Modularity of Galois representations, from Ramanujan to Fermat’s Last Theorem
July 7 at 17:15 – 18:30 CEST
9. Emil Artin Lecture
Prof. Dr. Chandrashekhar Khare, Department of Mathematics, UCLA
Ramanujan made a series of conjectures in his 1916 paper “On some arithmetical functions’’ on what is now called the Ramanujan $\tau$ function. Part of these conjectures were proved soon after Ramanujan formulated them by Mordell, while one of his conjectures (which is now the first of a vast web of conjectures in the theory of automorphic forms) took almost 6 decades to be settled (in work of Deligne). A congruence Ramanujan observed for $\tau(n)$ modulo 691 in the same paper, led to Serre and Swinnerton-Dyer developing a geometric theory of mod $p$ modular forms to explain some of Ramanujan’s observations. It was in the context of the theory of mod $p$ modular forms that Serre made his modularity conjecture, which was initially formulated in a letter of Serre to Tate in 1973.
I will narrate this story, starting from Ramanujan’s work in 1916, to the formulation of Serre’s conjecture in 1973, to its resolution in 2009 by Jean-Pierre Wintenberger and myself (using as a key ingredient the modularity lifting method developed by Wiles in his proof of Fermat’s Last Theorem). I will also try to indicate why this subject is very much alive and in spite of all the progress still in its infancy.