Geometry of the Wiman-Edge pencil and the Wiman curve

TL;DR

This paper provides an explicit uniformization and modular interpretation of the Wiman-Edge pencil and the Wiman curve, revealing their structure as quotients of the hyperbolic plane and their relation to Shimura curves.

Contribution

It offers the first explicit uniformization of the Wiman-Edge pencil and the Wiman curve, connecting them to hyperbolic geometry and Shimura varieties.

Findings

Uniformization of the parameter space as a non-congruence quotient of the hyperbolic plane.

Modular interpretation of degenerations into lines and conics with Petersen and $K_5$ graphs.

Identification of the Wiman curve as a Shimura curve of quaternionic type.

Abstract

The {\em Wiman-Edge pencil} is the universal family $C_t, t\in\mathcal B$ of projective, genus $6$, complex-algebraic curves admitting a faithful action of the icosahedral group $\Af_5$. The curve $C_0$, discovered by Wiman in 1895 \cite{Wiman} and called the {\em Wiman curve}, is the unique smooth, genus $6$ curve admitting a faithful action of the symmetric group $\Sf_5$. In this paper we give an explicit uniformization of $\mathcal B$ as a non-congruence quotient $\Gamma\backslash \Hf$ of the hyperbolic plane $\Hf$, where $\Gamma<\PSL_2(\Z)$ is a subgroup of index $18$. We also give modular interpretations for various aspects of this uniformization, for example for the degenerations of $C_t$ into $10$ lines (resp.\ $5$ conics) whose intersection graph is the Petersen graph (resp.\ $K_5$). In the second half of this paper we give an explicit arithmetic uniformization of the Wiman curve $C_0$ itself as the quotient $\Lambda\backslash \Hf$, where $\Lambda$ is a principal level $5$ subgroup of a certain "unit spinor norm" group of M\"{o}bius transformations. We then prove that $C_0$ is a certain moduli space of Hodge structures, endowing it with the structure of a Shimura curve of indefinite quaternionic type.