On The Image of Automorphic Galois Representations

Abstract

In this paper, we study extra-twists for automorphic representations of ${\mathrm{GL}}_n$ and use them to give a precise description of the image of the Galois representations associated with regular algebraic cuspidal automorphic representations of ${\mathrm{GL}}_3$ over totally real fields. We also formulate a conjecture for the ${\mathrm{GL}}_n$-case and show how it follows from some standard conjectures in the Langlands program. Finally, assuming the existence of a motive associated with the representation, we study the relation of our constructions with the Mumford-Tate group.