Fermat's Last Theorem (35 page)

In theory this new method could extend Wiles's argument from
the first piece of the elliptic equation to all pieces of the elliptic equation, and potentially it could work for every elliptic equation. Professor Kolyvagin had devised an immensely powerful mathematical method, and Matheus Flach had refined it to make it even more potent. Neither of them realised that Wiles intended to incorporate their work into the world's most important proof.

Wiles returned to Princeton, spent several months familiarising himself with his newly discovered technique, and then began the mammoth task of adapting it and implementing it. Soon for a particular elliptic equation he could make the inductive proof work – he could topple all the dominoes. Unfortunately the Kolyvagin–Flach method that worked for one particular elliptic equation did not necessarily work for another elliptic equation. He eventually realised that all the elliptic equations could be classified into various families. Once modified to work on one elliptic equation, the Kolyvagin–Flach method would work for all the other elliptic equations in that family. The challenge was to adapt the Kolyvagin–Flach method to work for each family. Although some families were harder to conquer than others, Wiles was confident that he could work his way through them one by one.

After six years of intense effort Wiles believed that the end was in sight. Week after week he was making progress, proving that newer and bigger families of elliptic curves must be modular. It seemed to be just a question of time before he would mop up the outstanding elliptic equations. During this final stage of the proof, Wiles began to appreciate that his whole proof relied on exploiting a technique which he had only discovered a few months earlier. He began to question whether he was using the Kolyvagin–Flach method in a fully rigorous manner.

‘During that year I worked extremely hard trying to make the Kolyvagin–Flach method work, but it involved a lot of sophisticated machinery that I wasn't really familiar with. There was a lot of hard algebra which required me to learn a lot of new mathematics. Then around early January of 1993 I decided that I needed to confide in someone who was an expert in the kind of geometric techniques I was invoking for this. I wanted to choose very carefully who I told because they would have to keep it confidential. I chose to tell Nick Katz.'

Professor Nick Katz also worked in Princeton University's Mathematics Department and had known Wiles for several years. Despite their closeness Katz was oblivious to what was going on literally just along the corridor. He recalls every detail of the moment Wiles revealed his secret: ‘One day Andrew came up to me at tea and asked me if I could come up to his office – there was something he wanted to talk to me about. I had no idea of what this could be. I went up to his office and he closed the door. He said he thought that he would be able to prove the Taniyama–Shimura conjecture. I was just amazed, flabbergasted – this was fantastic.

‘He explained that there was a big part of the proof that relied on his extension of the work of Flach and Kolyvagin but it was pretty technical. He really felt shaky on this highly technical part of the proof and he wanted to go through it with somebody because he wanted to be sure it was correct. He thought I was the right person to help him check it, but I think there was another reason why he asked me in particular. He was sure that I would keep my mouth shut and not tell other people about the proof.'

After six years in isolation Wiles had let go of his secret. Now it was Katz's job to get to grips with a mountain of spectacular calculations based on the Kolyvagin–Flach method. Virtually everything Wiles had done was revolutionary and Katz gave a great deal of thought as to the best way to examine it thoroughly: ‘What Andrew had to explain was so big and long that it wouldn't have
worked to try and just explain it in his office in informal conversations. For something this big we really needed to have the formal structure of weekly scheduled lectures, otherwise the thing would just degenerate. So, that's why we decided to set up a lecture course.'

They decided that the best strategy would be to announce a series of lectures open to the department's graduate students. Wiles would give the course and Katz would be in the audience. The course would effectively cover the part of the proof that needed checking but the graduate students would have no idea of this. The beauty of disguising the checking of the proof in this way was that it would force Wiles to explain everything step by step, and yet it would not arouse any suspicion within the department. As far as everyone else was concerned this was just another graduate course.

‘So Andrew announced this lecture course called “Calculations on Elliptic Curves”,' recalls Katz with a sly smile, ‘which is a completely innocuous title – it could mean anything. He didn't mention Fermat, he didn't mention Taniyama–Shimura, he just started by diving right into doing technical calculations. There was no way in the world that anyone could have guessed what it was really about. It was done in such a way that unless you knew what this was for, then the calculations would just seem incredibly technical and tedious. And when you don't know what the mathematics is for, it's impossible to follow it. It's pretty hard to follow it even when you do know what it's for. Anyway, one by one the graduate students just drifted away and after a few weeks I was the only person left in the audience.'

Katz sat in the lecture theatre and listened carefully to every step of Wiles's calculation. By the end of it his assessment was that the Kolyvagin–Flach method seemed to be working perfectly. Nobody else in the department realised what had been going on. Nobody
suspected that Wiles was on the verge of claiming the most important prize in mathematics. Their plan had been a success.

Once the lecture series was over Wiles devoted all his efforts to completing the proof. He had successfully applied the Kolyvagin–Flach method to family after family of elliptic equations, and by this stage only one family refused to submit to the technique. Wiles describes how he attempted to complete the last element of the proof: ‘One morning in late May, Nada was out with the children and I was sitting at my desk thinking about the remaining family of elliptic equations. I was casually looking at a paper of Barry Mazur's and there was one sentence there that just caught my attention. It mentioned a nineteenth-century construction, and I suddenly realised that I should be able to use that to make the Kolyvagin–Flach method work on the final family of elliptic equations. I went on into the afternoon and I forgot to go down for lunch, and by about three or four o'clock I was really convinced that this would solve the last remaining problem. It got to about tea-time and I went downstairs and Nada was very surprised that I'd arrived so late. Then I told her – I'd solved Fermat's Last Theorem.'

