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

Abstract:

Let $m$ be an integer greater than three and $\ell$ be an odd prime. In this paper we prove that at least one of the following groups: $\mathrm {P}\Omega ^\pm _{2m}(\mathbb {F}_{\ell ^s})$, $\mathrm {PSO}^\pm _{2m}(\mathbb {F}_{\ell ^s})$, $\mathrm {PO}_{2m}^\pm (\mathbb {F}_{\ell ^s})$, or $\mathrm {PGO}^\pm _{2m}(\mathbb {F}_{\ell ^s})$ 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 $\mathrm {GL}_{2m}(\mathbb {A}_\mathbb {Q})$.