Special Events Seminar

Calegari and Geraghty described an approach to modularity lifting theorems in the setting of locally symmetric spaces, where the basic numerology of the Taylor-Wiles method seemed to break down. A group of 10 mathematicians (Allen, Calegari, Caraiani, Gee, Helm, Le Hung, Newton, Scholze, Thorne and myself) were recently able to get this approach to work, the key ingredient being to systematically work in a derived framework. As applications we were able to prove the meromorphic continuation and functional equation of the L-series of elliptic curves over CM fields and to prove the Ramanujan conjecture for the action of Hecke operators on the cohomology of arithmetic hyperbolic three manifolds. I will describe these results and give an outline of the proof.

Sponsored by the Rainich Lecture Series Speaker(s): Richard Taylor (Institute for Advanced Study)