Modularity of Galois representations, from Ramanujan to Serre's conjecture and beyond

Abstract: 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’'.

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