Potential automorphy over CM fields
The Langlands reciprocity conjecture predicts that "nice" Galois representations come from automorphic forms. Potential automorphy is a slight weakening of this, which still suffice for many arithmetic applications, a famous example being the proof of the Sato-Tate conjecture for elliptic curves over totally real fields by Clozel-Harris-Shepherd-Barron-Taylor (and others). I will discuss the difficulties that arise when one tries to do the same for elliptic curves over imaginary CM fields, and how to bypass them. This is joint work with P. Allen, F. Calegari, A. Caraiani, T. Gee, D. Helm, J. Newton, P. Scholze, R. Taylor and J. Thorne.