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.’
The Lecture of the Century
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.
‘On 23 June Andrew began his third and final lecture,’ recalls John Coates. ‘What was remarkable was that practically everyone who contributed to the ideas behind the proof was there in the room, Mazur, Ribet, Kolyvagin, and many, many others.’
By this point the rumour was so persistent that everyone from the Cambridge mathematics community turned up for the final lecture. The lucky ones were crammed into the auditorium, while the others had to wait in the corridor, where they stood on tip-toe and peered through the window. Ken Ribet had made sure that he would not miss out on the most important mathematical announcement of the century: ‘I came relatively early and I sat in the front row along with Barry Mazur. I had my camera with me just to record the event. There was a very charged atmosphere and people were very excited. We certainly had the sense that we were participating in a historical moment. People had grins on their faces before and during the lecture. The tension had built up over the course of several days. Then there was this marvellous moment when we were coming close to a proof of Fermat’s Last Theorem.’
Barry Mazur had already been given a copy of the proof by Wiles, but even he wa
s astonished by the performance. ‘I’ve never seen such a glorious lecture, full of such wonderful ideas, with such dramatic tension, and what a build-up. There was only one possible punch line.’
After seven years of intense effort Wiles was about to announce his proof to the world. Curiously Wiles cannot remember the final moments of the lecture in great detail, but does recall the atmosphere: ‘Although the press had already got wind of the lecture, fortunately they were not at the lecture. But there were plenty of people in the audience who were taking photographs towards the end and the Director of the Institute certainly had come well prepared with a bottle of champagne. There was a typical dignified silence while I read out the proof and then I just wrote up the statement of Fermat’s Last Theorem. I said, “I think I’ll stop here”, and then there was sustained applause.’
The Aftermath
Strangely, Wiles was ambivalent about the lecture: ‘It was obviously a great occasion, but I had mixed feelings. This had been part of me for seven years: it had been my whole working life. I got so wrapped up in the problem that I really felt I had it all to myself, but now I was letting go. There was a feeling that I was giving up a part of me.’
Wiles’s colleague Ken Ribet had no such qualms: ‘It was a completely remarkable event. I mean, you go to a conference and there are some routine lectures, there are some good lectures and there are some very special lectures, but it’s only once in a lifetime that you get a lecture where someone claims to solve a problem that has endured for 350 years. People were looking at each other and saying, “My God, you know we’ve just witnessed an historical event.” Then people asked a few questions about technicalities of the proof and possible applications to other equations, and then there was more silence and all of a sudden a second round of applause. The next talk was given by one Ken Ribet, yours truly. I gave the lecture, people took notes, people applauded, and no one present, including me, has any idea what I said in that lecture.’
While mathematicians were spreading the good news via e-mail, the rest of the world had to wait for the evening news, or the following day’s newspapers. TV crews and science reporters descended upon the Newton Institute, all demanding interviews with the ‘greatest mathematician of the century’. The Guardian exclaimed, ‘The Number’s Up for Maths’ Last Riddle’, and the front page of Le Monde read, ‘Le théorèm de Fermat enfin résolu’. Journalists everywhere asked mathematicians for their expert opinion on Wiles’s work, and professors, still recovering from the shock, were expected to briefly explain the most complicated mathematical proof ever, or provide a soundbite which would clarify the Taniyama–Shimura conjecture.
The first time Professor Shimura heard about the proof of his own conjecture was when he read the front page of the New York Times — ‘At Last, Shout of “Eureka!” In Age-Old Math Mystery’. Thirty-five years after his friend Yutaka Taniyama had committed suicide, the conjecture which they had created together had now been vindicated. For many professional mathematicians the proof of the Taniyama–Shimura conjecture was a far more important achievement than the solution of Fermat’s Last Theorem, because it had immense consequences for many other mathematical theorems. The journalists covering the story tended to concentrate on Fermat and mentioned Taniyama–Shimura only in passing, if at all.
Shimura, a modest and gentle man, was not unduly bothered by the lack of attention given to his role in the proof of Fermat’s Last Theorem, but he was concerned that he and Taniyama had been relegated from being nouns to adjectives. ‘It is very curious that people write about the Taniyama–Shimura conjecture, but nobody writes about Taniyama and Shimura.’
This was the first time that mathematics had hit the headlines since Yoichi Miyaoka announced his so-called proof in 1988: the only difference this time was that there was twice as much coverage and nobody expressed any doubt over the calculation. Overnight Wiles became the most famous, in fact the only famous, mathematician in the world, and People magazine even listed him among ‘The 25 most intriguing people of the year’, along with Princess Diana and Oprah Winfrey. The ultimate accolade came from an international clothing chain who asked the mild-mannered genius to endorse their new range of menswear.
