The Ramanujan and Sato-Tate Conjectures for Bianchi modular forms

TL;DR

This paper proves the Ramanujan and Sato-Tate conjectures for Bianchi modular forms and more generally for certain automorphic representations over CM fields, using a new potential automorphy theorem.

Contribution

It introduces a new potential automorphy theorem for symmetric powers of Galois representations, enabling the proof of these conjectures in broader settings.

Findings

Proved Ramanujan and Sato-Tate conjectures for Bianchi modular forms.

Extended these results to all regular algebraic cuspidal automorphic representations of GL2 over CM fields.

Developed a new potential automorphy theorem for symmetric powers of Galois representations.

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 $\mathrm{GL}_2(\mathbf{A}_F)$ 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.