Fermat's Last Theorem (28 page)

Taniyama's only ally was Shimura, who believed in the power and depth of his friend's idea. Following the symposium he worked with Taniyama in an attempt to develop the hypothesis to a level where the rest of the world could no longer ignore their work. Shimura wanted to find more evidence to back up the relationship between the modular and elliptic worlds. The collaboration was temporarily halted when in 1957 Shimura was invited to attend the Institute for Advanced Study in Princeton. Following his two years as a visiting professor in America he intended to resume working with Taniyama, but this was never to happen. On 17 November 1958, Yutaka Taniyama committed suicide.

Shimura still keeps the postcard that Taniyama sent him when they first made contact over the library book. He also keeps the last letter Taniyama wrote to him while he was away in Princeton, but it contains not the merest hint as to what would happen just two months later. To this day Shimura has no understanding of what was behind Taniyama's suicide. ‘I was very much puzzled. Puzzlement may be the best word. Of course I was sad, but it was so sudden. I got his letter in September and he died in early November, and I was unable to make sense out of this. Of course, later I heard various things and I tried to reconcile myself with his death. Some people said that he lost confidence in himself but not mathematically.'

What was particularly confusing for Taniyama's friends was that he had just fallen in love with Misako Suzuki and planned to marry her later that year. In a personal tribute published in the
Bulletin of the London Mathematical Society
, Goro Shimura recollects Taniyama's engagement to Misako and the weeks which led up to his suicide:

When informed of their engagement, I was somewhat surprised, since I had vaguely thought she was not his type, but I felt no misgivings. I was told afterward that they had signed a lease for an apartment, apparently a better one, for their new home, had bought some kitchenware together, and had been preparing for their wedding. Everything looked promising for them and their friends. Then the catastrophe befell them.

On the morning of Monday, November 17, 1958, the superintendent of his apartment found him dead in his room with a note left on a desk. It was written on three pages of a notebook of the type he had been using for his scholastic work; its first paragraph read like this:

‘Until yesterday, I had no definite intention of killing myself. But more than a few must have noticed that lately I have been tired both physically and mentally. As to the cause of my suicide, I don't quite understand it myself, but it is not the result of a particular incident, nor of a specific matter. Merely may I say, I am in the frame of mind that I lost confidence in my future. There may be someone to whom my suicide will be troubling or a blow to a certain degree. I sincerely hope that this incident will cast no dark shadow over the future of that person. At any rate, I cannot deny that this is a kind of betrayal, but please excuse it as my last act in my own way, as I have been doing my own way all my life.'

He went on to describe, quite methodically, his wish of how his belongings should be disposed of, and which books and records were the ones he had borrowed from the library or from his friends, and so on. Specifically he says: ‘I would like to leave the records and the player to Misako Suzuki provided she will not be upset by me leaving them to her'. Also he explains how far he reached in the undergraduate courses on calculus and linear algebra he was teaching, and concludes the note with an apology to his colleagues for the inconveniences this act could cause.

Thus one of the most brilliant and pioneering minds of the time ended his life by his own will. He had attained the age of thirty-one only five days earlier.

A few weeks after the suicide, tragedy struck a second time. His fiancée, Misako Suzuki, also took her own life. She reportedly left a note which read: ‘We promised each other that no matter where we went, we would never be separated. Now that he is gone, I must go too in order to join him.'

During his short career Taniyama contributed many radical ideas to mathematics. The questions he handed out at the symposium contained his greatest insight, but it was so ahead of its time that he would never live to see its enormous influence on number theory. His intellectual creativity was to be sadly missed, along with his guiding role within the community of young Japanese scientists. Shimura clearly remembers Taniyama's influence: ‘He was always kind to his colleagues, especially to his juniors, and he genuinely cared about their welfare. He was the moral support of many of those who came into mathematical contact with him, including of course myself. Probably he was never concious of this role he was playing. But I feel his noble generosity in this respect even more strongly now than when he was alive. And yet nobody was able to give him any support when he desperately needed it. Reflecting on this, I am overwhelmed by the bitterest grief.'

Following Taniyama's death, Shimura concentrated all his efforts on understanding the exact relationship between elliptic equations and modular forms. As the years passed he struggled to gather more evidence and one or two pieces of logic to support the theory. Gradually he became increasingly convinced that every single elliptic equation must be related to a modular form. Other mathematicians were still dubious and Shimura recalls a conversation with an eminent colleague. The professor inquired, ‘I hear that you propose that some elliptic equations can be linked to modular forms.'

‘No, you don't understand,' replied Shimura. ‘It's not just
some
elliptic equations, it's
every
elliptic equation!'

Shimura could not prove that this was the case but every time
he tested the hypothesis it seemed to be true, and in any case it all seemed to fit in with his broad mathematical philosophy. ‘I have this philosophy of goodness. Mathematics should contain goodness. So in the case of the elliptic equation, one might call the equation good if it is parametrised by a modular form. I expect all elliptic equations to be good. It's a rather crude philosophy but one can always take it as a starting point. Then, of course, I had to develop various technical reasons for the conjecture. I might say that the conjecture stemmed from that philosophy of goodness. Most mathematicians do mathematics from an aesthetic point of view and that philosophy of goodness comes from my aesthetic viewpoint.'

Eventually Shimura's accumulation of evidence meant that his theory about elliptic equations and modular forms became more widely accepted. He could not prove to the rest of the world that it was true, but at least it was now more than mere wishful thinking. There was enough evidence for it to be worthy of the title of conjecture. Initially it was referred to as the Taniyama–Shimura conjecture in recognition of the man who inspired it and his colleague who went on to develop it fully.

