The Ramanujan and Sato–Tate Conjectures for Bianchi modular forms

Source abstract

Abstract We prove the Ramanujan and Sato–Tate conjectures for Bianchi modular forms of weight at least $2$ . More generally, we prove these conjectures for all regular algebraic cuspidal automorphic representations of $\operatorname {\mathrm {GL}}_2(\mathbf {A}_F)$ of parallel weight, where F is any CM field. We deduce these theorems from a new potential automorphy theorem for the symmetric powers of $2$ -dimensional compatible systems of Galois representations of parallel weight.