Automorphic Galois representations and the inverse Galois problem for certain groups of type $D_m$

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