Fermat's Last Theorem/Andrew Wiles

Andrew Wiles
Andrew Wiles was born in the UK, the 11th of April 1953. From a young age he demonstrated a strong interest in mathematical enigmas and problems. When he was still a small child he loved to go into public libraries in search of books containing problems and enigmas. At the age of 10 years he found in a library the book The Last Problem by Eric Temple Bell. In this book the author described what came to be known as Fermat's Last Theorem from the Greeks up to the discoveries at the end of the 1800s. The young Wiles remained fascinated by the problem. An equation so simple to enunciate had eluded some of the most able mathematicians of the world and the boy then began to fantasise hoping to find the elementary proof that had eluded the others. Wiles hypothesised that Fermat did not have a knowledge in the mathematical field superior to his own and he therefore sought a proof with his limited knowledge. Obviously he was not able to find it, he would pursue Fermat’s Last Theorem for a good part of his life. Wiles became a professional mathematician and had to temporarily abandon his quest since finding a proof was thought to be too difficult for a young mathematician and that he didn’t hold that it could produce interesting mathematics in the immediate future. Wiles’ dissertation advisor was John Coates, and he directed Wiles in the study of elliptic curves. That was Wiles’ good fortune, since Wiles' was able to use his knowledge of elliptic curves to the proof of Fermat’s Last Theorem.

Elliptic curves and modular arithmetic
Specifically Wiles analysed some elliptic curves in modular arithmetic. While classic arithmetic deals with an infinite number of numbers, in modular arithmetic one utilises only a subset of values. Modular arithmetic was also called clock arithmetic, given that this utilised a modular arithmetic with modulo equal to 24. Everyone knows that if it is 18 hundred and one waits for 8 hours one does not find oneself at 26 hundred but at 2 hundred, given that when the clock arrives at 24 it starts to count again from 0. Modular arithmetic is a complete arithmetic like that classic, only that the numbers utilised are limited. Modular arithmetic furthermore has at its disposal some very interesting properties that render it particularly useful in some fields of mathematics. When a mathematician analyses an elliptic curve with modular arithmetic he extracts a series of solutions that are called L-series. Wiles worked much on elliptic curves and on their L-series accumulating experience that would become useful to him in the future.

The Taniyama-Shimura Conjecture
Towards the end of the 1950s the Japanese mathematicians Yutaka Taniyama and Goro Shimura formulated a conjecture known as the Taniyama-Shimura conjecture. This conjecture stated that every L-series of an elliptic curve could be associated with a specific M-series of a modular form. In substance this conjecture asserted that every modular form could be put in bi-unequivocal correspondence with an elliptic curve or, given in other terms, the modular forms and the elliptic curves were the same mathematical object seen from different points of view. This conjecture from the point of view of a mathematician is very important, in fact, if it were proved true, it would have meant that problems of elliptic curves centuries old would be able to be transferred into their modular forms and tackled with new mathematical tools. Obviously the converse would also be valid.

The convention of 1984
In 1984 a event happened that revolutionised the life of Wiles: during a convention in Germany Gerhard Frey demonstrated that whoever proved the Taniyama-Shimura conjecture would also automatically have proved Fermat's last theorem. Frey showed with a series of not too complex steps that in a hypothetical counterexample to Fermat's theorem (that is a valid solution of the equation an+bn=cn) one would be able to write it as an elliptic curve so particular and atypical as not to be able to be associated to any M-series of a modular form. Therefore proving the Taniyama-Shimura conjecture proved that this degenerate equation did not exist and that therefore Fermat's theorem was true. In reality almost everything tied to Fermat's theorem is not as simple as it seems, so much so that the development of a proof that tied Fermat's Last Theorem in an indissoluble manner to the Taniyama-Shimura conjecture caused the mathematicians of half the world to struggle for more than two years, and in fact Frey's initial proof was incomplete.

