Potential automorphy of some non-self dual Galois representations

Potential automorphy of some non-self dual Galois representations

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Richard Taylor , IAS
Fine Hall 214

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 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.