Fermat, Taniyama–Shimura–Weil and Andrew Wiles John Rognes UniversityofOslo,Norway May 13th and 20th 2016 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. SirAndrewJ.Wiles Sketch proof of Fermat’s Last Theorem: (cid:73) Frey (1984): A solution ap +bp = cp to Fermat’s equation gives an elliptic curve y2 = x(x −ap)(x +bp). (cid:73) Ribet (1986): The Frey curve does not come from a modular form. (cid:73) Wiles (1994): Every elliptic curve comes from a modular form. (cid:73) Hence no solution to Fermat’s equation exists. Point counts and Fourier expansions: Elliptic curve Hasse–Weil (cid:40)(cid:40) L(cid:54)(cid:54) -function Mellin Modular form Modularity: Elliptic curve (cid:40)(cid:40) ◦ ? L(cid:54)(cid:54) -function (cid:15)(cid:15) Modular form Wiles’ Modularity Theorem: Semistable elliptic curve defined over Q (cid:41)(cid:41) Wiles ◦ (cid:53)(cid:53)L-function (cid:15)(cid:15) Weight 2 modular form Wiles’ Modularity Theorem: Semistable elliptic curve over Q of conductor N (cid:41)(cid:41) Wiles ◦ (cid:53)(cid:53)L-function (cid:15)(cid:15) Weight 2 modular form of level N Frey Curve (and a special case of Wiles’ theorem): Solution to Fermat’s equation Frey (cid:15)(cid:15) Semistable elliptic curve over Q with peculiar properties (cid:41)(cid:41) Wiles ◦ (cid:53)(cid:53)L-function (cid:15)(cid:15) Weight 2 modular form with peculiar properties (A special case of) Ribet’s theorem: Solution to Fermat’s equation Frey (cid:15)(cid:15) Semistable elliptic curve over Q with peculiar properties (cid:42)(cid:42) Wiles ◦ (cid:52)(cid:52)L-function (cid:15)(cid:15) Weight 2 modular form with peculiar properties (cid:79)(cid:79) Ribet Weight 2 modular form of level 2