> By the tIme Katz found the error, other mathematicians throughout
> the world were aware of the exact same problem with Wiles' proof.
> There simply was no Euler System here, and there was nothing
> doing. And without the Euler System -- supposedly a
> generalization of the earlier work of Flach and Kolyvagm -- there
> was no Class Number Formula. Without the Class Number Formula, it
> was impossible to "count" the Galois representations of the
> elliptic curves against the modular forms, and Shimura-Taniyama
> was not established. And without the Shimura-Taniyama conjecture
> proved as correct, there was no proof of Fermat's Last Theorem. In
> short, the hole in the Euler System made everything collapse like
> a house of cards.
> The Agony
> Andrew Wiles returned to Princeton in the fall of 1993. He was
> embarrassed, he was upset, he was angry, frustrated, humiliated.
> He had promised the world a proof of Fermat's Last Theorem -- but
> he couldn't deliver. In mathematics, as in almost any other field,
> there are no real "second prizes" or "also ran" awards. The
> crestfallen Wiles was back in his attic trying to fix the proof.
> "At this point, he was hiding a secret from the world," recalled
> Nick Katz, "and I think he must have felt pretty uncomfortable
> about it." Other colleagues had tried to help Wiles, including his
> former student Richard Taylor who was teaching at Cambridge but
> joined Wiles at Princeton to help him try to fix the proof
> "The first seven years, working all alone, I enjoyed every minute
> of it," Wiles recalled, "no matter how hard or seemingly
> impossible a hurdle I faced. But now, doing mathematics in this
> over-exposed way was certainly not my style. I have no wish to
> ever repeat this experience." And the bad experience lasted and
> lasted. Richard Taylor, his sabbatical leave over, returned to
> Cambridge and still Wiles saw no end in sight. His colleagues
> looked at him with a mixture of anticipation, hope, and pity, and
> his suffering was clear to everyone around him. People wanted to
> know. They wanted to hear good news, but none of his colleagues
> dared ask him how he was doing with the proof. Outside his
> department, the rest of the world was curious, too. Sometime in
> the night of December 4, 1993, Andrew Wiles posted an e-mail
> message to the computer news group Sci math, to which several
> number theorists and other mathematicians belonged.
Originally Posted by
Andrew Wiles
> > [i]In view of the speculation on the status of my work on the
> > Taniyama-Shimura conjecture and Fermat's Last Theorem I will
> > give a brief account of the situation. During the review
> > process a number of problems emerged, most of which have been
> > resolved, but one In particular I have not yet settled I
> > believe that I wIll be able to finish this In the near future
> > using the ideas explained in my Cambridge lectures. The fact
> > that a lot of work remains to be done on the manuscript makes
> > it unsuitable for release as a preprint. In my course in
> > Princeton beginning in February I will give a full account of
> > thIs work.
> > Andrew Wiles[i]
> The Post-Mortem
> But Andrew Wiles was prematurely optimistic. And whatever course
> he may have planned to offer at Princeton would not yield any
> solution. When more than a year passed since his short-lived
> triumph in Cambridge, Andrew Wiles was about to give up all hope
> and to forget his crippled proof.
> On Monday morning, September 19, 1994, Wiles was sitting at his
> desk at Princeton University, piles of paper strewn all around
> him. He decided he would take one last look at his proof before
> chucking it all and abandoning all hope to prove Fermat's Last
> Theorem. He wanted to see exactly what it was that was preventing
> him from constructing the Euler System. He wanted to know -- just
> for his own satisfaction -- why he had failed. Why was there no
> Euler System? He wanted to be able to pinpoint precisely which
> technical fact was making the whole thing fail. If he was going
> to give up, he felt, then at least he was owed an answer to why he
> had been wrong
> Wiles studied the papers in front of him, concentrating very hard
> for about twenty minutes. And then he saw exactly why he was
> unable to make the system work Finally, he understood what was
> wrong. "It was the most important moment in my entire working
> life," he later described the feeling "Suddenly, totally
> unexpectedly, I had this incredible revelation. Nothing I'll ever
> do again will" at that moment tears welled up and Wiles was
> choking with emotion. What Wiles realized at that fateful moment
> was "so indescribably beautiful, it was so simple and so elegant
> and I just stared in disbelief." Wiles realized that exactly what
> was making the Euler System fail is what would make the Horizontal
> Iwasawa Theory approach he had abandoned three years earlier work.
> Wiles stared at his paper for a long time. He must be dreaming, he
> thought, this was just too good to be true. But later he said it
> was simply too good to be false. The discovery was so powerful, so
> beautiful, that it had to be true.
> Wiles walked around the department for several hours. He didn't
> know whether he was awake or dreaming. Every once in a while, he
> would return to his desk to see if his fantastic finding was
> still there -- and it was. He went home. He had to sleep on it
> -- maybe in the morning he would find some flaw in this new
> argument. A year of pressure from the entire world, a year of one
> frustrated attempt after another had shaken Wiles' confidence. He
> came back to his desk in the morning, and the incredible gem he
> had found the day before was still there, waiting for him.
> Wiles wrote up his proof using the corrected Horizontal Iwasawa
> Theory approach. Finally, everything fell perfectly into place.
> The approach he had used three years earlier was the correct one.
> And that knowledge came to him from the failing of the Flach and
> Kolyvagin route he had chosen in midstream. The manuscript was
> ready to be shipped out. Elated, Andrew Wiles logged into his
> computer account. He sent email messages across the Internet to a
> score of mathematicIans around the world "Expect a Federal Express
> package in the next few days," the messages read.