L\"ubeck's classification of representations of finite simple groups of Lie type and the inverse Galois problem for some orthogonal groups

TL;DR

This paper demonstrates that for certain integers, specific orthogonal groups over finite fields can be realized as Galois groups over the rationals, using classification of Lie type representations and automorphic Galois representations.

Contribution

It establishes new cases of the inverse Galois problem for orthogonal groups by leveraging L"ubeck's classification and automorphic Galois representations.

Findings

Orthogonal groups of specific dimensions are Galois groups over  for infinitely many primes and integers.

Utilizes classification of small degree representations of finite simple groups of Lie type.

Connects automorphic Galois representations to inverse Galois realizations.

Abstract

In this paper we prove that for each integer of the form $n=4\varpi$ (where $\varpi$ is a prime between $17$ and $73$) at least one of the following groups: $P\Omega^+_n(\mathbb{F}_{\ell^s})$, $PSO^+_n(\mathbb{F}_{\ell^s})$, $PO_n^+(\mathbb{F}_{\ell^s})$ or $PGO^+_n(\mathbb{F}_{\ell^s})$ is a Galois group of $\mathbb{Q}$ for almost all primes $\ell$ and infinitely many integers $s > 0$. This is achieved by making use of the classification of small degree representations of finite simple groups of Lie type in defining characteristic of F. L\"ubeck and a previous result of the author on the image of the Galois representations attached to RAESDC automorphic representations of $GL_n(\mathbb{A}_\mathbb{Q})$.