## Abstract

The Wiman–Edge pencil is the universal family C_{t}, t∈ B of projective, genus 6, complex-algebraic curves admitting a faithful action of the icosahedral group A_{5}. The curve C, 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 S_{5}. In this paper we give an explicit uniformization of B as a non-congruence quotient Γ \ H of the hyperbolic plane H, where Γ<PSL2(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 itself as the quotient Λ \ H, where Λ is a principal level 5 subgroup of a certain “unit spinor norm” group of Möbius transformations. We then prove that C is a certain moduli space of Hodge structures, endowing it with the structure of a Shimura curve of indefinite quaternionic type.

Original language | English |
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Pages (from-to) | 197-220 |

Number of pages | 24 |

Journal | Geometriae Dedicata |

Volume | 208 |

Issue number | 1 |

DOIs | |

Publication status | Published - 1 Oct 2020 |