Computation of Hurwitz spaces and new explicit polynomials for almost simple Galois groups

TL;DR

This paper computes explicit polynomials with specific complex Galois groups over $Q(t)$ and $Q$, utilizing Hurwitz space techniques, revealing new examples and rational points not previously known.

Contribution

It provides the first explicit polynomials for several almost simple Galois groups and demonstrates the use of Hurwitz space computations to find rational points.

Findings

Explicit polynomials for new Galois groups over $Q(t)$ and $Q$

Identification of rational points on Hurwitz spaces

Novel computational techniques for Galois group realization

Abstract

We compute the first explicit polynomials with Galois groups $G=P\Gamma L_3(4)$, $PGL_3(4)$, $PSL_3(4)$ and $PSL_5(2)$ over $\mathbb{Q}(t)$. Furthermore we compute the first examples of totally real polynomials with Galois groups $PGL_2(11)$, $PSL_3(3)$, $M_{22}$ and $Aut(M_{22})$ over $\mathbb{Q}$. All these examples make use of families of covers of the projective line ramified over four or more points, and therefore use techniques of explicit computations of Hurwitz spaces. Similar techniques were used previously e.g. by Malle, Couveignes, Granboulan and Hallouin. Unlike previous examples, however, some of our computations show the existence of rational points on Hurwitz spaces that would not have been obvious from theoretical arguments.