quoting, but not proving the Artin representation of the zeta function for E, the author uses this to motivate the `L-function' for E. The Birch-Swinnerton-Dyer conjecture comes in when considering the Mordell-Weil group of E, and asserts that the rank of this group is equal to the order of the zero of the L-function at 1.

In the very last section of the book, the author discusses some new areas and concepts in mathematics that were generated by the solution of Fermat's last theorem. One of these concerns a new definition of the ring of p-adic integers, and arises when considering the reduction of an elliptic curve modulo a prime number. For p = 3 or 5, showing that the impossibility of the case of Fermat's theorem for these values of the exponent must be done by the considering, not the congruence modulo p, but the congruence module p^2. The same holds for p = 7, where no h-th power of p will give the result modulo p^h. The author therefore considers infinite powers of p, which brings in the notion of a `projective limit.' Infinite products of the integers modulo prime powers, taken with the Tychonoff topology, gives a local ring on which one can define a p-adic valuation. The author then considers the fraction field of this ring, which is locally compact under the p-adic distance, is the completion of the rational numbers under the p-adic distance, and is isomorphic to the field of p-adic numbers.

The author then generalizes this construction by starting with an elliptic curve E over a field K, and for a prime number not equal to the characteristic L of K, he shows how to construct the `Tate module' T(E;L) of E at L. Taking projective limits in this case shows that T(E; L) is a free Z(L)-module of rank 2. For the Galois group G of the algebraic closure of K, the Tate module is also shown to be a G-module over Z(L). Given a prime number p, the Tate module T(E; L) allows one to do arithmetic just as easily, or just as hard, as one does arithmetic in a finite field F[L], if one views the arithmetic in the context of an elliptic curve over Q (one is thus justified in setting L = p). The elliptic curve and the Tate module allow one to know just how many points are in the reduced elliptic curve E in F[p], this following from an understanding of the representations of the Galois group for a fixed L (these representations are related to each other, and thus serves to make the prime arithmetic more manageable). This line of thought is continued by putting the loxodromic parametrization of elliptic curves into this context, resulting in "Tate curves" E[q] for a p-adic number q. The author ends this section by discussing briefly some conjectures that he feels will be major unsolved problems in the years. One of these, called `Szpiro's Conjecture', postulates that the minimal discriminant of an elliptic curve over Q is bounded by its conductor. The other, called the `abc Conjecture' conjectures that the maximum of the valuations of three relatively prime integers is bounded by the radical of the product of these integers. Consequences of these conjectures are briefly discussed, including an interesting generalization of Fermat's equation.

A very helpful historical summary of the "elliptic curve approach" to Fermat's Last Theorem is given in the appendix.