Fermat's Last Theorem (38 page)

Despite the pressure Wiles refused to release the manuscript. After seven years of devoted effort he was not ready to sit back and watch someone else complete the proof and steal the glory. The person who proves Fermat's Last Theorem is not the person that puts in the most work, it's the person who delivers the final and complete proof. Wiles knew that once the manuscript was published in its flawed state he would immediately be swamped by questions and demands for clarification from would-be gap-fixers, and these distractions would destroy his own hopes of mending the proof while giving others vital clues.

Wiles attempted to return to the same state of isolation which had allowed him to create the original proof, and reverted to his habit of studying intensely in his attic. Occasionally he would wander down by the Princeton lake, as he had done in the past. The joggers, cyclists and rowers who had previously passed him by with a brief wave now stopped and asked him whether there was any progress with the gap. Wiles had appeared on front pages around the world, he had been featured in
People
magazine and he had even been interviewed on CNN. The previous summer Wiles had become the world's first mathematical celebrity, and already his image was tarnished.

Meanwhile back in the mathematics department the gossip continued. Princeton mathematician Professor John H. Conway remembers the atmosphere in the department's tea-room: ‘We'd gather for tea at 3 o'clock and make a rush for the cookies. Sometimes we'd discuss mathematical problems, sometimes we'd discuss the O.J. Simpson trial, and sometimes we'd discuss
Andrew's progress. Because nobody actually liked to come out and ask him how he's getting on with the proof, we were behaving a little bit like Kremlinologists. So somebody would say: “I saw Andrew this morning” – “Did he smile?” – “Well, yes, but he didn't look too happy.” We could only gauge his feelings by his face.'

As winter deepened, hopes of a breakthrough faded, and more mathematicians argued that it was Wiles's duty to release the manuscript. The rumours continued and one newspaper article claimed that Wiles had given up and that the proof had irrevocably collapsed. Although this was an exaggeration, it was certainly true that Wiles had exhausted dozens of approaches which might have circumvented the error and he could see no other potential routes to a solution.

Wiles admitted to Peter Sarnak that the situation was getting desperate and that he was on the point of accepting defeat. Sarnak suggested that part of the difficulty was that Wiles had nobody he could trust on a day-to-day basis; there was nobody he could bounce ideas off or who could inspire him to explore more lateral approaches. He suggested that Wiles took somebody into his confidence and try once more to fill the gap. Wiles needed somebody who was an expert in manipulating the Kolyvagm—Flach method and who could also keep the details of the problem secret. After giving the matter prolonged thought, he decided to invite Richard Taylor, a Cambridge lecturer, to Princeton to work alongside him.

Taylor was one of the referees responsible for verifying the proof and a former student of Wiles, and as such he could be doubly trusted. The previous year he had been in the audience at the Isaac Newton Institute watching his former supervisor present the proof of the century. Now it was his job to help rescue the flawed proof.

By January Wiles, with the help of Taylor, was once again tirelessly exploring the Kolyvagin–Flach method, trying to find a way out of the problem. Occasionally after days of effort they would enter new territory, but inevitably they would find themselves back where they started. Having ventured further than ever before and failing over and over again, they both realised that they were in the heart of an unimaginably vast labyrinth. Their deepest fear was that the labyrinth was infinite and without exit, and that they would be doomed to wander aimlessly and endlessly.

Then in the spring of 1994, just when it looked as though things could not get any worse, the following e-mail hit computer screens around the world:

Noam Elkies was the Harvard professor who back in 1988 had found a counter-example to Euler's conjecture, thereby proving that it was false:

Now he had apparently discovered a counter-example to Fermat's Last Theorem, proving that it too was false. This was a tragic blow for Wiles – the reason he could not fix the proof was that the so-called error was a direct result of the falsity of the Last Theorem. It was an even greater blow for the mathematical community at large, because if Fermat's Last Theorem was false, then Frey had already shown that this would lead to an elliptic equation which was
not
modular, a direct contradiction to the Taniyama-Shimura conjecture. Elkies had not only found a counter-example to Fermat, he had indirectly found a counter-example to Taniyama–Shimura.

The death of the Taniyama–Shimura conjecture would have devastating repercussions throughout number theory, because for two decades mathematicians had tacitly assumed its truth. In
Chapter 5
it was explained that mathematicians had written dozens of proofs which began with ‘Assuming the Taniyama–Shimura conjecture', but now Elkies had shown that this assumption was wrong and all those proofs had simultaneously collapsed. Mathematicians immediately began to demand more information and bombarded Elkies with questions, but there was no response and no explanation as to why he was remaining tight-lipped. Nobody could even find the exact details of the counter-example.

