The Norwegian Academy of Science and Letters has awarded the 2016 Abel Prize to Sir Andrew J. Wiles of the 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.” The Abel Prize is awarded by the Norwegian Academy of Science and Letters. It recognizes contributions of extraordinary depth and influence to the mathematical sciences and has been awarded annually since 2003. It carries a cash award of 6,000,000 Norwegian krone (approximately US$700,000).
Number theory, an old and beautiful branch of mathematics,is concerned with the study of arithmetic properties ofthe integers. In its modern form the subject is fundamentallyconnected to complex analysis, algebraic geometry,and representation theory. Number theoretic results playan important role in our everyday lives through encryptionalgorithms for communications, financial transactions,and digital security.
Fermat’s Last Theorem, first formulated by Pierre deFermat in the seventeenth century, is the assertion thatthe equation x^n+y^n=z^n has no solutions in positive integersfor n>2. Fermat proved his claim for n=4, Leonhard Eulerfound a proof for n=3, and Sophie Germain proved thefirst general result that applies to infinitely many primeexponents. Ernst Kummer’s study of the problem unveiledseveral basic notions in algebraic number theory, such asideal numbers and the subtleties of unique factorization.The complete proof found by Andrew Wiles relies on threefurther concepts in number theory, namely elliptic curves,modular forms, and Galois representations.
Elliptic curves are defined by cubic equations in twovariables. They are the natural domains of definition ofthe elliptic functions introduced by Niels Henrik Abel.Modular forms are highly symmetric analytic functionsdefined on the upper half of the complex plane, andnaturally factor through shapes known as modular curves.An elliptic curve is said to be modular if it can be parametrizedby a map from one of these modular curves. Themodularity conjecture, proposed by Goro Shimura, YutakaTaniyama, and Andre Weil in the 1950s and 1960s, claimsthat every elliptic curve defined over the rational numbersis modular.
In 1984, Gerhard Frey associated a semistable ellipticcurve to any hypothetical counterexampleto Fermat’s Last Theorem, and stronglysuspected that this elliptic curve wouldnot be modular. Frey’s nonmodularity wasproven via Jean-Pierre Serre’s epsilon conjectureby Kenneth Ribet in 1986. Hence, aproof of the Shimura-Taniyama-Weil modularityconjecture for semistable ellipticcurves would also yield a proof of Fermat’sLast Theorem. However, at the time the modularity conjecturewas widely believed to be completely inaccessible. Itwas therefore a stunning advance when Andrew Wiles, ina breakthrough paper published in 1995, introduced hismodularity lifting technique and proved the semistablecase of the modularity conjecture.
The modularity lifting technique of Wiles concernsthe Galois symmetries of the points of finite order in theabelian group structure on an elliptic curve. Building uponBarry Mazur’s deformation theory for such Galois representations,Wiles identified a numerical criterion whichensures that modularity for points of order p can be liftedto modularity for points of order any power of p, where pis an odd prime. This lifted modularity is then sufficientto prove that the elliptic curve is modular. The numericalcriterion was confirmed in the semistable case by using animportant companion paper written jointly with RichardTaylor. Theorems of Robert Langlands and Jerrold Tunnellshow that in many cases the Galois representation givenby the points of order three is modular. By an ingeniousswitch from one prime to another, Wiles showed that inthe remaining cases the Galois representation given by thepoints of order five is modular. This completed his proofof the modularity conjecture, and thus also of Fermat’s Last Theorem.
The new ideas introduced by Wiles were crucial to manysubsequent developments, including the proof in 2001 ofthe general case of the modularity conjecture by Christophe Breuil, Brian Conrad, Fred Diamond, and RichardTaylor. As recently as 2015, Nuno Freitas, Bao V. Le Hung,and Samir Siksek proved the analogous modularity statementover real quadratic number fields. Few results haveas rich a mathematical history and as dramatic a proof asFermat’s Last Theorem.
* News from Notices of the AMS(http://dx.doi.org/10.1090/noti1386)