Quebec Mathematical Sciences Colloquium

September 16, 2022 from 15:30 to 16:30 (Montreal/EST time)

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

Colloquium presented by Chandrashekhar Khare (UCLA)

Ramanujan made a series of influential conjectures in his 1916 paper ``On some arithmetical functions’' on what is now called the Ramanujan τ\tauτ function. A congruence Ramanujan observed for τ(n)\tau(n)τ(n) modulo 691 in the paper led to Serre and Swinnerton-Dyer developing a geometric theory of mod ppp modular forms. It was in the context of the theory of mod ppp 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.


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