## Modular elliptic curves and Fermat's Last Theorem. [with] Ring-theoretic properties of certain Hecke algebras.

Princeton: Princeton University Press, 1995.

First edition, journal issue, of his proof of Fermat’s Last Theorem, which was perhaps the most celebrated open problem in mathematics. In a marginal note in the section of his copy of Diophantus’ Arithmetica (1621) dealing with Pythagorean triples (positive whole numbers x, y, z satisfying x2 + y2 = z2 – of which an infinite number exist), Fermat stated that the equation xn + yn = zn, where n is any whole number greater than 2, has no solution in which x, y, z are positive whole numbers. Fermat followed this assertion with what is probably the most tantalising comment in the history of mathematics: ‘I have a truly marvellous demonstration of this proposition which this margin is too narrow to contain.’ Fermat believed he could prove his theorem, but he never committed his proof to paper. After his death, mathematicians across Europe, from the enthusiastic amateur to the brilliant professional, tried to rediscover the proof of what became known as Fermat’s Last Theorem, but for more than 350 years none succeeded, nor could anyone disprove the theorem by finding numbers x, y, z which did satisfy Fermat’s equation. When the great German mathematician David Hilbert was asked why he never attempted a proof of Fermat’s Last Theorem, he replied, “Before beginning I should have to put in three years of intensive study, and I haven’t that much time to squander on a probable failure.” Soon after the Second World War computers helped to prove the theorem for all values of n up to five hundred, then one thousand, and then ten thousand. In the 1980’s Samuel S. Wagstaff of the University of Illinois raised the limit to 25,000 and more recently mathematicians could claim that Fermat’s Last Theorem was true for all values of n up to four million. But no general proof was found until 1995.

“Between 1954 and 1986 a chain of events of occurred which brought Fermat’s Last Theorem back into the mainstream. The incident which began everything happened in post-war Japan, when Yutaka Taniyama and Goro Shimura, two young academics, decided to collaborate on the study of elliptic curves and modular forms. These entities are from opposite ends of the mathematical spectrum, and had previously been studied in isolation.

“Elliptic curves, which have been studied since the time of Diophantus, concern cubic equations of the form:

y2 = (x + a).(x + b).(x + c), where a, b and c can be any whole number, except zero.

The challenge is to identify and quantify the whole solutions to the equations, the solutions differing according to the values of a, b, and c.

“Modular forms are a much more modern mathematical entity, born in the nineteenth century. They are functions, not so different to functions such as sine and cosine, but modular forms are exceptional because they exhibit a high degree of symmetry. For example, the sine function is slightly symmetrical because 2p can be added to any number, x, and yet the result of the function remains unchanged, i.e., sin x = sin (x + 2p). However, for modular forms the number x can be transformed in an infinite number of ways and yet the outcome of the function remains unchanged, hence they are said to be extraordinarily symmetric …

“Despite belonging to a completely different area of the mathematics, Shimura and Taniyama began to suspect that the elliptic curves might be related to modular forms in a fundamental way. It seemed that the solutions for any one of the infinite number of elliptic curves could be derived from one of the infinite number of modular forms. Each elliptic curve seemed to be a modular form in disguise. This apparent unification became known as the Shimura-Taniyama conjecture, reflecting the fact that mathematicians were confident that it was true, but as yet were unable to prove it. The conjecture was considered important because if it were true problems about elliptic curves, which hitherto had been insoluble, could potentially be solved by using techniques developed for modular forms, and vice versa …

“Even though the Shimura-Taniyama conjecture could not be proved, as the decades passed it gradually became increasingly influential, and by the 1970s mathematicians would begin papers by assuming the Shimura-Taniyama conjecture and then derive some new result. In due course many major results came to rely on the conjecture being proved, but these results could themselves only be classified as conjectures, because they were conditional on the proof of the Shimura-Taniyama conjecture. Despite its pivotal role, few believed it would be proved this century.

“Then, in 1986, Kenneth A Ribet of the University of California at Berkeley, building on the work of Gerhard Frey of the University of Saarlands, made an astonishing breakthrough. He was unable to prove the Shimura-Taniyama conjecture, but he was able to link it with Fermat’s Last Theorem. The link occurred by contemplating the unthinkable – what would happen if Fermat’s Last Theorem was not true? This would mean that there existed a set of solutions to Fermat’s equation, and therefore this hypothetical combination of numbers could be used as the basis for constructing a hypothetical elliptic curve. Ribet demonstrated that this elliptic curve could not possibly be related to a modular form, and as such it would defy the Shimura-Taniyama conjecture. Running the argument backwards, if somebody could prove the Shimura-Taniyama conjecture then every elliptic curve must be related to a modular form, hence any solution to Fermat’s equation is forbidden to exist, and hence Fermat’s Theorem must be true. If somebody could prove the Shimura-Taniyama conjecture, then this would immediately imply the proof of Fermat’s Last Theorem. By proving one of the most important conjectures of the twentieth century, mathematicians could solve a riddle from the seventeenth century.

