On The Image of Automorphic Galois Representations

TL;DR

This paper investigates the images of Galois representations linked to automorphic forms on GL_3 over totally real fields, introduces a conjecture for GL_n, and explores connections with Mumford-Tate groups under motive assumptions.

Contribution

It provides a detailed description of Galois representation images for GL_3 automorphic forms and proposes a general conjecture for GL_n, connecting automorphic and Galois data.

Findings

Explicit description of Galois image for GL_3 automorphic representations.

Formulation of a conjecture for the image of Galois representations for GL_n.

Analysis of the relation between automorphic Galois representations and Mumford-Tate groups.

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.