Back to Fermat's Last Theorem: Conference Report

There are several points to take note of with respect to Wiles proof of Fermat's Last Theorem:

(A) The doubts that arose with respect to his proof that were raised at the time of his initial announcement on June 23, 1993 have definitively been laid to rest. For the resolution of several critical problems he was indebted to the assistance of Mr. Richard Taylor who can be reached through the mathematics department of Cambridge University in England. The proof is therefore properly known as the Wiles-Taylor proof, although it is understood that Wiles did most of the work while Taylor filled in the gaps.

It is important to know this, because one still hears of some 19-year old prodigy who found some mistake, or of some lingering doubts, or some abiding suspicion that a complete proof might be impossible by some application of Gödel's Theorem about undecidable propositions, and the like. Wiles proved it all right and if the guardians of the Wolfskehl Prize in Gottingen abide by their promises, he should be receiving something like 7,000 Deutschmarks 2 years from now.

(B) His proof does not rule out the possibility of a proof at some point by the methods of classical number theory, those in use before 1984. This, by the way , is not just a matter of advanced high school algebra , but already, even before Wiles, included all of Complex Variable, Abstract Algebra, Fourier Analysis, Graph Theory, and so on.

In particular it was asserted by several people at the conference that it is very likely that a proof of the First Case of Fermat's Theorem will be accomplished through classical methods. The distinction between the First and the Second Case runs through the whole history of the Fermat Theorem starting with Sophie Germain in 1823:

Consider the 4 variables x,y,z,p appearing in the Fermat Equation x**p + y**p = z**p . [Ed: ** is exponentiation, eg. a**2 is a squared.] It is very quickly shown that x, y and z can be taken to be relatively prime to one another, ( no common factors), and that p can be assumed to be a prime number. Then:

(I) If p does not divide x or y or z, this is known as the first case.

(II) If p does divide any one of these variables, we call this the second case.

Virtually all of the surprising discoveries about general classes of prime exponents that satisfy Fermat's Theorem that were made up through the 1980's , dealt with the first case only. Almost nothing was known about the second case, save for instances of particular primes . So much indeed was known about the first case , that it is very reasonable to expect that a little more time and work will lead to a classical first case solution .

(C) With one exception, Wiles proof makes no use whatsoever of the partial results on Fermat's Last Theorem discovered by hundreds of mathematicians since 1637 ! It is as if someone had lost his keys and spent many years looking for them in his living-room, until one day someone ran at his doorbell to tell him that they had been found lying around outside. We are speaking here of the efforts of just about every famous mathematician of the 18th and 19th centuries. ( No evidence has come down to us that Isaac Newton ever bothered with the problem, but it is not at all far-fetched to surmise that he must have glanced at it in his leisure time. )

That exception is in the work of the two German mathematicians, Kummer and Dirichlet, in the 19th century . It is instructive to learn that Kummer was led to an intensive study of Fermat's Last Theorem through his being shown, by Dirichlet, that his earlier proof of the theorem was erroneous. The kind of error was itself prescient. He had thought that all prime numbers could be uniquely factored into pairs of complex algebraic numbers, but this was incorrect. Many of the mistaken proofs of Fermat's Last Theorem have hinged on this fact.

Kummer re-examined the situation and , after many years of work, produced the theory of rings, ideals, and classes that is one of the cornerstones of modern or abstract algebra. By applying this to the theory of prime numbers, he was able to show that Fermat's Last Theorem is satisfied by all regular primes. The definition of a regular prime is much too technical for an article of this sort, but although it appears that regular primes are much more frequent than irregular primes, it is still not known if their number be finite or infinite. It is however known that the number of irregular primes is infinite, another one of those odd facts of mathematics that you can bring up at parties or other gatherings.

The theory of rings, ideals, classes and what are called analytic class numbers, is very much at the foundation of Wiles proof. Once again the distinction between regular and irregular primes does not appear. Likewise the historically very important distinction between the first and the second case doesn't even come up in Wiles proof.