After seven years of single-minded effort Wiles had completed a proof of the Taniyama–Shimura conjecture. As a consequence, and after thirty years of dreaming about it, he had also proved Fermat's Last Theorem. It was now time to tell the rest of the world.

‘So by May 1993, I was convinced that I had the whole of Fermat's Last Theorem in my hands,' recalls Wiles. ‘I still wanted
to check the proof some more but there was a conference which was coming up at the end of June in Cambridge, and I thought that would be a wonderful place to announce the proof – it's my old home town, and I'd been a graduate student there.'

The conference was being held at the Isaac Newton Institute. This time the institute had planned a workshop on number theory with the obscure title ‘
L
-functions and Arithmetic'. One of the organisers was Wiles's Ph.D. supervisor John Coates: ‘We brought people from all around the world who were working on this general circle of problems and, of course, Andrew was one of the people that we invited. We'd planned one week of concentrated lectures and originally, because there was a lot of demand for lecture slots, I only gave Andrew two lecture slots. But then I gathered he needed a third slot, and so in fact I arranged to give up my own slot for his third lecture. I knew that he had some big result to announce but I had no idea what.'

When Wiles arrived in Cambridge he had two and a half weeks before his lectures began and wanted to make the most of the opportunity: ‘I decided I would check the proof with one or two experts, in particular the Kolyvagin–Flach part. The first person I gave it to was Barry Mazur. I think I said to him, “I have a manuscript here with a proof to a certain theorem.” He looked very baffled for a while, and then I said, “Well, have a look at it.” I think it then took him some time to register. He appeared stunned. Anyway I told him that I was hoping to speak about it at the conference, and that I'd really like him to try and check it.'

One by one the most eminent figures in number theory began to arrive at the Newton Institute, including Ken Ribet whose calculation in 1986 had inspired Wiles's seven-year ordeal. ‘I arrived at this conference on
L
-functions and elliptic curves and it didn't seem to be anything out of the ordinary until people started telling
me that they had been hearing weird rumours about Andrew Wiles's proposed series of lectures. The rumour was that he had proved Fermat's Last Theorem, and I just thought this was completely nuts. I thought it couldn't possibly be true. There are lots of cases when rumours start circulating in mathematics, especially through electronic mail, and experience shows that you shouldn't put too much stock in them. But the rumours were very persistent and Andrew was refusing to answer questions about it and he was behaving very very queerly. John Coates said to him, “Andrew, what have you proved? Shall we call the press?” Andrew just kind of shook his head and sort of kept his lips sealed. He was really going for high drama.

‘Then one afternoon Andrew came up to me and started asking me about what I'd done in 1986 and some of the history of Frey's ideas. I thought to myself, this is incredible, he must have proved the Taniyama–Shimura conjecture and Fermat's Last Theorem, otherwise he wouldn't be asking me this. I didn't ask him directly if this was true, because I saw that he was behaving very coyly and I knew I wouldn't get a straight answer. So I just kind of said, “Well Andrew, if you have occasion to speak about this work, here's what happened.” I sort of looked at him as though I knew something, but I didn't really know what was going on. I was still just guessing.'

Wiles's reaction to the rumours and the mounting pressure was simple: ‘People would ask me, leading up to my lectures, what exactly I was going to say. So I said, well, come to my lectures and see.'

Back in 1920 David Hilbert, then aged fifty-eight, gave a public lecture in Göttingen on the subject of Fermat's Last Theorem. When asked if the problem would ever be solved, he replied that he would not live to see it, but perhaps younger members of the
audience might witness the solution. Hilbert's estimate for the date of the solution was proving to be fairly accurate. Wiles's lecture was also well timed in relation to the Wolfskehl Prize. In his will Paul Wolfskehl had set a deadline of 13 September 2007.

The title of Wiles's lecture series was ‘Modular Forms, Elliptic Curves and Galois Representations'. Once again, as with the graduate lectures he had given earlier in the year for the benefit of Nick Katz, the title of the lectures was so vague that it gave no hint of his ultimate aim. Wiles's first lecture was apparently mundane, laying the foundations for his attack on the Taniyama–Shimura conjecture in the second and third. The majority of his audience were completely unaware of the gossip, did not appreciate the point of the lectures, and paid little attention to the details. Those in the know were looking for the slightest clue which might give credence to the rumours.

Immediately after the lecture ended the rumour mill started again with renewed vigour, and electronic mail flew around the world. Professor Karl Rubin, a former student of Wiles, reported back to his colleagues in America:

By the following day more people had heard the gossip, and so the audience for the second lecture was significantly larger. Wiles teased them with an intermediate calculation which showed that he was clearly trying to tackle the Taniyama–Shimura conjecture, but the audience was still left wondering if he had done enough to prove it and, as a consequence, conquer Fermat's Last Theorem. A new batch of e-mails bounced off the satellites.