Geometry of the Wiman–Edge pencil and the Wiman curve

Abstract

The 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 $$\mathfrak {A}_5$$ . The curve $$C_0$$ , discovered by Wiman in 1895 (Ueber die algebraische Curven von den Geschlecht $$p=4,5$$ and 6 welche eindeutige Transformationen in sich besitzen) and called the Wiman curve, is the unique smooth, genus 6 curve admitting a faithful action of the symmetric group $$\mathfrak {S}_5$$ . In this paper we give an explicit uniformization of $${\mathcal {B}}$$ as a non-congruence quotient $$\Gamma \backslash \mathfrak {H}$$ of the hyperbolic plane $$\mathfrak {H}$$ , where $$\Gamma <{{\,\mathrm{PSL}\,}}_2(\mathbb {Z})$$ 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 $$\Lambda \backslash \mathfrak {H}$$ , where $$\Lambda $$ is a principal level 5 subgroup of a certain “unit spinor norm” group of Möbius 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.