Irreducibility and Monodromy of Automorphic Galois Representations of $\mathrm{GL}(4)$

TL;DR

This paper proves irreducibility of certain Galois representations attached to automorphic forms on GL(4) over totally real fields and determines their monodromy groups, advancing understanding of their algebraic and arithmetic properties.

Contribution

It establishes irreducibility for non-self-dual automorphic Galois representations of GL(4) and develops a theory of extra-twists to compute their monodromy groups.

Findings

Irreducibility of Galois representations over totally real fields.

Computation of monodromy groups for these representations.

Results on big image properties of the Galois representations.

Abstract

We prove that over totally real fields, the $p$-adic Galois representations attached to non-self-dual regular algebraic cuspidal automorphic representations of $\mathrm{GL}(4)$ are irreducible. We then develop the theory of extra-twists in a general setting and use it to compute the monodromy group (over $\mathbb{Q}$) of these Galois representations, in both self-dual and non-self-dual settings, and prove $p$-adic and residual big image results.