Potential automorphy of some compatible systems over CM fields

Potential automorphy of some compatible systems over CM fields - Patrick Allen

Patrick Allen
University of Illinois at Urbana-Champaign; Member, School of Mathematics
November 7, 2017

Abstract:  We will discuss joint work with Calegari, Caraiani, Gee, Helm, Le Hung, Newton, Scholze, Taylor, and Thorne that proves potential automorphy of symmetric powers of rank two compatible systems of weight zero.  As a consequence, we deduce the Sato-Tate conjecture for elliptic curves over CM fields a swell as the Ramanujan conjecture for weight zero cohomological automorphic representations of GL(2) over CM fields.  Our proof follows the framework of Calegari and Geraghty.  To implement it unconditionally, we establish cases of local-global compatibility for Galois representations attached to torsion classes, the result at p relying on forthcoming work of Caraiani-Scholze, and a derived version of Taylor's Ihara avoidance.