Remarks on the inverse Galois problem over function fields

TL;DR

This paper advances the inverse Galois problem over function fields by constructing Galois representations with maximal images for groups of Lie type, utilizing Galois theoretic and automorphic techniques.

Contribution

It introduces new methods to realize finite groups of Lie type as Galois groups over function fields through compatible systems of $ll$-adic representations.

Findings

Constructed compatible systems with maximal Galois images

Proved new instances of the inverse Galois problem over function fields

Applied Larsen's classical results to ensure maximality

Abstract

In this paper, we prove new instances of the inverse Galois problem over global function fields for finite groups of Lie type. This is done by constructing compatible systems of $\ell$-adic Galois representations valued in a semisimple group $G$ using Galois theoretic and automorphic methods, and then proving that the Galois images are maximal for a set of primes of positive density using a classical result of Larsen on Galois images for compatible sytems.