After one or two days of turmoil some mathematicians took a second look at the e-mail and began to realise that, although it was typically dated 2 April or 3 April, this was a result of having received it second or third hand. The original message was dated 1 April. The e-mail was a mischievous hoax perpetrated by the Canadian number theorist Henri Darmon. The rogue e-mail served as a suitable lesson for the Fermat rumour-mongers, and for a while the Last Theorem, Wiles, Taylor and the damaged proof were left in peace.

That summer Wiles and Taylor made no progress. After eight years of unbroken effort and a lifetime's obsession Wiles was prepared to admit defeat. He told Taylor that he could see no point in continuing with their attempts to fix the proof. Taylor had already planned to spend September in Princeton before returning to Cambridge, and so despite Wiles's despondency, he suggested they persevere for one more month. If there was no sign of a fix by the end of September, then they would give up, publicly acknowledge their failure and publish the flawed proof to allow others an opportunity to examine it.

Although Wiles's battle with the world's hardest mathematical problem seemed doomed to end in failure, he could look back at the last seven years and be reassured by the knowledge that the bulk of his work was still valid. To begin with Wiles's use of Galois groups had given everybody a new insight into the problem. He had shown that the first element of every elliptic equation could be paired with the first element of a modular form. Then the challenge was to show that if one element of the elliptic equation was modular, then so must the next piece be modular, and so must they all be modular.

During the middle years Wiles wrestled with the concept of extending the proof. He was trying to complete an inductive approach and had wrestled with Iwasawa theory in the hope that this would demonstrate that if one domino fell then they all would. Initially Iwasawa theory seemed powerful enough to cause the required domino effect but in the end it could not quite live up to his expectation. He had devoted two years of effort to a mathematical dead end.

In the summer of 1991, after a year in the doldrums, Wiles encountered the method of Kolyvagin and Flach and he abandoned Iwasawa theory in favour of this new technique. The following year the proof was announced in Cambridge and he was proclaimed a hero. Within two months the Kolyvagin–Flach method was shown to be flawed, and ever since the situation had only worsened. Every attempt to fix Kolyvagin–Flach had failed.

All of Wiles's work apart from the final stage involving the Kolyvagin–Flach method was still worthwhile. The Taniyama–Shimura conjecture and Fermat's Last Theorem might not have
been solved; nevertheless he had provided mathematicians with a whole series of new techniques and strategies which they could exploit to prove other theorems. There was no shame in Wiles's failure and he was beginning to come to terms with the prospect of being beaten.

As a consolation he at least wanted to understand why he had failed. While Taylor re-explored and re-examined alternative methods, Wiles decided to spend September looking one last time at the structure of the Kolyvagin–Flach method to try and pinpoint exactly why it was not working. He vividly remembers those final fateful days: ‘I was sitting at my desk one Monday morning, 19 September, examining the Kolyvagin–Flach method. It wasn't that I believed I could make it work, but I thought that at least I could explain why it didn't work. I thought I was clutching at straws, but I wanted to reassure myself. Suddenly, totally unexpectedly, I had this incredible revelation. I realised that, although the Kolyvagin–Flach method wasn't working completely, it was all I needed to make my original Iwasawa theory work. I realised that I had enough from the Kolyvagin–Flach method to make my original approach to the problem from three years earlier work. So out of the ashes of Kolyvagin–Flach seemed to rise the true answer to the problem.'

Iwasawa theory on its own had been inadequate. The Kolyvagin–Flach method on its own was also inadequate. Together they complemented each other perfectly. It was a moment of inspiration that Wiles will never forget. As he recounted these moments, the memory was so powerful that he was moved to tears: ‘It was so indescribably beautiful; it was so simple and so elegant. I couldn't understand how I'd missed it and I just stared at it in disbelief for twenty minutes. Then during the day I walked around the department, and I'd keep coming back to
my desk looking to see if it was still there. It was still there. I couldn't contain myself, I was so excited. It was the most important moment of my working life. Nothing I ever do again will mean as much.'

This was not only the fulfilment of a childhood dream and the culmination of eight years of concerted effort, but having been pushed to the brink of submission Wiles had fought back to prove his genius to the world. The last fourteen months had been the most painful, humiliating and depressing period of his mathematical career. Now one brilliant insight had brought an end to his suffering.

‘So the first night I went back home and slept on it. I checked through it again the next morning and by 11 o'clock I was satisfied, and I went down and told my wife, “I've got it! I think I've found it.” And it was so unexpected that she thought I was talking about a children's toy or something, and she said, “Got what?” I said, “I've fixed my proof. I've got it.'”

The following month Wiles was able to make up for the promise he had failed to keep the previous year. ‘It was coming up to Nada's birthday again and I remembered that last time I could not give her the present she wanted. This time, half a minute late for our dinner on the night of her birthday, I was able to give her the complete manuscript. I think she liked that present better than any other I had ever given her.'