A family of polynomials with Galois group $PSL_5(2)$ over $\mathbb{Q}(t)$

TL;DR

This paper constructs an explicit polynomial over $Q(t)$ with Galois group $PSL_5(2)$, providing the first known example and confirming that this group can occur as a monodromy group over $Q$.

Contribution

It presents the first explicit polynomial with Galois group $PSL_5(2)$ over $Q(t)$ and demonstrates that $PSL_5(2)$ can be realized as a monodromy group over $Q$.

Findings

First explicit polynomial with group $PSL_5(2)$ over $Q(t)$.

Confirms $PSL_5(2)$ as a monodromy group over $Q$.

Identifies a unique genus zero Hurwitz family with four branch points.

Abstract

We compute a family of coverings with four ramification points, defined over $\mathbb{Q}$, with regular Galois group $PSL_5(2)$. On the one hand, this is (to my knowledge) the first explicit polynomial with group $PSL_5(2)$ over $\mathbb{Q}(t)$. On the other hand, it also positively answers the question whether $PSL_5(2)$ is the monodromy group of a rational function over $\mathbb{Q}$. At least this does not follow from considering class triples in $PSL_5(2)$, as there are no rigid, rational genus-zero triples. Also, for 4-tuples, our family is the only one with a Hurwitz curve of genus zero (however it does not seem immediately clear without explicit computations whether this curve can be defined as a rational curve over $\mathbb{Q}$). There are also genus zero families with five branch points, and maybe their Hurwitz spaces can be shown to have rational points; however, so far I have not seen such arguments.