The Laureate 2016

Sir Andrew J. Wiles ,

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 for semistable elliptic curves, opening a new era in 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 , such as ideal 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 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, , , 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 . points of finite order in the abelian mathematical history and as dramatic a opening a new era in Modular forms are highly symmetric analytic structure on an . Building upon proof as Fermat’s Last Theorem. number theory.” functions defined on the upper half of the ’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 , 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 , Yutaka modularity is then sufficient to prove that geometry, and . 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, associated a paper written jointly with Richard Taylor. transactions, and digital security. semistable elliptic curve to any hypothetical Theorems of and Jerrold Fermat’s Last Theorem, first formulated counterexample to Fermat’s Last Theorem, Tunnell show that in many cases the Galois by 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 – 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 ’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 , 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 . 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 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 , 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 . 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 , 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 , 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 . 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 . 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. He extremely unusual to go back and fix

8 9 A glimpse of the Laureate’s work But why is the statement true for n≥3? Is Then, in the mid 1980s, the German there a mysterious connection between Gerhard Frey asserted that Hanc marginis powers and sums of powers? Or are there if TSW was true, then FLT would follow as just too few integers? a consequence. Frey suggested that if FLT exiguitas non caperet Among the first 10,000 numbers was false, then there would exist a semi- there are 2,691 sums of two squares, 100 stable elliptic curve that was not modular. Arne B. Sletsjøe squares and 42 numbers that are both a However, the TSW conjecture says just the square and a sum of squares. In contrast, opposite, that all elliptic curves are modular. there are only 202 sums of two cubes, 21 So when a few years later cubes and, according to FLT, none of these proved Frey’s assertion, the only obstacle are sums of cubes. The two properties, to proving FLT was to prove the TSW being a sum of cubes and being a cube conjecture. Many experts considered this “Cubum autem in duos cubos, all attempts to find a rigorous proof of the itself, are so rare that it is unlikely that to be a challenge for the distant future. But aut quadratoquadratum in duos statement have failed. However, it was not any number would be both. Nevertheless, Andrew Wiles dug into the problem anyway, quadratoquadratos et generaliter nullam all in vain: a large body of knowledge has according to Wiles’ work, the reason for the and within the next seven years he came in infinitum ultra quadratum potestatem been created on the roads into the many lack of concurrence between powers and up with a proof. He kept his discoveries in duos eiusdem nominis fas est blind alleys, and on the one successful road sums of power is much more subtle. hidden from the mathematical community, dividere cuius rei demonstrationem toward the final solution. In the 1950s, two young Japanese but during a conference in Cambridge in mirabilem sane detexi. Hanc marginis Today very few people believe that mathe­maticians, Yutaka Taniyama and Goro the summer of 1993 there were rumours of exiguitas non caperet.” Fermat actually had a proof of the theorem. Shimura, were studying certain sequences an upcoming sensation. Tension built up, Pierre de Fermat One can be fairly certain that it would of numbers. They considered the number and the number of curious colleagues in the have been extremely difficult or even of solutions of a type of equations, called audience increased during the lecture series (It is impossible to separate a cube into two impossible to provide a complete proof elliptic curves, and compared them to Wiles gave. In his final lecture he concluded cubes, or a fourth power into two fourth of the assertion using the mathematical specific expressions of a class of functions, that Fermat’s Last Theorem had finally powers, or in general, any power higher than tools and techniques on hand in the 17th called modular forms. Taniyama and got a proof. the second, into two like powers. I have century. Even for a brilliant mathematician Shimura discovered that the sequences of The proof of TSW is a major discovered a truly marvellous proof of this, like Pierre de Fermat, the proof of FLT numbers were very similar and concluded mathematical masterpiece and is not easily which this margin is too narrow to contain.) given by Andrew Wiles would have been that this could not be a coincidence. accessible if you are not a true expert. highly inaccessible had he been given the They conjectured that there was a deeper Wiles writes in his This is the famous comment, written by possibility to read it. In 1637 it would still connection between elliptic curves and article from 1995 where he presents the Pierre de Fermat around 1637 in his copy be centuries before the concepts of elliptic modular forms, producing two identical proof: “Let f be an eigenform associated to of ’ book Arithmetica. Fermat curves and modular forms were to emerge. sequences of numbers in apparently the congruence subgroup Γ1(N) of SL2(Z) was a French lawyer with a passion for Today Fermat’s marginal comment different mathematical subfields. About 10 of weight k≥2 and character c. Thus if Tn is mathematics. The statement he refers to, is phrased: years later these ideas were considered the associated to an integer n n n known as Fermat’s Last Theorem (it was The equation x +y =z , where n≥3 has in a publication by the influential French n there is an algebraic integer c(n,f) such no non-trivial integer solutions. not his last assertion, but the last one to mathematician André Weil. The conjecture that Tnf=c(n,f)f for each n. We let Kf be the be proved), or just FLT, is one of history’s Notice that Fermat requires the exponent was promptly hailed as hot stuff, now under number field generated over Q by ...” the longest lasting puzzles, easy to formulate n to be greater or equal to 3. For n=2 the the name of the Taniyama-Shimura-Weil proof is truly marvellous, but unfortunately – but equally difficult to crack. Numerous statement is false, since the equation conjecture (or TSW for short). In spite of Hanc marginis exiguitas non caperet. 2 2 2 great mathematical thinkers have taken up x +y =z has many non-trivial integer numerous attempts to crack the puzzle, no 2 2 2 the challenge, but for more than 350 years solutions, the most famous being 3 +4 =5 . one managed to come up with a proof.