In due course André Weil, one of the godfathers of twentieth-century number theory, was to adopt the conjecture and publicise it in the West. Weil investigated the idea of Shimura and Taniyama, and found even more solid evidence in favour of it. As a result, the hypothesis was often referred to as the Taniyama–Shimura–Weil conjecture, sometimes as the Taniyama–Weil conjecture and occasionally as the Weil conjecture. In fact there has been much debate and controversy over the official naming of the conjecture. For those of you interested in combinatorics there are 15 possible permutations given the three names involved, and it is quite probable that every one of those combinations has appeared
in print over the years. However, I will refer to the conjecture by its original title, the Taniyama–Shimura conjecture.

Professor John Coates, who guided Andrew Wiles when he was a student, was himself a student when the Taniyama–Shimura conjecture became a talking point in the West. ‘I began research in 1966 when the conjecture of Taniyama and Shimura was sweeping through the world. Everyone was amazed and began to look seriously at the issue of whether all elliptic equations could be modular. This was a tremendously exciting time; the only problem, of course, was that it seemed very hard to make progress. I think it's fair to say that beautiful though this idea was it seemed very difficult to actually prove, and that's what we're primarily interested in as mathematicians.'

During the late sixties hoards of mathematicians repeatedly tested the Taniyama–Shimura conjecture. Starting with an elliptic equation and its
E
-series they would search for a modular form with an identical
M
-series. In every single case the elliptic equation did indeed have an associated modular form. Although this was good evidence in favour of the Taniyama–Shimura conjecture, it was by no means a proof. Mathematicians suspected that it was true, but until somebody could find a logical proof it would remain merely a conjecture.

Barry Mazur, a professor at Harvard University, witnessed the rise of the Taniyama–Shimura conjecture. ‘It was a wonderful conjecture – the surmise that every elliptic equation is associated with a modular form – but to begin with it was ignored because it was so ahead of its time. When it was first proposed it was not taken up because it was so astounding. On the one hand you have the elliptic world, and on the other you have the modular world. Both these branches of mathematics had been studied intensively but separately. Mathematicians studying elliptic equations might not
be well versed in things modular, and conversely. Then along comes the Taniyama–Shimura conjecture which is the grand surmise that there's a bridge between these two completely different worlds. Mathematicians love to build bridges.'

The value of mathematical bridges is enormous. They enable communities of mathematicians who have been living on separate islands to exchange ideas and explore each other's creations. Mathematics consists of islands of knowledge in a sea of ignorance. For example, there is the island occupied by geometers who study shape and form, and then there is the island of probability where mathematicians discuss risk and chance. There are dozens of such islands, each one with its own unique language, incomprehensible to the inhabitants of other islands. The language of geometry is quite different to the language of probability, and the slang of calculus is meaningless to those who speak only statistics.

The great potential of the Taniyama–Shimura conjecture was that it would connect two islands and allow them to speak to each other for the first time. Barry Mazur thinks of the Taniyama–Shimura conjecture as a translating device similar to the Rosetta stone, which contained Egyptian demotic, ancient Greek and hieroglyphics. Because demotic and Greek were already understood, archaeologists could decipher hieroglyphics for the first time. ‘It's as if you know one language and this Rosetta stone is going to give you an intense understanding of the other language,' says Mazur. ‘But the Taniyama–Shimura conjecture is a Rosetta stone with a certain magical power. The conjecture has the very pleasant feature that simple intuitions in the modular world translate into deep truths in the elliptic world, and conversely. What's more, very profound problems in the elliptic world can get solved sometimes by translating them using this Rosetta stone into the modular world, and discovering that we have the insights and tools
in the modular world to treat the translated problem. Back in the elliptic world we would have been at a loss.'

If the Taniyama–Shimura conjecture was true it would enable mathematicians to tackle elliptic problems which had remained unsolved for centuries by approaching them through the modular world. The hope was that the fields of elliptic equations and modular forms could be unified. The conjecture also inspired the hope that links might exist between various other mathematical subjects.

During the 1960s Robert Langlands, at the Institute for Advanced Study, Princeton, was struck by the potency of the Taniyama–Shimura conjecture. Even though the conjecture had not been proved, Langlands believed that it was just one element of a much grander scheme of unification. He was confident that there were links between all the main mathematical topics and began to look for these unifications. Within a few years a number of links began to emerge. All these other unification conjectures were much weaker and more speculative than Taniyama–Shimura, but they formed an intricate network of hypothetical connections between many areas of mathematics. Langlands's dream was to see each of these conjectures proved one by one, leading to a grand unified mathematics.

Langlands discussed his plan for the future and tried to persuade other mathematicians to take part in what became known as the Langlands programme, a concerted effort to prove his myriad of conjectures. There seemed to be no obvious way to prove such speculative links, but if the dream could be made a reality then the reward would be enormous. Any insoluble problem in one area of mathematics could be transformed into an analogous problem in another area, where a whole new arsenal of techniques could be brought to bear on it. If a solution was still elusive, the problem could be transformed and transported to yet another area of
mathematics, and so on, until it was solved. One day, according to the Langlands programme, mathematicians would be able to solve their most esoteric and intractable problems by shuffling them around the mathematical landscape.

There were also important implications for the applied sciences and engineering. Whether it is modelling the interactions between colliding quarks or discovering the most efficient way to organise a telecommunications network, often the key to the problem is performing a mathematical calculation. In some areas of science and technology the complexity of the calculations is so immense that progress in the subject has been severely hindered. If only mathematicians could prove the linking conjectures of the Langlands programme, then there would be short cuts to solving real-world problems, as well as abstract ones.