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

TL;DR

This paper demonstrates that for certain groups of type D_{m}, infinitely many can be realized as Galois groups over the rationals, using automorphic Galois representations and a modified group theory result.

Contribution

It establishes the existence of infinitely many Galois groups of specific type D_{m} groups over , extending previous automorphic Galois representation results.

Findings

At least one of the specified groups is a Galois group over  for infinitely many s.

Uses a modified group theory result of Khare, Larsen, and Savin.

Builds on previous work on Galois representations attached to automorphic forms.

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: $\mbox{P}\Omega^\pm_{2m}(\mathbb{F}_{\ell^s})$, $\mbox{PSO}^\pm_{2m}(\mathbb{F}_{\ell^s})$, $\mbox{PO}_{2m}^\pm(\mathbb{F}_{\ell^s})$ or $\mbox{PGO}^\pm_{2m}(\mathbb{F}_{\ell^s})$ 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 $\mbox{GL}_{2m}(\mathbb{A}_\mathbb{Q})$..