Potential automorphy of some non-self dual Galois representations

Joint IAS/Princeton University Number Theory Seminar
Topic:Potential automorphy of some non-self dual Galois representations
Speaker:Richard Taylor
Affiliation:Robert and Luisa Fernholz Professor, School of Mathematics
Date:Thursday, May 18
Time/Room:4:30pm - 5:30pm/Fine 214, Princeton University

The IAS recently organized an Emerging Topics Working Group on "Applications to modularity of recent progress on the cohomology of Shimura varieties". Based on ideas of Calegari and Geraghty, and using recent results of Khare-Thorne and Caraiani-Scholze, the group (Allen, F.Calegari, Caraiani, Gee, Helm, Le Hung, Newton, Scholze, Thorne and myself) were able to prove modularity lifting theorems for n-dimensional Galois representations over CM (or totally real) fields without a (conjugate) self-duality hypothesis. Most notably our theorems apply in the 'non-minimal' case. As applications we prove that elliptic curves over CM fields become modular after a finite base change. We also prove that cohomological (for trivial coefficients) automorphic forms on $\mathrm{GL}(2)$ over a CM field satisfy the Ramanujan conjecture. We are not able to reduce the Ramanujan conjecture to the Weil conjectures, rather we deduce it from the potentially automorphy of symmetric powers.