10 11 12 13 About the Abel Prize

The Abel Prize is an international award Michael Holmboe Memorial Prize for for outstanding scientific work in the field excellence in teaching mathematics in of mathematics, including mathematical . In addition, national mathematical aspects of computer science, mathematical contests, and various other projects and physics, probability, numerical analysis, activities are supported in order to stimulate scientific computing, statistics, and also interest in mathematics among children and applications of mathematics in the sciences. youth. The Norwegian Academy of Science and Letters awards the Abel Prize based — upon recommendations from the Abel Committee. The Prize is named after the Call for nominations 2016: exceptional Norwegian mathematician Niels The Norwegian Academy of Science and Henrik Abel (1802–1829). According to the Letters hereby calls for nominations for statutes of the Abel Prize, the objective is the Abel Prize 2017, and invite you (or both to award the annual Abel Prize, and your society or institution) to nominate to contribute towards raising the status of candidate(s). Nominations are confidential mathematics in society and stimulating the and a nomination should not be made interest of children and young people in known to the nominee. mathematics. The prize carries a cash award of 6 million NOK (about 600,000 Euro or Deadline for nominations for the Abel Prize about 700,000 USD) and was first awarded 2017 is September 15, 2016. Please consult in 2003. Among initiatives supported are www.abelprize.no for more information. 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

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The Abel Prize Laureates

2015 John Forbes Nash, Jr. and “for striking and seminal contributions to the theory of nonlinear partial differential equations and its applications to .”

Vigeland’s Abel monument in the Palace Park in .

2014 2013 2012 2011 2010 2009 Yakov G. Sinai Endre Szemerédi John Torrence Tate Mikhail Leonidovich Gromov “for his fundamental contributions “for seminal contributions to “for his fundamental contributions “for pioneering discoveries in “for his vast and lasting impact to dynamical systems, ergodic and for their to dis­crete mathematics and topology, geometry and algebra.” on the theory of numbers.” “for his revolutionary contributions theory, and mathematical transformative impact on number theoretical computer science, to geometry.” physics.” theory, representation theory, and in recognition of the profound and related fields.” and lasting impact of these contributions on additive number theory and .”

2008 2007 2006 2005 2004 2003 John Griggs Thompson Srinivasa S. R. Varadhan Peter D. Lax Sir Michael Francis Atiyah Jean-Pierre Serre and and Isadore M. Singer “for his fundamental contributions “for his profound and seminal “for his groundbreaking “for playing a key role in shaping “for their profound achievements to and in contributions to harmonic contributions to the theory and “for their discovery and proof the modern form of many parts in algebra and in particular for particular for creating a unified analysis and the theory of application of partial differential of the index theorem, bringing of mathemat­ics, including shaping modern .” theory of large deviations.” smooth dynamical systems.” equations and to the computation together topology, geometry topology, algebraic geometry of their solutions.” and analysis, and their outstand ­ and number theory.” ing role in building new bridges between mathematics and theoretical physics.”

18 19 Programme Abel Week 2016

May 23 Abel Banquet at Akershus Castle Front page photo: © in honor of the Abel Laureate — Hosted by the Norwegian Government Holmboe Prize Award Ceremony (by invitation only) The Minister of Education and Research presents the Bernt Michael Holmboe Alain Goriely Memorial Prize for teachers of mathematics at Oslo katedralskole May 25 — — The Abel Lectures Wreath-laying at the Abel Monument Laureate Lecture, Science Lecture, and by the Abel Prize Laureate in the other lectures in the field of the Laureate’s Palace Park work at Georg Sverdrups Hus, Aud. 1, May 24 — The Abel Party — at The Norwegian Academy of Science Abel Prize Award Ceremony and Letters (by invitation only) His Royal Highness Crown Prince Haakon presents the Abel Prize in the University Aula, University of Oslo — May 26 — Reception and interview Abel day for school children at the with the Abel Laureate University of Agder, Kristiansand TV host and journalist Nadia Hasnaoui Activities for middle school children, interviews the Abel Laureate at and Laureate lecture at the Det Norske Teatret University of Agder

The Norwegian Academy Press contact: For other information: of Science and Letters Anne-Marie Astad Trine Gerlyng [email protected] [email protected] +47 22 84 15 12 +47 415 67 406

Register online at: www.abelprize.no from mid-April, or contact [email protected], facebook.com/Abelprize