“The Shimura-Taniyama conjecture had remained unproven since the 1950s and so there was little optimism that it was a realistic route to a proof of Fermat’s Last Theorem. Some mathematicians joked that, if anything, the Shimura-Taniyama conjecture was even further out of reach, because, by definition, anything that led to a proof of the Last Theorem must be impossible. But for Wiles, anything that would lead to the Last Theorem was worth pursuing. He knew that this might be his only chance to realise his childhood dream and he had the audacity to attack the Shimura-Taniyama conjecture. As a graduate student at Cambridge University, he had concentrated on studying elliptic curves, and then as a professor at Princeton University he had continued his research, putting him in an ideal position for attempting a proof.

“As he embarked on his proof, Wiles made the extraordinary decision to conduct his research in complete secrecy. He did not want the pressure of public attention, nor did he want to risk others copying his ideas and stealing the prize. In order not to arouse suspicion Wiles devised a cunning ploy that would throw his colleagues off the scent. During the early 1980s he had been working on a major piece of research on a particular type of elliptic curve, which he was about to publish in its entirety until the discoveries of Ribet and Frey made him change his mind. Wiles decided to publish his research bit by bit, releasing another minor paper every six months or so. This apparent productivity would convince his colleagues that Wiles was still continuing with his usual research. For as long as he could maintain this charade Wiles could continue working on his true obsession without revealing any of his breakthroughs. For the next seven years he worked in isolation, and his colleagues were oblivious to what he was doing. The only person who knew of his secret project was his wife – he told her during their honeymoon.

“The number of elliptic curves and modular forms is infinite, and the Shimura-Taniyama conjecture claimed each elliptic curve could be matched with a modular. However, to succeed Wiles did not have to prove the full Shimura-Taniyama conjecture. Instead he only had to show that a particular subset of elliptic curves (one which would include the hypothetical Fermat elliptic curve) is modular. However, this subset is still infinite in size and it includes the majority of interesting curves.

“To prove that something is true for an infinite number of cases required Wiles to pull together some of the most recent breakthroughs in number theory, and in addition invent new techniques of his own. He adopted a strategy loosely based on a method known as induction. Proof by induction can prove something for an infinite number of cases by invoking a domino toppling approach, i.e., to knock down an infinite number of dominoes, one merely has to ensure that knocking down any domino will always topple the next domino. In other words, Wiles had to develop an argument in which he could prove the first case, and then be sure that proving any one case would implicitly prove the next one.

“At each stage Wiles could never be sure that he could complete his proof. He realised that even if he did have the correct strategy, the mathematical techniques required might not yet exist – he might be on the right track, but living in the wrong century. Eventually, in 1993, Wiles felt confident that his proof was reaching completion. The opportunity arose to announce his proof of a major section of the Shimura-Taniyama conjecture, and hence Fermat’s Last Theorem, at a special conference to be held at the Isaac Newton Institute in Cambridge, England. Because this was his home town, where he had encountered the Last Theorem as a child, he decided to make a concerted effort to complete the proof in time for the conference. On June 23rd he announced his seven-year calculation to a stunned audience.

“His secret research programme had apparently been a success, and the mathematical community and the world’s press rejoiced. The front page of the New York Times exclaimed “At Last, Shout of ‘Eureka!’ in Age-Old Math Mystery”, and Wiles appeared on television stations around the world … While the media circus continued, the official peer review process began. Over the summer the 200-page proof was examined line by line by a team of referees. The manuscript was split into seven chapters, and each chapter was sent to a pair of expert examiners. Wiles checked and double-checked the proof before releasing it to the referees, so he was expecting little more than the mathematical equivalent of grammatical and typographic errors, trivial mistakes that he could fix immediately. However, gradually it emerged that there was a fundamental and devastating flaw in one stage of the argument.

“Essentially, the inductive argument used by Wiles could not guarantee that if one domino toppled, then so would the next. Over the course of the next year his childhood dream turned into a nightmare. Each attempt to fix the error ended in failure, each attempt to by-pass the error ended in a dead-end. And throughout this period the manuscript had only been seen by the small team of referees and Wiles himself. There were calls from the mathematics community to publish the flawed proof, which would allow others to try and fix it, but Wiles steadfastly refused. He believed that he deserved the first chance to correct a piece of work that had already taken him seven years.

“After months of failure Wiles did take into his confidence Richard Taylor, a former student of his, hoping that this would give him someone to bounce ideas off, someone who could inspire him to consider alternative strategies. By September 1994 they were at the point of admitting defeat, ready to release the flawed proof so that others could try and fix it. Then on September 19th they made the vital breakthrough. Many years earlier, when he was working in secrecy, Wiles had considered using an alternative approach, but it floundered and so he had abandoned it. Now they realised that what was causing the more recent method to fail was exactly what would make the abandoned approach succeed.

Wiles recalls his reaction to the discovery: “It was so indescribably beautiful, it was so simple and so elegant. The first night I went back home and slept on it. I checked through it again the next morning and, and I went down and told me 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'” (Simon Singh, ‘The Whole Story,’ simonsingh.net/books/fermats-last-theorem/the-whole-story/).

In: Annals of Mathematics, Vol. 141, No. 3, May 1995, pp. 443-551 & pp. 553-572. 8vo (254 x 178 mm), original printed wrappers, in virtually mint condition. Custom cloth box.

Item #4866

Price: \$4,500.00