Fermat's Last Theorem

by Simon Singh

  This was an astonishing discovery because, for no apparent reason, this modular form could be related to an elliptic equation through their respective M-series and E-series – these series were identical. The mathematical DNA which made up these two entities was exactly the same. This was a doubly profound discovery. First, it suggested that deep down there was a fundamental relationship between the modular form and the elliptic equation, objects which come from opposite ends of mathematics. Second, it meant that mathematicians, who already knew the M-series for the modular form, would not have to calculate the E-series for the corresponding elliptic equation because it would be the same as the M-series.

  Relationships between apparently different subjects are as creatively important in mathematics as they are in any discipline. The relationship hints at some underlying truth which enriches both subjects. For instance, originally scientists had studied electricity and magnetism as two completely separate phenomena. Then, in the nineteenth century, theorists and experimentalists realised that electricity and magnetism were intimately related. This resulted in a deeper understanding of both of them. Electric currents generate magnetic fields, and magnets can induce electricity in wires passing close to them. This led to the invention of dynamos and electric motors, and ultimately the discovery that light itself is the result of magnetic and electric fields oscillating in harmony.

  Taniyama examined a few other modular forms and in each case the M-series seemed to correspond perfectly with the E-series of an elliptic equation. He began to wonder if it could be that every single modular form could be matched with an elliptic equation. Perhaps every modular form has the same DNA as an elliptic equation: perhaps each modular form is an elliptic equation in disguise? The questions he handed out at the symposium were related to this hypothesis.

  The idea that every elliptic equation was related to a modular form was so extraordinary that those who glanced at Taniyama’s questions treated them as nothing more than a curious observation. Sure enough Taniyama had demonstrated that a few elliptic equations could be related to particular modular forms, but they claimed that this was nothing more than a coincidence. According to the sceptics Taniyama’s claim of a more general and universal relationship seemed to be largely unsubstantiated. The hypothesis was based on intuition rather than on any real evidence.

  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.

  Death of a Genius

  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.’

  Philosophy of Goodness

  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 wit
h 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.

  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 conjectu
res. 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.