Fermat’s Last Theorem Proof Secures Mathematics’ Top Prize For Sir Andrew Wiles

Fermat’s Last Theorem proof secures mathematics’ top prize for Sir Andrew Wiles

Sir Andrew Wiles
source: www.ox.ac.uk
The solution to a 300-year-old mystery has landed Oxford University Professor Sir Andrew Wiles the top international prize for mathematics.

Sir Andrew has been awarded the 2016 Abel Prize, regarded as mathematics’ equivalent of the Nobel Prize, ‘for his stunning proof of Fermat’s Last Theorem by way of the modularity conjecture for semistable elliptic curves, opening a new era in number theory’.

Sir Andrew, Royal Society Research Professor of Mathematics at Oxford, will receive the Prize from Crown Prince Haakon of Norway at a ceremony in Oslo in May.

Learning of the award today, Sir Andrew said: ‘It is a tremendous honour to receive the Abel Prize and to join the previous Laureates who have made such outstanding contributions to the field. Fermat’s equation was my passion from an early age, and solving it gave me an overwhelming sense of fulfilment. It has always been my hope that my solution of this age-old problem would inspire many young people to take up mathematics and to work on the many challenges of this beautiful and fascinating subject.’

Fermat’s Last Theorem had been widely regarded by many mathematicians as seemingly intractable. First formulated by the French mathematician Pierre de Fermat in 1637, it states:

There are no whole number solutions to the equation xn + yn = zn when n is greater than 2, unless xyz=0.

Fermat himself claimed to have found a proof for the theorem but said that the margin of the text he was making notes on was not wide enough to contain it. Sir Andrew first became fascinated with the problem as a boy. After seven years of intense study in private at Princeton University, he announced he had found a proof in 1993, combining three complex mathematical fields – modular forms, elliptic curves and Galois representations.

Sir Andrew not only solved the long-standing puzzle of the Theorem, but in doing so he created entirely new directions in mathematics, which have proved invaluable to other scientists in the years since his discovery. The Norwegian Academy of Science and Letters, which presents the Abel Prize, said in its citation: ‘Few results have as rich a mathematical history and as dramatic a proof as Fermat’s Last Theorem.’

The Vice-Chancellor of the University of Oxford, Professor Louise Richardson, said of the award: ‘The work of Oxford mathematicians lays the foundation of remarkable science – helping to address fundamental questions and enabling stunning innovation. At the same time, our mathematicians rightly remind us that they “seek truth, beauty and elegance in mathematics itself”. Very few have done so with the creativity, tenacity and sheer brilliance of Sir Andrew. The recognition he has received today is a source of immense pride to our University and we send him our warmest congratulations.’

Professor Martin Bridson, Head of Oxford’s Mathematical Institute, which is based in the Andrew Wiles Building, said: ‘I got to know Andrew in Princeton in the early 1990s and witnessed first-hand his struggle to tame his proof in the year 1993-94. The way in which he prevailed under such extraordinary pressure is the most compelling thing I have seen in my professional life. It was a joy to see how the appreciation of his triumph spread so widely beyond mathematics, to the enormous benefit of our subject, and it is a further joy to see it recognised with the award of the Abel Prize today.’

Professor Bridson added: ‘We are immensely proud to have Andrew as a colleague at the Mathematical Institute in Oxford; he is the living embodiment of the excellence that is at the core of our identity. Andrew continues to inspire current and future generations of mathematicians through his public lectures in Oxford, and the excitement he generates among school children and students is extraordinary to behold.’

Sir Andrew is still an active member of the research community at Oxford, where he is a member of the number theory research group. In his current research he is developing new ideas in the context of the Langland’s Program, a set of far-reaching conjectures connecting number theory to algebraic geometry and the theory of automorphic forms. His longer term focus is on the Birch/Swinnerton-Dyer Conjecture.

The Abel Prize is named after the Norwegian mathematician Niels Henrik Abel (1802-29). Abel himself did some of the early work on the properties of elliptical functions. Previous winners of the Prize include Britain’s Sir Michael Atiyah and the late US mathematician John Nash. It will be presented to Sir Andrew at the University of Oslo on May 24. The prize carries a cash award of 6 million NOK (about 500,000 GBP, 600,000 Euro or 700,000 USD).

