Arithmeticity of the monodromy of the Wiman-Edge pencil

TL;DR

This paper proves that the monodromy group of the Wiman-Edge pencil, a family of genus 6 curves with icosahedral symmetry, is arithmetic and relates it to Hilbert modular groups, providing a modular interpretation and uniformization.

Contribution

It establishes the arithmeticity of the monodromy of the Wiman-Edge pencil and offers a modular interpretation and uniformization of its base.

Findings

Monodromy group is commensurable with a Hilbert modular group

Provides a modular interpretation of the family

Offers a uniformization of the base space

Abstract

The {\em Wiman-Edge pencil} is the universal family $\Cs/\mathcal B$ of projective, genus $6$, complex-algebraic curves admitting a faithful action of the icosahedral group $\Af_5$. The goal of this paper is to prove that the monodromy of $\Cs/\mathcal B$ is commensurable with a Hilbert modular group; in particular is arithmetic. We then give a modular interpretation of this, as well as a uniformization of $\mathcal B$.