Fermat's Last Theorem (29 page)

By the 1970s the Langlands programme had become a blueprint for the future of mathematics, but this route to a problem-solver's paradise was blocked by the simple fact that nobody had any real idea how to prove any of Langlands's conjectures. The strongest conjecture within the programme was still Taniyama–Shimura, but even this seemed out of reach. A proof of the Taniyama–Shimura conjecture would be the first step in the Langlands programme, and as such it had become one of the biggest prizes in modern number theory

Despite its status as an unproven conjecture, Taniyama–Shimura was still mentioned in hundreds of mathematical research papers speculating about what would happen if it could be proved. The papers would begin by clearly stating the caveat Assuming that the Taniyama–Shimura conjecture is true …', and then they would continue to outline a solution for some unsolved problem. Of course, these results could themselves only be hypothetical, because they relied on the Taniyama–Shimura conjecture being
true. These new hypothetical results were in turn incorporated into other results until there existed a plethora of mathematics which relied on the truth of the Taniyama–Shimura conjecture. This one conjecture was a foundation for a whole new architecture of mathematics, but until it could be proved the whole structure was vulnerable.

At the time, Andrew Wiles was a young researcher at Cambridge University, and he recalls the trepidation that plagued the mathematics community in the 1970s: ‘We built more and more conjectures which stretched further and further into the future, but they would all be ridiculous if the Taniyama–Shimura conjecture was not true. So we had to prove Taniyama–Shimura to show that this whole design we had hopefully mapped out for the future was correct.'

Mathematicians had constructed a fragile house of cards. They dreamed that one day someone would give their architecture the solid foundation it needed. They also had to live with the nightmare that one day someone might prove that Taniyama and Shimura were in fact wrong, causing two decades' worth of research to crash to the ground.

During the autumn of 1984 a select group of number theorists gathered for a symposium in Oberwolfach, a small town in the heart of Germany's Black Forest. They had been brought together to discuss various breakthroughs in the study of elliptic equations, and naturally some of the speakers would occasionally report any minor progress that they had made towards proving the Taniyama–Shimura conjecture. One of the speakers, Gerhard Frey, a
mathematician from Saarbrücken, could not offer any new ideas as to how to attack the conjecture, but he did make the remarkable claim that if anyone could prove the Taniyama–Shimura conjecture then they would also immediately prove Fermat's Last Theorem.

When Frey got up to speak he began by writing down Fermat's equation:

Fermat's Last Theorem claims that there are no whole number solutions to this equation, but Frey explored what would happen if the Last Theorem were false, i.e. that there is at least one solution. Frey had no idea what his hypothetical, and heretical, solution might be and so he labelled the unknown numbers with the letters
A, B
and
C:

Frey then proceeded to ‘rearrange' the equation. This is a rigorous mathematical procedure which changes the appearance of the equation without altering its integrity. By a deft series of complicated manoeuvres Frey fashioned Fermat's original equation, with the hypothetical solution, into

Although this rearrangement seems very different from the original equation, it is a direct consequence of the hypothetical solution. That is to say if, and it is a big ‘if, there is a solution to Fermat's equation and Fermat's Last Theorem is false, then this rearranged equation must also exist. Initially Frey's audience was not particularly impressed by his rearrangement, but then he pointed out that this new equation was in fact an elliptic equation, albeit a rather
convoluted and exotic one. Elliptic equations have the form

but if we let

then it is easier to appreciate the elliptical nature of Frey's equation.

By turning Fermat's equation into an elliptic equation, Frey had linked Fermat's Last Theorem to the Taniyama–Shimura conjecture. Frey then pointed out to his audience that his elliptic equation, created from the solution to the Fermat equation, is truly bizarre. In fact, Frey claimed that his elliptic equation is so weird that the repercussions of its existence would be devastating for the Taniyama–Shimura conjecture.

Remember that Frey's elliptic equation is only a phantom equation. Its existence is conditional on that fact that Fermat's Last Theorem is false. However, if Frey's elliptic equation does exist, then it is so strange that it would be seemingly impossible for it ever to be related to a modular form. But the Taniyama–Shimura conjecture claims that
every
elliptic equation must be related to a modular form. Therefore the existence of Frey's elliptic equation defies the Taniyama–Shimura conjecture.

