The Abel Prize Laureate 2016 Sir Andrew J. Wiles University of Oxford, England www.abelprize.no Sir Andrew J. Wiles receives the Abel Prize for 2016 “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.” Citation The Abel Committee The Norwegian Academy of Science and unveiled several basic notions in algebraic However, at the time the modularity The new ideas introduced by Wiles were Letters has decided to award the Abel Prize number theory, such as ideal numbers and conjecture was widely believed to be crucial to many subsequent developments, for 2016 to the subtleties of unique factorization. The completely inaccessible. It was therefore a including the proof in 2001 of the general complete proof found by Andrew Wiles stunning advance when Andrew Wiles, in case of the modularity conjecture by Sir Andrew J. Wiles, Mathematical Institute, relies on three further concepts in number a breakthrough paper published in 1995, Christophe Breuil, Brian Conrad, Fred University of Oxford theory, namely elliptic curves, modular introduced his modularity lifting technique Diamond, and Richard Taylor. As recently forms, and Galois representations. and proved the semistable case of the as 2015, Nuno Freitas, Bao V. Le Hung, “for his stunning proof of Elliptic curves are defined by cubic modularity conjecture. and Samir Siksek proved the analogous Fermat’s Last Theorem by way equations in two variables. They are the The modularity lifting technique of Wiles modularity statement over real quadratic of the modularity conjecture natural domains of definition of the elliptic concerns the Galois symmetries of the number fields. Few results have as rich a for semistable elliptic curves, functions introduced by Niels Henrik Abel. points of finite order in the abelian group mathematical history and as dramatic a opening a new era in Modular forms are highly symmetric analytic structure on an elliptic curve. Building upon proof as Fermat’s Last Theorem. number theory.” functions defined on the upper half of the Barry Mazur’s deformation theory for such complex plane, and naturally factor through Galois representations, Wiles identified Number theory, an old and beautiful branch shapes known as modular curves. An elliptic a numerical criterion which ensures that of mathematics, is concerned with the study curve is said to be modular if it can be modularity for points of order p can be lifted of arithmetic properties of the integers. In its parametrized by a map from one of these to modularity for points of order any power modern form the subject is fundamentally modular curves. The modularity conjecture, of p, where p is an odd prime. This lifted connected to complex analysis, algebraic proposed by Goro Shimura, Yutaka modularity is then sufficient to prove that geometry, and representation theory. Taniyama, and André Weil in the 1950s and the elliptic curve is modular. The numerical Number theoretic results play an important 60s, claims that every elliptic curve defined criterion was confirmed in the semistable role in our everyday lives through encryption over the rational numbers is modular. case by using an important companion algorithms for communications, financial In 1984, Gerhard Frey associated a paper written jointly with Richard Taylor. transactions, and digital security. semistable elliptic curve to any hypothetical Theorems of Robert Langlands and Jerrold Fermat’s Last Theorem, first formulated counterexample to Fermat’s Last Theorem, Tunnell show that in many cases the Galois by Pierre de Fermat in the 17th century, is and strongly suspected that this elliptic representation given by the points of order the assertion that the equation xn + yn = zn curve would not be modular. Frey’s non- three is modular. By an ingenious switch has no solutions in positive integers for n>2. modularity was proven via Jean-Pierre from one prime to another, Wiles showed Fermat proved his claim for n=4, Leonhard Serre’s epsilon conjecture by Kenneth Ribet that in the remaining cases the Galois Euler found a proof for n=3, and Sophie in 1986. Hence, a proof of the Shimura- representation given by the points of order Germain proved the first general result that Taniyama-Weil modularity conjecture five is modular. This completed his proof of applies to infinitely many prime exponents. for semistable elliptic curves would also the modularity conjecture, and thus also of Ernst Kummer’s study of the problem yield a proof of Fermat’s Last Theorem. Fermat’s Last Theorem. 