"MODULARITY OF GALOIS REPRESENTATIONS, FROM RAMANUJAN TO SERRE’S CONJECTURE"
Prof. Dr. Chandrashekhar Khare
(University of California, LA)
Thursday, July 7, 2022, 16:45h
Mathematikon, Hörsaal, Im Neuenheimer Feld 205
Ramanujan made a series of influential conjectures in his
1916 paper »On some arithmetical functions« on what is now called
the Ramanujan τ-function. A congruence Ramanujan observed
for τ(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.