Invitation to the Mathematics of Fermat-Wiles [Hardcover]

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.

This is an exciting book, but beware!

Hellegouarch claims to major in giving examples rather than proving "the basic structure theorems" in this book, and, this he does very well. The examples are beautiful. The proofs that he does offer are unusually elegant and instructive. This book is perfect for the serious student of mathematics who has had the usual undergraduate course covering things like the theory of rings and ideals, Galois theory, complex analysis, that sort of thing.

What really makes this book so perfect are the holes in the proofs. They are just exactly the right size to fill in with enough difficulty to strengthen your muscles but not to break your back. This book would be absolutely fantastic for a first or second year graduate course that used the Texas method to introduce the students to arithmetic geometry or albegraic number theory or modular forms, or any of a number of other sub-fields of mathematics.

A very strange thing, though, is that some of the holes in the proofs that are labelled "exercises" are trivial to fill in when compared with the real holes. I must qualify this statement by mentioning that I have only worked through the first 17 pages in detail. Although, I have read the whole book several times without paying too much attention to detail.

For example, on page 13, there is the statement, "since x and y are odd, p**2 + 3*q*2 must be odd." (x,y,p,q, are all integers, x=p+q, y=p-q, and GCD(x,y) = (x,y) = 1.) Now, using elementary number-theory odd-even type arguments, this is not obvious. However, computing modulo 2 makes it easy: p**2 + 3*q*2 == p+q = x == 1 modulo 2. Note also that the result does not depend on "y" being odd, as Hellegouarch's statement would have you believe. Figuring out mis-leading statements like this are a great way to prepare a young graduate student to become a research mathematician. In real research problems, you are not usually told which theorems to invoke to prove your results.

Two sentences later, he mentions that (p,q) = 1, which again requires a little thought. In the next sentence he applies a result (Corollary 1.6.1) proved for the Gaussian integers on the previous page, but, he claims that it is a Z[squareroot(-3)] form of this result that is really being used. Not so. It is the Gaussian integer [Z(i)] form that is being used. Furthermore, in applying Corollary 1.6.1, he uses not only the Corollary but side results that appear in the proof of the Corollary. Furthermore, he applies the Corollary in the highly special case when b=0 but doesn't tell you this.

For a professional PhD mathematician (like myself) figuring all this out was great fun, but, then, to further confuse the issue, when Hellegouarch gets to the bottom of the proof, he claims that the filling in of the final details are left to the reader as an "exercise." But, the final deatils are not an "exercise," they are immediately obvious, especially for the reader who has jumped the hurdles required to get to the end.

Another example of a hole which is a great exercise is the statement on the bottom of page 15 that if p is prime over the integers and reducible over the Gaussian integers, then the reduction is essentially unique. In other words, p can be written as x*y over the Gaussian integers where neither x nor y are units in essentially one and only one way. BTW, x is the conjugate of y in such a reduction. The proof follows easily by applying the norm function N(a+bi) = a**2 + b**2, but he doesn't tell you this. He doesn't even tell you that this is a hole that needs to be filled in. Noticing holes like this one are a great way for a young mathematician to be prepared for a career in research mathematics. Sometimes, such holes are not just little annoyances, but, real holes, and part of the work of a research mathematician is being able to find them. This was the case for Wiles first proof (1993) of Fermat's Last Theorem. It had a "real" whole and it tood a "real mathematician" to find it.

All this makes for great fun, but beware!

In working through most of the first 17 pages with a fine-toothed comb, I was struck by the lack of typos. I don't remember seeing any. Certainly not any that forced me to run a computation to decide whether or not it was a typo. But then, on page 18, I found the following statement,

"Euler introduced the ring Z[j] where j = exp(2pi*i/3) is a primitive root of unity, in order to study the Fermat equationn of degree 3; he accepted the fact that the fundamental theorem of arithmetic extends to Z[j] (fortunately for him this is actually the case, although it is not for the ring Z[squareroot(-3)])."

This statement sure looks false as it stands. There are various possibile explanation for it, but, the most likely is that "-3" is a typo and it should be "-5" or any other negative integer except for the nine integers that constitute H. M. Stark's 1967 solution to the "Gaussian number problem."

(The 9 values of "D" for which Q[sqr(D)] and hence Z[sqr(D)] are UFDs are -1, -2, -3, -7, -11, -19, -43, -67, and -163. Being a UFD is the usual interpertation of the phrase "the fundamental theorem of arithmetic extends to Z[sqr(D)]." Reference: Stewart and Tall (S&T) "ANT," 3rd edition, 2002, page 86. Although S&T do not call it "the Gaussian number problem, many other books do.)

Naturally, it would be nice if there were an errata sheet for a book like this that neither gives definitions of many of its terms nor gives proofs of its basic structure theorems (from which the definitions could be deduced). However, I could not find such a list on the web. John G. Aiken, PhD in 1972 in C* and W* algebras.