Chandrashekhar Khare, UCLA
Colloquium Colloque des sciences math茅matiques du Qu茅bec
Title:听Modularity of Galois representations, from Ramanujan to Serre's conjecture and beyond.
础产蝉迟谤补肠迟:听Ramanujan made a series of influential conjectures in his 1916 paper ``On some arithmetical functions鈥�' on what is now called the Ramanujan $ au$ function. A congruence Ramanujan observed for $ au(n)$ modulo 691 in the paper led to Serre and Swinnerton-Dyer developing a geometric theory of mod $p$ modular forms. 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 describe the path from Ramanujan's work in 1916, to the formulation of a first version of Serre's conjecture in 1973, to its resolution in 2009 by Jean-Pierre Wintenberger and myself. I will also try to indicate why this subject is very much alive and, in spite of all the progress, still in its infancy.