While the media circus continued and while mathematicians made the most of being in the spotlight, the serious work of checking the proof was under way. As with all scientific disciplines each new piece of work has to be thoroughly examined, before it could be accepted as accurate and correct. Wiles’s proof had to be submitted to the ordeal of trial by referee. Although Wiles’s lectures at the Isaac Newton Institute had provided the world with an outline of his calculation, this did not qualify as official peer review. Academic protocol demands that any mathematician submits a complete manuscript to a respected journal, the editor of which then sends it to a team of referees whose job it is to examine the proof line by line. Wiles had to spend the summer anxiously waiting for the referees’ report, hoping that eventually he would get their blessing.
7
A Slight Problem
A problem worthy of attack
Proves its worth by fighting back.
Piet Hein
As soon as the Cambridge lecture was over, the Wolfskehl committee was informed of Wiles’s proof. They could not award the prize immediately because the rules of the contest clearly demand verification by other mathematicians and official publication of the proof:
The Königliche Gesellschaft der Wissenschaften in Göttingen … will only take into consideration those mathematical memoirs which have appeared in the form of a monograph in the periodicals, or which are for sale in the bookshops … The award of the Prize by the Society will take place not earlier than two years after the publication of the memoir to be crowned. The interval of time is intended to allow German and foreign mathematicians to voice their opinion about the validity of the solution published.
Wiles submitted his manuscript to the journal Inventiones Mathematicae, whereupon its editor Barry Mazur began the process of selecting the referees. Wiles’s paper involved such a variety of mathematical techniques, both ancient and modern, that Mazur made the exceptional decision to appoint not just two or three referees, as is usual, but six. Each year thirty thousand papers are published in journals around the world, but the sheer size and importance of Wiles’s manuscript meant that it would undergo a unique level of scrutiny. To simplify matters the 200-page proof was divided into six sections and each of the referees took responsibility for one of these chapters.
Chapter 3 was the responsibility of Nick Katz, who had already examined that part of Wiles’s proof earlier in the year: ‘I happened to be in Paris for the summer to work at the Institut des Hautes Etudes Scientifiques, and I took with me the complete 200-page proof – my particular chapter was seventy pages long. When I got there I decided I wanted to have serious technical help, and so I insisted that Luc Illusie, who was also in Paris, become a joint referee on this chapter. We would meet a few times a week throughout that summer, basically lecturing to each other to try and understand this chapter. Literally we did nothing but look through this manuscript line by line to try and make sure that there were no mistakes. Sometimes we got confused by things and so every day, sometimes twice a day, I would e-mail Andrew with a question – I don’t understand what you say on this page or it seems to be wrong on this line. Typically I would get a response that day or the next day which clarified the matter and then we’d go on to the next problem.’
The proof was a gigantic argument, intricately constructed from hundreds of mathematical calculations glued together by thousands of logical links. If just one of the calculations was flawed or if one of the links became unstuck then the entire proof was potentially worthless. Wiles, who was now back in Princeton, anxiously waited for the referees to complete their task. ‘I don’t like to celebrate full out until I have the paper completely off my hands. In the meantime I had my work cut out dealing with the questions I was getting via e-mail from the referees. I
was still pretty confident that none of these questions would cause me much trouble.’ He had already checked and double-checked the proof before releasing it to the referees, so he was expecting little more than the mathematical equivalent of grammatical or typographic errors, trivial mistakes which he could fix immediately.
‘These questions continued relatively uneventfully through till August,’ recalls Katz, ‘until I got to what seemed like just one more little problem. Sometime around 23 August I e-mail Andrew, but it’s a little bit complicated so he sends me back a fax. But the fax doesn’t seem to answer the question so I e-mail him again and I get another fax which I’m still not satisfied with.’
Wiles had assumed that this error was as shallow as all the others, but Katz’s persistence forced him to take it seriously: ‘I couldn’t immediately resolve this one very innocent looking question. For a little while it seemed to be of the same order as the other problems, but then sometime in September I began to realise that this wasn’t just a minor difficulty but a fundamental flaw. It was an error in a crucial part of the argument involving the Kolyvagin–Flach method, but it was something so subtle that I’d missed it completely until that point. The error is so abstract that it can’t really be described in simple terms. Even explaining it to a mathematician would require the mathematician to spend two or three months studying that part of the manuscript in great detail.’