Shall we say that this huge labor of the centuries, enough to fill skyscrapers with treatises, journals, notebooks, self-published pamphlets, transactions, communications, letters, etc., was all a colossal waste of time? Naturally not. Permanence is not a virtue associated with scientific work. Scientists are rightly praised for the intelligence with which they developed the resources at their command, not for the ultimate rightness or eternal worth of their conclusions.

(D) At the same time, the powerful methods of algebraic geometry that were brought to bear on the proof of Fermat's Last Theorem seem, at the present moment, to be of little use in the resolution of many, if not most, of the other unsolved conjectures of the theory of numbers: the Goldbach conjecture, the twin primes problem, the aliquot parts problem, the number of primes of the form 1 + a**2 , and dozens of others. This situation must certainly change, but for the moment the Wiles proof appears to be a one-shot deal. Even as I speak this may turn out to be false, although , being at the Fermat Conference, I am relaying in some sense the latest information available on this matter.

The combination of (C) and (D) contrive to perpetuate the reputation of Fermat's Last Theorem as one of the greatest anomalies in the history of science. It may be that we will someday find some very deep reasons connecting all of these scattered facts, or perhaps we will continue forever to discover a host of disconnected truths established only by very prolonged and difficult work by the most advanced professionals in the field.

Certainly one cannot doubt that Andrew J. Wiles invested an incredible amount of sustained effort in his proof. One does not have to be schooled in contemporary algebraic geometry to understand the gist of his remarks in the introduction to his paper in the Annals of Mathematics.

Summarizing with commentary ( Annals of Math, May 1995, pgs. 443-454):

" An elliptic curve is said to be modular if it has a finite covering by a modular curve of the form X0(N)"

Without knowing what this means, we recognize that it is relating 2 different families of curves ( think of simple curves from analytic geometry, "drawn" on surfaces generalized in various ways) : elliptic curves and modular curves. Elliptic curves are just polynomials in two variables x, y, with x up to the 3rd degree, and y to the 2nd. Modular curves are more difficult to define. He goes on:

" A well known conjecture which grew out of the work of Shimura and Taniyama in the 1950's and 1960's asserts that every elliptic curve over Q is modular."

Q is the set of all rational numbers, that is to say, fractions.⁽¹⁾ What this conjecture states is that the elliptic curves are a subset of the modular curves.

" In 1985, [Gerhart] Frey made the remarkable observation that this conjecture should imply Fermat's Last Theorem. The precise mechanism relating the two was formulated by Serre as the e - conjecture [Ed. Note: MS-Word version of article uses epsilon, not e, for epsilon-conjecture.] and this was then proved by Ribet."

We need not worry here about the e- conjecture or its proof. In fact, Ribet proved something more, as Wiles goes onto say:

" Ribet's result only requires one to prove the conjecture for semistable elliptic curves in order to deduce Fermat's Last Theorem."

There are, in fact, 3 classes of elliptic curves: stable, semistable and unstable. The distinction has to do with the singularities ( loosely speaking, places where the equation breaks down) that arise from a process of 'reducing' the curve in a certain fashion.

Notice how the discussion is exclusively in terms of families of curves over various kinds of algebraic domains, with few references to number theory. On the next page, Wiles explains how he has brought together two widely separated branches of number theory through algebraic geometry:

" The key development in the proof is a new and surprising link between two strong but distinct traditions in number theory, the relationship between Galois representations and modular forms on the one hand and on the interpretation of special values of L-functions on the other. The former tradition is of course more recent...."

" The second tradition goes back to the famous analytic class number formula of Dirichlet, but owes its modern revival to the conjecture of Birch and Swinnerton-Dyer. In practice however, it is the ideas of Iwasawa in this field on which we attempt to draw.... "

In other words, he does admit some indebtedness to Dirichlet, but immediately explains that it is really their modern interpretation and development by Swinnerton, Dyer and Iwasawa that were useful to him. Wiles had no need of opening any textbook or studying any paper in number theory before the 1980's in the construction of his proof.

The next few pages of his introduction then restate what has just been said in the technical language familiar to mathematicians in the field.

On page 449 he gives us a chronology of the proof:

" I began working on these problems in the late summer of 1986 immediately on learning of Ribet's result"

" ..... After several months ....I made the first real breakthrough..."

