Arithmeticity of the monodromy of the Wiman-Edge pencil

Abstract

The Wiman-Edge pencil is the universal family C /B of projective, genus 6, complexalgebraic curves admitting a faithful action of the icosahedral group A 5 . The goal of this paper is to prove that the monodromy of C /B 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 B. Contents1 Introduction 12 Some algebra of ZA 5 -modules 3 2.1 Irreducible ZS 5 -modules of degree six .

Geometry of the Wiman-Edge monodromy

“…The proof of Theorem A is in §3. Other than the explicit matrix generators for Γ given in [6], our proof that the monodromy has finite index in SL 2 (O) is completely independent of Farb and Looijenga's argument. We use a presentation for SL 2 (O) due to Yoshida [10] along with the computer algebra program Magma [4] to describe Γ.…”

“…In this paper, we give a complete description of the monodromy by congruence conditions modulo 4 and 5. The congruence condition modulo 4 is new, and this answers a question of Farb-Looijenga [6,Qn. 4.4].…”

“…There is a unique smooth member, the Wiman sextic, whose automorphism group is the symmetric group S 5 . In a recent series of papers, Dolgachev, Farb, and Looijenga [5] then Farb and Looijenga [6,7] studied this pencil from a modern perspective. See these papers for basic facts about the pencil.…”

“…In [6], Farb and Looijenga proved that the monodromy of the Wiman-Edge pencil is commensurable with the Hilbert modular group SL 2 (Z[ √ 5]).…”

“…Question 1. Can one interpret the mod 4 condition on Γ in terms of properties of the Wiman-Edge pencil analogous to the description of the mod p 5 condition described in [6,Cor. 4…”