The secret work
In 1986 Wiles came to know that the link between the Taniyama-Shimura conjecture and Fermat's Last Theorem had been demonstrated. This seemed to Wiles a golden opportunity to use advanced mathematics (seeking to prove the Taniyama-Shimura conjecture) to attain the mathematical ambition of his life (proving Fermat's Last Theorem). Wiles decided to work in absolute secrecy, contrary to the prevailing ethos. While in many disciplines it is common when working on a project to maintain strict secrecy, in order to protect patent rights, in mathematics one follows the opposite approach. Mathematicians talked among themselves constantly, successive comparison is an optimum method for facing problems that seem insoluble and in theoretical mathematics the problem of the industrial secret practically does not exist. Wiles prepared himself carefully, he abandoned all his non-obligatory duties, took an impressive research that he was about to publish, divided it into a good number of articles in such a way as to be able to furnish a constant flux of works while he was working on the conjecture and sought to assimilate as much as possible on modular forms and elliptic curves. Wiles revealed the secret of his work only to his wife. In the first two years of work Wiles made progress utilising the group theory of Galois, although he was still a long way from the solution. The 8th of March 1988 Wiles was stunned to read in the Washington Post that the Japanese mathematician Yoichi Miyaoka had proved Fermat’s conjecture. In reality Miyaoka had not produced a complete proof but utilising differential geometry, a new branch of geometry he had put forward some optimum bases for the definitive solution. Unfortunately for Miyaoka his proof had a gap that although at the beginning did not appear grave, subsequently proved itself to be catastrophic and rendered it impossible to carry out a proof of Fermat’s Last Theorem using that technique. Finally Wiles, utilising a method called Kolyvagin-Flach, succeeded in associating a particular elliptic curve to its modular form. The problem for Wiles was that the method could not be extended to all elliptic curves. Wiles then classified all the elliptic curves into families and modified the method in such a way as to adapt it to single families.

Wiles, in order to utilise the Kolyvagin-Flach method, had to profoundly expand his knowledge of algebraic geometry and not being very expert decided in the end to confide himself to a trusted mathematical expert in the sector. Therefore Wiles contacted Nick Katz, a mathematician in his department. Given that the proof was very complex and voluminous an informal discussion in Katz’s study would not have been sufficient in order to sort out all the doubts, the two decided therefore to disguise their meetings as some post graduate lessons. Wiles would have given the lessons while Katz would have participated among the public. Wiles presented the lessons in a very technical and boring manner in such a way as to eventually discourage students. In fact after a few weeks only Katz remained of the class. Wiles thanks to the help of Katz went over his proof again which effectively seemed correct. Finally Wiles working speedily eliminated the remaining families and finally in May 1993 Wiles finished the proof laying low even the final recalcitrant family.

The Cambridge conference
At the end of June at Cambridge a conference was held on L-functions and their arithmetic. In this context, surrounded by some of the most brilliant mathematicians of the planet Wiles presented his proof in a series of three conferences. The 23rd of June the third and final conference took place and a lengthy applause concluded Wiles’ final conference. Fermat’s Last Theorem had been proved. All the world’s papers spoke of the event and in an afternoon Wiles became the most famous mathematician on the planet.

An error and its resolution
Wiles sent the proof to a journal in such a way that the editor of the journal could subject the proof to verification by a qualified commission. The editor of the journal, given the importance and complexity of the proof, subdivided the sheaf into six parts and entrusted them to as many commissioners. These analysed the manuscript piece by piece and contacted Wiles in order to obtain clarifications on unclear passages and presumed errors. One of the commissioners was Katz, the same mathematician that Wiles had contacted in order to verify the correct application of the Kolyvagin-Flach method. Unfortunately for Wiles, Katz identified an error.

Initially it seemed one of the many errors of little account that stud a complex proof. These errors are comparable to oversights and usually they are corrected in the space of a few hours but that error even if very subtle was also very insidious, in fact Wiles did not succeed in eliminating it. With the passage of time ever more persistent news of the gap spread within the mathematical community and also in the general papers such as the New York Times. Finally Wiles by means of e-mail communicated to the mathematical community that effectively there was a gap in the proof but that he expected to be able sort it out in a few weeks. After months of not being able to do so, and on the advice of a friend he decided to get help from an expert on the Kolyvagin-Flach method and he therefore contacted Richard Taylor. Taylor was one of the reviewers of the proof and was an ex-student of Wiles. They worked on the problem for many months, and towards the end of summer Wiles was demoralised to such a point as to propose to Taylor publicly declaring defeat, but Taylor convinced him to persevere at least until the end of September.

The 19th of September Wiles was analysing the Kolyvagin-Flach method seeking to understand why the method failed when he became aware that, although the method was insufficient in order to obtain a proof, it permitted a method called Iwasawa’s method to function. Iwasawa’s method had initially been utilised by Wiles for the proof but had been abandoned as insufficient. The same method, instead, utilised in conjunction with the Kolyvagin-Flach method, furnished a valid proof.

The proof
The 25th of October Wiles gave two manuscripts to the press, in the first there was the proof of Fermat’s Last Theorem and it carried his signature. The second manuscript specified some properties of some elliptic curves and was signed by Wiles and Taylor. The second manuscript served to prove a fundamental passage of the first manuscript. The publication of the manuscripts put an end to one of the most complex and difficult proofs that mathematics had ever developed.

Wiles and Taylor did not totally prove the Taniyama-Shimura conjecture, in fact the proof for all the cases arrived in 1999 by Christophe Breuil, Brian Conrad, Fred Diamond, and the same Taylor, that starting from the work of Wiles incrementally proved the remaining cases.