Automorphic Galois repesentations and the inverse Galois problem for certain groups of type $D_{2m}$

Adrian Zenteno

Published 2019-11-05Version 1

Let $\ell$ be an odd prime and $m$ be a positive integer greater than two. In this paper, we prove that at least one of the following groups: $\mbox{P}\Omega^\pm_{4m}(\mathbb{F}_{\ell^s})$, $\mbox{PSO}^\pm_{4m}(\mathbb{F}_{\ell^s})$, $\mbox{PO}_{4m}^\pm(\mathbb{F}_{\ell^s})$ or $\mbox{PGO}^\pm_{4m}(\mathbb{F}_{\ell^s})$ is a Galois group of $\mathbb{Q}$ for infinitely many integers $s > 0$. This is achieved by making use of a slight modification of a group theory result of Khare, Larsen and Savin, and previous results of the author on the images of the Galois representations attached to cuspidal automorphic representations of $\mbox{GL}_{4m}(\mathbb{A}_\mathbb{Q})$.