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

Madoka Horie; Takuya Yamauchi

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arXiv · 2608.30798

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

Madoka Horie·Takuya Yamauchi

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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 $\Gamma_{\widetilde{W}} \subset {\rm SL}_2(\mathbb{Z})$. As an application, we give a direct proof of the unbounded denominators conjecture in weight $2$ for $\Gamma_{\widetilde{W}}$. 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.

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Madoka Horie, Takuya Yamauchi. 2026-08-31. An Explicit Belyi Map for the Wiman Sextic and Cusp Forms for a Noncongruence Subgroup. https://arxiv.org/abs/2608.30798

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