In other words, Frey's argument was as follows:

(1) If (and only if) Fermat's Last Theorem is wrong, then Frey's elliptic equation exists.

(2) Frey's elliptic equation is so weird that it can never be modular.

(3) The Taniyama–Shimura conjecture claims that every elliptic equation must be modular.

(4) Therefore the Taniyama–Shimura conjecture must be false!

Alternatively, and more importantly, Frey could run his argument backwards:

(1) If the Taniyama–Shimura conjecture can be proved to be true, then every elliptic equation must be modular.

(2) If every elliptic equation must be modular, then the Frey elliptic equation is forbidden to exist.

(3) If the Frey elliptic equation does not exist, then there can be no solutions to Fermat's equation.

(4) Therefore Fermat's Last Theorem is true!

Gerhard Frey had come to the dramatic conclusion that the truth of Fermat's Last Theorem would be an immediate consequence of the Taniyama–Shimura conjecture being proved. Frey claimed that if mathematicians could prove the Taniyama–Shimura conjecture then they would automatically prove Fermat's Last Theorem. For the first time in a hundred years the world's hardest mathematical problem looked vulnerable. According to Frey, proving the Taniyama–Shimura conjecture was the only hurdle to proving Fermat's Last Theorem.

Although the audience was impressed by Frey's brilliant insight, they were also struck by an elementary blunder in his logic. Almost everyone in the auditorium, except Frey himself, had spotted it. The mistake did not appear to be serious: nonetheless as it stood Frey's work was incomplete. Whoever could correct the error first would take the credit for linking Fermat and Taniyama–Shimura.

Frey's audience dashed out of the lecture theatre and headed for the photocopying room. Often the importance of a talk can be gauged by the length of the queue waiting to run off copies of the lecture. Once they had a complete outline of Frey's ideas, they returned to their respective institutes and began to try and fill in the gap.

Frey's argument depended on the fact that his elliptic equation derived from Fermat's equation was so weird that it was not modular. His work was incomplete because he had not quite demonstrated that his elliptic equation was sufficiently weird. Only when somebody could prove the
absolute
weirdness of Frey's elliptic equation would a proof of the Taniyama–Shimura conjecture then imply a proof of Fermat's Last Theorem.

Initially mathematicians believed that proving the weirdness of Frey's elliptic equation would be a fairly routine process. At first sight Frey's mistake seemed to have been elementary and everyone who had been at Oberwolfach assumed that it was going to be a race to see who could shuffle the algebra most quickly. The expectation was that somebody would send out an e-mail within a matter of days describing how they had established the true weirdness of Frey's elliptic equation.

A week passed and there was no such e-mail. Months passed and what was supposed to be a mathematical mad dash was turning into a marathon. It seemed that Fermat was still teasing and tormenting his descendants. Frey had outlined a tantalising strategy for proving Fermat's Last Theorem, but even the first elementary step, proving that Frey's hypothetical elliptic equation was not modular, was baffling mathematicians around the globe.

To prove that an elliptic equation is not modular, mathematicians were looking for invariants similar to those described in
Chapter 4
. The knot invariant showed that one knot could not be transformed into another, and Loyd's puzzle invariant showed that his 14–15 puzzle could not be transformed into the correct arrangement. If number theorists could discover an appropriate invariant to describe Frey's elliptic equation, then they could prove that, no matter what was done to it, it could never be transformed into a modular form.

One of those toiling to prove and complete the connection between the Taniyama–Shimura conjecture and Fermat's Last Theorem was Ken Ribet, a professor at the University of California at Berkeley. Since the lecture at Oberwolfach, Ribet had become obsessed with trying to prove that Frey's elliptic equation was too weird to be modular. After eighteen months of effort he, along with everybody else, was getting nowhere. Then, in the summer of 1986, Ribet's colleague Professor Barry Mazur was visiting Berkeley to attend the International Congress of Mathematicians. The two friends met up for a cappuccino at the Café Strada and began sharing bad luck stories and grumbling about the state of mathematics.