" In the late 1980's , I translated these ideas into ring-theoretic language....."

" In the fall of 1989, I set Ramakrishna, then a student of mine at Princeton, the task of proving the existence of a deformation theory associated to representations arising from finite flat group schemes over Zp .......the work of Ramakrishna was not completed until 1991...."

"... The turning point in this and indeed in the whole proof came in the spring of 1991.... It was only in reading Section 6 of [ a paper by J. Tilouine] that I learned that it followed from Tate's account of Grothendieck duality theory for complete intersections that these two invariants were equal for such rings."

Eventually there arrives the extended paw-print of the great Grothendieck. To paraphrase the German mathematician Jens Gamst to whom I spoke at a reception given in the Sherman student union of Boston University on the evening of Saturday, August 12th: " All the people that you see here are residents of the house that Grothendieck built. " This reception , by the way, as well as the whole conference was underwritten by the National Science Foundation, the National Security Agency, and a wealthy Texas oilman names James Vaughn who has been fascinated with Fermat's Last Theorem all his life. The manifest presence of military funding would, in itself, have guaranteed that Alexandre Grothendieck himself , even before his withdrawal from society, ( which has now become total) would never have come to the conference.

It is clear, isn't it, that the world uses great minds for what it considers valuable in them, then ignores anything else they have to tell us. One isn't required to share the reasons of a Grothendieck for rejecting the role of the military in modern science, but why don't any of his mathematical disciples at least listen to what he has to say?

Anyway - back to Wiles.

"... The impact of this result on the main problem was enormous...."

"....Then unexpectedly in May 1993, on reading of a construction of twisted forms of modular curves in a paper of Mazur ....I made a crucial and surprising breakthrough...."

" Believing now that the proof was complete, I sketched the whole theory in three lectures in Cambridge, England on June 21-23. However, it became clear to me in the fall of 1993 that the construction of the Euler system...............was incomplete and possibly flawed.."

"... in January 1994, R. Taylor had joined me in the attempt to repair the Euler system argument... spring of 1994.... I suddenly came to a marvelous revelation : I saw in a flash on September 19th, 1994, that de Shalit's theory, if generalised, could be used...............I had unexpectedly found the missing key to m y old abandoned approach...."

" After I communicated the argument to Taylor, we spent the next few days making sure of the details."

He concludes the introduction by giving acknowledgments for the help of Nicholas Katz, Taylor, F. Diamond, Conrad, de Shalit, Faltings, Ribet, Rubin, Skinner and H. Darmon.

Anyone who knows how difficult mathematics research is, how hard it is even to read and thoroughly digest papers written by others, even in the same field, can read between the lines of this account and realize that he is describing a collosal labor of love sustained over 7 years. It does not in the least diminish from the credit due him to state that the picture of him sitting all alone in an attic in Princeton for these long years before emerging with the proof full blown, is an considerable exaggeration ,(of the sort that is permitted , of course, in the formation of a legend) . The solution of Fermat's Last Theorem has been the work of dozens of mathematicians, some famous, others with little celebrity attached to their names, some graduate students like Ramakrishna, a collective effort of the entire community. Wiles has the credit of being the one man who put together everything that the others had done, to forge a sharp instrument customized to the task of solving the most historically celebrated outstanding problem in mathematics.

A few more words about the role of Grothendieck. Alexandre Grothendieck is filled with a kind of righteous indignation at what he believes to be the 'misappropriation' of his ideas by the mathematics community. He charges them with using pieces of his program to solve difficult and prestigious problems for the purpose of aggrandizing themselves, without any understanding of how these pieces fit into the grand picture of his ultimate goal: the unification of all branches of mathematics, geometry, algebra, analysis and logic on the basis of a few simple but very fundamental ideas.

Grothendieck may have a point, but speaking just for myself, it was not until my attendance at this conference that I realized that I am under an obligation, if I wish to continue to relate, ( in whatever capacity), to the mathematics of our time , to study the ideas, language and methods of his "post-modern" mathematics as I have called it. I have put off doing so because I have believed so far that algebraic geometry was an obscure, arcane discipline whose appeal was limited to a few hardy souls who enjoyed talking and living above the heads of the rest of us at all times. Earlier I quoted the observations of René Thom, now, with due respect to him, I have to say that Thom is wrong.

