An Explicit Belyi Map for the Wiman Sextic and Cusp Forms for a Noncongruence Subgroup

Abstract

In this paper, we explicitly determine an algebraic Belyi function on a unique smooth projective model $\widetilde{W}$ of the Wiman sextic curve $W$ and describe its complex uniformization in terms of modular functions associated with a certain noncongruence subgroup $Γ_{\widetilde{W}} \subset {\rm SL}_2(\mathbb{Z})$. As an application, we give a direct proof of the unbounded denominators conjecture in weight $2$ for $Γ_{\widetilde{W}}$. The conjecture is now known in full generality by the work of Calegari, Dimitrov, and Tang, following earlier progress including work of Dong, Lin, and Ng. Our proof, however, uses a degeneration of $\widetilde{W}$ over $\mathbb{F}_{5}$ together with explicit Puiseux series expansions and is substantially different from the methods employed in their work.