4 5 © Abelprisen/DNVA/Calle Huth © Alain Goriely decided that he would return to the an error like this, because of the mental A biography problem that so excited him as a child. “The exhaustion from trying it the first time challenge proved irresistible,” he said. around. No gaps were found in the revised of Andrew Wiles Wiles made the unusual choice to work proof and it was published in Annals of on Fermat alone, rather than collaborating Mathematics in 1995, with the title Modular Alexander Bellos with colleagues. Since the problem was so elliptic curves and Fermat’s Last Theorem. famous, he was worried that news he was As well as the attention of the global working on it would attract too much media, Wiles received many awards. Andrew Wiles is one of the very few connection with measuring the lengths attention and he would lose focus. The only They include the Rolf Schock Prize, the mathematicians – if not the only – whose of planetary orbits. Together they made person he confided in was his wife, Nada, Ostrowski Prize, the Wolf Prize, the Royal proof of a theorem has been international the first progress on one of the field’s who he married shortly after embarking on Medal of the Royal Society, the U.S. headline news. In 1994 he cracked Fermat’s fundamental conjectures, the Birch and the proof. National Academy of Science’s Award in Last Theorem, which at the time was the Swinnerton-Dyer conjecture, proving it for After seven years of intense and secret Mathematics, and the Shaw Prize. The most famous, and long-running, unsolved certain special cases. Wiles was awarded study, Wiles believed he had a proof. He International Mathematical Union presented problem in the subject’s history. his PhD in 1980 for the thesis Reciprocity decided to go public during a lecture series him with a silver plaque, the only time they Wiles’ proof was not only the high point laws and the conjecture of Birch and at a seminar in Cambridge, England. He have ever done so. He was awarded the of his career – and an epochal moment for Swinnerton-Dyer. did not announce it beforehand. The title of inaugural Clay Research Award. In 2000 he mathematics – but also the culmination of Between 1977 and 1980 Wiles was an his talk, Modular Forms, Elliptic Curves and was given a knighthood. a remarkable personal journey that began Assistant Professor at Harvard University, Galois Representations, gave nothing away, Wiles was at Princeton between 1982 three decades before. In 1963, when he where he started to study modular forms, although rumour had spread around the and 2010, except for short periods of leave. was a ten-year-old boy growing up in a separate field from elliptic curves. There mathematical community and two hundred In 2010 he returned to Oxford as a Royal Cambridge, England, Wiles found a copy he began a collaboration with Barry Mazur, people were packed in the lecture theatre to Society Research Professor. His address of a book on Fermat’s Last Theorem in his which resulted in their 1984 proof of the hear him. When he wrote the theorem up as at the Mathematical Institute is the Andrew local library. He became captivated by the main conjecture of Iwasawa theory, a field the conclusion to the talk, the room erupted Wiles Building, which opened in 2013 and problem – that there are no whole number within number theory. In 1982 he was made in applause. was named in his honour. solutions to the equation xn + yn = zn when a professor at Princeton University. Later that year, however, a referee n is greater than 2 – which was easy During the early years of Wiles’ checking the details of his proof found an Sources: to understand but which had remained academic career he was not actively trying error in it. It was devastating for Wiles to Fermat’s Last Theorem by Simon Singh. Wikipedia unsolved for three hundred years. “I knew to solve Fermat’s Last Theorem – nor was contemplate the idea that he had not, in Notices of the AMS from that moment that I would never let it anyone else, since the problem was fact, solved Fermat’s Last Theorem. He set Shawprize.org go,” he said. “I had to solve it.” generally regarded as too difficult, and to work trying to fix the issue, enlisting one BBC Horizon. Wiles studied mathematics at possibly unsolvable. A turning point came of his former students, Richard Taylor, to Merton College, Oxford, and returned in 1986 when it was shown that the three- help him with the task. After a year’s work, to Cambridge, at Clare College, for century-old problem could be rephrased Wiles found a way to correct the error. postgraduate studies. His research area using the mathematics of elliptic curves and “I had this incredible revelation,” a tearful was number theory, the mathematical field modular forms. It was an amazing twist of Wiles told a BBC documentary. “It was the that investigates the properties of numbers. fate that two subjects that Wiles had most important moment of my working life.” Under the guidance of his advisor John specialized in turned out to be exactly the Not only is it rare to announce the Coates, Wiles studied elliptic curves, a areas that were needed to tackle Fermat’s proof of a famous theorem, but it is also type of equation that was first studied in Last Theorem with modern tools.