What has convinced me, in fact, is precisely that phenomena against which Grothendieck rails so vehemently: a piece of his grand picture was appropriated and used to prove Fermat's Last Theorem! We all need to be convinced by something: even prophets have recourse to miracles. I have no intention whatsoever of doing research in this area, but the time has come in which we will all have to sit down and really learn what it is all about.

As to the whereabouts and present circumstances of Alexandre Grothendieck: In 1991, he wrote a letter to 300 associates, ( myself among them), in which he prophesied the coming of the Apocalypse, the Millennium and God himself to the world. Around the same time he wrote a 2000 page sequel to an earlier, equally lengthy autobiography ( Recoltes et Semailles) entitled La Clé des Songes. I haven't seen it but have been told that these Messianic ideas are further elaborated in this book.

Not long after that he sent out another letter in which he wrote that he had been seduced by the devil and that we should all disregard his previous communication. Shortly after that he disappeared, and it seemed that no-one knew where he was.

During the conference I ran into another friend of his, the Roumanian, Nicholas Raduliscu. He said that Alexandre's whereabouts are in fact known to a very small circle of friends, and that he is living somewhere in Arriège. He said that he helped Alexandre move from his former home in the Vaucluse, in 1993, and that his mental state then was deplorable, a mixture of suspicions, wild accusations, religious pontification, hallucinations and visions . This is my latest news as to the fate of the man who has "built the house all mathematicians ( do, or shortly will) live in."

From Pierre de Fermat to Alexandre Grothendieck to Andrew J. Wiles to Malvina Baica. What is mathematics? What is the mind? What is truth?

Unlike Pontius Pilate, we ought to at least wait around for the answer.

For the following description of the underlying ideas of Wiles' proof, I am indebted to discussions with conference speakers, Michael Rosen , Gregory Call, Rene Schoof, Jaap Top, and with participants Corneliu Hoffman, Jeffrey Hooper, Lisa Fastenberg, Jens Gamst, Zasen Chen and others. Most valuable were the survey lectures given by Glenn Stevens, Joe Silverman, David Gross and Barry Mazur. I also benefited from a conversation with Arkady Berenstein, a mathematician and friend who teaches at Northeastern University.

The critical event, what one might call the watershed, which has led more or less directly to the proof of Fermat's Last Theorem, was the proof in 1984, by Faltings , of the Mordell-Weil conjecture for elliptic curves. It is not difficult to explain this result to a mathematically-minded audience. An elliptic curve is given by an equation of the form:

E: y**2 + a1 xy + a3 y = x**3 + a2x**2 + a4 + a6 , where the coefficients may be real numbers, complex numbers, or elements of any field or ring.

It is easy to show that the points on an elliptic curve form a natural geometric group: if one draws a line through the curve, it will intersect it in 3 points. If x1, x2 and x3 are the intersection points then we can define our group operation (+) by letting x1 (+) x2 (+) x3 = 0.

Observe that elliptic curves are of the 3rd degree, while the Fermat Equation in general is of the nth degree, where we may fix n at any integer >2. It turns out that there is an ingenious, but simple way, to "encode" the triple of integers A = u**n , B = v**n , C = w**n , into an elliptic curve, so that information about the group of the curve gives information about the equation A +B = C

The Mordell- Weil conjecture states that the rational points on every elliptic curve form an Abelian group that is finitely generated. The rational points are those locations on the curve (x,y) at which both x and y are fractions. From the proof of the Mordell -Weil conjecture, it is a simple matter to prove the following:

For any fixed n>2 the number of (relatively prime) solutions of the equation u**n +v**n - w**n = 0 , is finite.

Thus, by 1984, we were almost home. The trick, then, was to replace the statement "is finite", by the statement " is equal to zero"!

