Arithmeticity of the monodromy of the Wiman–Edge pencil

Abstract

The Wiman-Edge pencil is the universal family of projective, genus 6, complex-algebraic curves endowed with a faithful action of the icosahedral group. The goal of this paper is to prove that its monodromy group is commensurable with a Hilbert modular group; in particular is arithmetic. We then give a modular interpretation of this, as well as a uniformization of its base.Résumé. -Le pinceau de Wiman-Edge est une famille universelle de courbes projectives non singulières de genre 6 et munie d'une action fidèle du groupe icosahédral. Le but principal de ce travail est la détermination de son groupe de monodromie. Nous montrons que ce groupe est arithmétique et commensurable avec un groupe modulaire de Hilbert. Nous donnons une interprétation modulaire de ce fait et décrivons en plus une uniformisation de la base.

On the Monodromy and Period Map of the Winger Pencil

“…The quotient SL 2 (O o )/H 2 is then a algebraic surface called Hilbert's modular surface. We will need the Theorem 4.6 in [5] listed below which is also a special case of the main theorem of [1]. Since the two models Σ and Π gives the same singular fiber at the "edge" ends, it is clear that ρ Σ,edge and ρ Π,edge should conjugate to each other by a transformation in Sp 1 (O o ) ∼ = SL 2 (O o ).…”

“…We will always denote E Q to be the vector space ∧ 2 V o ⊗ Q. Using the similar notion of [5] we will take f i,j as the image of e i ∧ e j in ∧ 2 V o . Let φ : Z 5 → Z be the morphism of taking the coordinate sum.…”

“…For example we may take E ⊂ E o to be the subspace which has even coefficients sum with respect to the basis in 1.4. It is a sublattice of index two in E o and it was proved in the Lemma 2.8 of [5] that…”

Geometry of the Wiman–Edge pencil and the Wiman curve

“…As explained by Edge [6] (see [5] for a more modern treatment), the family C t of curves appears naturally on a quintic del Pezzo surface S, with the Wiman curve "a uniquely special canonical curve of genus 6" on S: the standard action of S 5 on S leaves C t invariant and leaves invariant exactly the curve C 0 . The base B of the Wiman pencil appears also as the moduli space of K3 surfaces with (a certain) faithful µ 2 × A 5 action; see §5.3 of [7]. For a number of recent papers on the Wiman-Edge pencil, see [2,3,5,7,12].…”

“…The base B of the Wiman pencil appears also as the moduli space of K3 surfaces with (a certain) faithful µ 2 × A 5 action; see §5.3 of [7]. For a number of recent papers on the Wiman-Edge pencil, see [2,3,5,7,12]. The problem of finding uniformizations of moduli spaces is a classical one, but it is typically a difficult task.…”