Sir Andrew Wiles

Fermat’s Last Theorem

Fermat’s Last Theorem states that:

There are no whole number solutions to the equation xn + yn = zn when n is greater than 2.

The French mathematician Pierre de Fermat first expressed the theorem in the margin of a book around 1637, together with the words: ‘I have a truly marvellous demonstration of this proposition which this margin is too narrow to contain.’ Over the next three centuries, generations of mathematicians tried and failed to find the proof that Fermat claimed to have discovered. By the time Andrew Wiles was a boy, proving the theorem was generally considered well beyond the reaches of available conceptual tools.

The proof that Andrew Wiles discovered in 1994 was certainly not the one that Fermat was thinking of when he scribbled in his margin. Most people now believe that the Frenchman was mistaken in thinking he had a proof. Instead Wiles’ work builds on two concepts that were introduced to mathematics in the eighteenth and nineteenth centuries: elliptic curves and modular forms.

An elliptic curve is an equation of the form y2 = x3 + ax + b, where a and b are constants. Mathematicians began to study these equations in order to calculate the distances planets moved along their elliptical paths. By the beginning of the nineteenth century, however, they were of interest for their own properties, and were the subject of work by Niels Henrik Abel among others.

Modular forms are a more abstract kind of mathematical object. They are a certain type of mapping on a certain type of graph exhibiting an extremely high number of symmetries. Elliptic curves and modular forms were thought to have no connection to each other until the 1950s when two Japanese mathematicians, Yutaka Taniyama and Goro Shimura, as well as the French mathematician Andre Weil, first had the idea that on a deep level the fields were equivalent. They suggested that every elliptic curve could be associated with its own modular form, a claim known as the Taniyama-Shimura conjecture, a radical proposition which no one had any idea how to prove.

In 1984 the German mathematician Gerhard Frey first linked Fermat’s Last Theorem to the Taniyama-Shimura conjecture. Frey showed that if you assume Fermat to be false, you can create an elliptic curve so weird that it seems to have no associated modular form. Two years later the American Ken Ribet proved that Frey’s hunch was correct: if Fermat’s Last Theorem is false, there is an elliptic curve that has no associated modular form. In other words, if Fermat is false, the Taniyama-Shimura conjecture is also false.

Frey and Ribet’s work revealed that all that was needed for a proof of Fermat’s Last Theorem was a proof of the Taniyama-Shimura conjecture. Andrew Wiles, a specialist both in elliptic curves – the subject of his PhD – and modular forms, realised he had the right background to engage with the problem. After eight intense years of study, he proved that a restricted case of the Taniyama-Shimura Conjecture was true, which included the case that would imply the truth of Fermat’s Last Theorem.

The Wiles proof is considered one of the greatest triumphs of contemporary mathematics. In outline, it shows that each elliptic curve has a sequence of numbers that defines it, as does each modular form. Wiles then showed that every sequence belonging to an elliptic curve could be exactly matched with the sequence belonging to a modular form. To do this he devised a toolkit based on the work of the 19th-century mathematician Évariste Galois, who discovered the symmetries that arise from the solutions of certain equations.

The impact of Wiles’ work on mathematics has been immense. He demonstrated a fundamental structural connection between elliptic curves and modular forms, a rich and important result within number theory with many profound consequences. He also devised a powerful conceptual toolkit that has been used over the past two decades by other mathematicians in many spectacular and significant ways.

The Abel Prize

The Abel Prize is an international award for outstanding scientific work in the field of mathematics, including mathematical aspects of computer science, mathematical physics, probability, numerical analysis, scientific computing, statistics, and also applications of mathematics in the sciences.

