Geometry of the Wiman-Edge pencil and the Wiman curve

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.