Chandrashekhar Khare
UCLA
October 6, 2022
Modularity of Galois representations, from Ramanujan to Serre's conjecture and beyond: Ramanujan made a series of influential conjectures in his 1916 paper ``On some arithmetical functions'' on what is now called the Ramanujan $\tau$ function. A congruence Ramanujan observed for $\tau(n)$ 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.
UCLA
October 6, 2022
Modularity of Galois representations, from Ramanujan to Serre's conjecture and beyond: Ramanujan made a series of influential conjectures in his 1916 paper ``On some arithmetical functions'' on what is now called the Ramanujan $\tau$ function. A congruence Ramanujan observed for $\tau(n)$ 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. I will end with some questions about counting mod $p$ Galois representations, and the use of Serre's conjecture in the`` computational Langlands program''.