The Norwegian Academy of Science and Letters awards the Abel Prize based upon recommendations from the Abel Committee. The Prize is named after the exceptional Norwegian mathematician Niels Henrik Abel (1802–1829). According to the statutes of the Abel Prize, the objective is both to award the annual Abel Prize, and to contribute towards raising the status of mathematics in society and stimulating the interest of children and young people in mathematics. The prize carries a cash award of 6 million NOK (about 500,000 GBP, 600,000 Euro or 700,000 USD) and was first awarded in 2003. Among initiatives supported are the Abel Symposium, the International Mathematical Union’s Commission for Developing Countries, the Abel Conference at the Institute for Mathematics and its Applications in Minnesota, and The Bernt Michael Holmboe Memorial Prize for excellence in teaching mathematics in Norway. In addition, national mathematical contests, and various other projects and activities are supported in order to stimulate interest in mathematics among children and youth.

2016 Abel Prize – full citation

The Norwegian Academy of Science and Letters has decided to award the Abel Prize for 2016 to Sir Andrew J. Wiles, University of Oxford ‘for his stunning proof of Fermat’s Last Theorem by way of the modularity conjecture for semistable elliptic curves, opening a new era in number theory’.

Number theory, an old and beautiful branch of mathematics, is concerned with the study of arithmetic properties of the integers. In its modern form the subject is fundamentally connected to complex analysis, algebraic geometry, and representation theory. Number theoretic results play an important role in our everyday lives through encryption algorithms for communications, financial transactions, and digital security.

Fermat’s Last Theorem, first formulated by Pierre de Fermat in the 17th century, is the assertion that the equation xn + yn = zn has no solutions in positive integers for n>2. Fermat proved his claim for n=4, Leonhard Euler found a proof for n=3, and Sophie Germain proved the first general result that applies to infinitely many prime exponents. Ernst Kummer’s study of the problem unveiled several basic notions in algebraic number theory, such as ideal numbers and the subtleties of unique factorization. The complete proof found by Andrew Wiles relies on three further concepts in number theory, namely elliptic curves, modular forms, and Galois representations.

Elliptic curves are defined by cubic equations in two variables. They are the natural domains of definition of the elliptic functions introduced by Niels Henrik Abel. Modular forms are highly symmetric analytic functions defined on the upper half of the complex plane, and naturally factor through shapes known as modular curves. An elliptic curve is said to be modular if it can be parametrized by a map from one of these modular curves. The modularity conjecture, proposed by Goro Shimura, Yutaka Taniyama, and André Weil in the 1950s and 60s, claims that every elliptic curve defined over the rational numbers is modular.

In 1984, Gerhard Frey associated a semistable elliptic curve to any hypothetical counterexample to Fermat’s Last Theorem, and strongly suspected that this elliptic curve would not be modular. Frey’s nonmodularity was proven via Jean-Pierre Serre’s epsilon conjecture by Kenneth Ribet in 1986. Hence, a proof of the Shimura-Taniyama-Weil modularity conjecture for semistable elliptic curves would also yield a proof of Fermat’s Last Theorem. However, at the time the modularity conjecture was widely believed to be completely inaccessible. It was therefore a stunning advance when Andrew Wiles, in a breakthrough paper published in 1995, introduced his modularity lifting technique and proved the semistable case of the modularity conjecture.

The modularity lifting technique of Wiles concerns the Galois symmetries of the points of finite order in the abelian group structure on an elliptic curve. Building upon Barry Mazur’s deformation theory for such Galois representations, Wiles identified a numerical criterion which ensures that modularity for points of order p can be lifted to modularity for points of order any power of p, where p is an odd prime. This lifted modularity is then sufficient to prove that the elliptic curve is modular. The numerical criterion was confirmed in the semistable case by using an important companion paper written jointly with Richard Taylor. Theorems of Robert Langlands and Jerrold Tunnell show that in many cases the Galois representation given by the points of order three is modular. By an ingenious switch from one prime to another, Wiles showed that in the remaining cases the Galois representation given by the points of order five is modular. This completed his proof of the modularity conjecture, and thus also of Fermat’s Last Theorem.

The new ideas introduced by Wiles were crucial to many subsequent developments, including the proof in 2001 of the general case of the modularity conjecture by Christophe Breuil, Brian Conrad, Fred Diamond, and Richard Taylor. As recently as 2015, Nuno Freitas, Bao V. Le Hung, and Samir Siksek proved the analogous modularity statement over real quadratic number fields. Few results have as rich a mathematical history and as dramatic a proof as Fermat’s Last Theorem.