Geometry of the Wiman-Edge pencil and the Wiman curve

The {em Wiman-Edge pencil} is the universal family $C_t, tinmathcal B$ 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 $Gammabackslash Hf$ 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 $Lambdabackslash Hf$, 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.