Galois Realisations of $\operatorname{PSL}_2(\mathbb{F}_{p^2})$ via non-unirational Hilbert Irreducibility

TL;DR

This paper proves new cases of the inverse Galois problem for certain simple groups by establishing non-unirational Hilbert irreducibility results for specific Hilbert modular surfaces of K3 type.

Contribution

It introduces non-unirational Hilbert irreducibility techniques for Hilbert modular surfaces of K3 type, enabling new Galois realizations of PSL_2 groups over finite fields.

Findings

New Galois realizations of PSL_2(𝔽_{p^2}) groups

Non-unirational Hilbert irreducibility results for K3 type surfaces

Applications to the inverse Galois problem

Abstract

We establish non-unirational versions of Hilbert Irreducibility for all Hilbert modular surfaces which are of K3 type. As an application we prove new instances of the regular Inverse Galois Problem for the simple groups $\operatorname{PSL}_2(\mathbb{F}_{p^2})$ subject to congruence conditions on $p$.