Appears in collections : A Conference in Arithmetic Algebraic Geometry in Memory of Jan Nekovář, ECM 2024 Invited Speakers

I'll discuss potential automorphy results which imply the Ramanujan and Sato-Tate conjectures for regular algebraic cuspidal automorphic representations of GL(2) over an imaginary quadratic field. This generalizes results in the "10 author paper" which restricted to cohomological weight 0, and is joint work with George Boxer, Frank Calegari, Toby Gee and Jack Thorne. New ingredients include an application of my work with Ana Caraiani on local-global compatibility, pieces of the cohomology of Dwork hypersurfaces previously investigated by Lie Qian, and a generic reducedness result for mod p fibres of local deformation rings.