In 1985, Gerhart Frey observed that it was possible to combine Faltings result with the Modularity Conjecture of Taniyama, Shimura and Weil, to produce a proof of Fermat's Last Theorem. This conjecture has already been discussed above, and one need only add that modular curves are derived from modular forms, which can be understood as generalizations of the "fractional-linear transformations" that underlie the projective geometry of the upper half plane. The transformations are 2x2 matrices with integral coefficients, with various conditions on the entries.

A remarkable breakthrough occur with the proof of the e -conjecture of Jean-Pierre Serre by Ken Ribet in 1986. What this showed was, that for the purposes of proving Fermat's Last Theorem, one did not have to consider all elliptic curves. One only had to examine the special case of a single family of curves :

E A,B,C : y**2 = x(x-A)(x-B) = x**3 -Cx**2 + AB, where A, B and C are defined as above . When A+B = C the curve one derives that has been described as " possessing properties that are too good to be true." Without going into the details, this meant in particular that it would have a "Weight 2 Hecke eigenform" that could not possibly exist, for very technical reasons that are described in the last paragraph of the outline of Glenn Stevens' talk.

If you want to see this last paragraph, you should head over to Boston University, where you may still find several participants wearing a T-shirt made especially for this conference, holding this paragraph on the front side, and a complete list of references on the back. The T-shirt is guaranteed to stop any conversation dead in its tracks , and the only reason I didn't buy it was because I already know how to do that without benefit of T-shirt.

By proving the Modularity conjecture for semi-stable elliptic curves, he completed the proof of Fermat's Last Theorem. The proof of this conjecture was a major advance in mathematics in its own right. As Arkady Berenstein told me, " It is amazing that all this algebraic geometry developed in such a way that it could all come together at around the same time to give a proof of Fermat's theorem."

Coincidence? The Unseen Hand? Divine Intervention? Deep Structures? The Collective Unconscious? Who is to say? How important is the question of the importance of a question?

I would like to end this report with a few thoughts of my own on the subject of the likelihood that Pierre de Fermat really did have a proof in 1637 when he wrote his enigmatic marginal notes. Number Theory as we know it was created by Fermat. One must go back 1300 years to Diophantus of Alexandria to find comparable work in this field. This is not to discredit the discoveries of Middle Eastern, Indian and Chinese mathematicians from the 6th to the 12th century. They were not primarily interested in developing number theory as a branch of mathematics, and much of the research relates to solutions of equations that arise in the course of making astronomical observations.( In the same way, special techniques similar to those used in the calculus were used by many people, including Archimedes, for the solution of specific problems, but Isaac Newton is rightly deemed the inventor of the calculus. )

Because Fermat was entering into a domain that was relatively new and unexplored, he had no way of knowing how difficult some of its challenges might be. No-one had ever encountered a problem that could take 300 years to solve; one can suggest that the scientists of that time didn't believe that such problems existed. If you take someone who has been climbing tall hills all of his life, and you bring him to the base of, say, Mt. McKinley, he might well say, " It's a little bigger, a little harder, but I can do it." This is probably what happened with Pierre de Fermat. He probably spent several hours on it, found that it did not yield to standard methods, many of which he had himself invented, and put it aside. " A little more work", he must have said, "when I find the time." Then he forgot about it.

For this is the ultimate anomaly of Fermat's Last Theorem: is the result of any value at all in and of itself? Carl Friedrich Gauss, who is perhaps the greatest 19th century mathematician, but about whom it can be arguably maintained that his reputation is inflated, but of whom it is generally felt that he did a few good things in his life that, well, might prove to have enduring value when the chips are down, though there is a school of thought who feel that he wasted much of his time because....

Anyway, C.F. Gauss refused to spend any real time on Fermat's Last Theorem because to him it had no intrinsic worth. This characterizes the attitude of many mathematicians who picked it up, proved a few things about it, found it very difficult and thought, "Well, so what?" before going onto something else.

Indeed one can surmise that it was only among the amateurs, the cranks, and the deluded that one found people who devoted their whole career to the proof of Fermat's Last Theorem. Even Wiles was after much bigger game: the proof of the Modularity conjecture. And herein, lies the most challenging question of them all:

What is mathematics, that we should be mindful of her ?

Roy Lisker
August 15, 1995
150 Kisor Rd
Highland, NY 12528

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