9. Emil Artin Vorlesung
Modularity of Galois representations, from Ramanujan to Serre's Conjecture

Ramanujan made a series of conjectures in his 1916 paper ``On some arithmetical functions'' on what is now called the Ramanujan $\tau$ function. Part of these conjectures were proved soon after Ramanujan formulated them by Mordell, while one of his conjectures (which is now the first of a vast web of conjectures in the theory of automorphic forms) took almost 6 decades to be settled (in work of Deligne). A congruence Ramanujan observed for $\tau(n)$ modulo 691 in the same paper, led to Serre and Swinnerton-Dyer developing a geometric theory of mod $p$ modular forms to explain some of Ramanujan's observations. 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 narrate this story, starting from Ramanujan's work in 1916, to the formulation of Serre's conjecture in 1973, to its resolution in 2009 by Jean-Pierre Wintenberger and myself (using as a key ingredient the modularity lifting method developed by Wiles in his proof of Fermat's Last Theorem). I will also try to indicate why this subject is very much alive and in spite of all the progress still in its infancy.

Donnerstag, den 7. Juli 2022 um 17.15 Uhr, in INF205, HS Mathematikon Donnerstag, den 7. Juli 2022 at 17.15, in INF205, HS Mathematikon

Der Vortrag folgt der Einladung von The lecture takes place at invitation by Prof. Dr. O. Venjakob