Lübeck's classification of representations of finite simple groups of Lie type and the inverse Galois problem for some orthogonal groups

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: $P{\Omega }_n^{+}({\mathbb{F}}_{{\ell }^s})$, $PS{O}_n^{+}({\mathbb{F}}_{{\ell }^s})$, $P{O}_n^{+}({\mathbb{F}}_{{\ell }^s})$ or $PG{O}_n^{+}({\mathbb{F}}_{{\ell }^s})$ 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. Lübeck and a previous result of the author on the image of the Galois representations attached to RAESDC automorphic representations of $G{L}_n({\mathbb{A}}_{\mathbb{Q}})$.

This paper replaces "Lübeck's classification of Chevalley groups representations and the inverse Galois problem for some orthogonal groups" arXiv:1712.05528v2. Revised version - referees' comments added. The final version is to appear in J. Number Theory

References in corpus (3)

• Compatible systems of symplectic Galois representations and the inverse Galois problem II. Transvections and huge image
• On the images of the Galois representations attached to certain RAESDC automorphic representations of $\mbox{GL}_n(\mathbb{A}_{\mathbb{Q}})$
• Explicit open image theorems for abelian varieties with trivial endomorphism ring

Cited by in corpus (1)

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