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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 W̃ of the Wiman sextic curve W and describe its complex uniformization in terms of modular functions associated with a certain noncongruence subgroup ΓW̃ ⊂ SL2(ℤ). As an application, we give a direct proof of the unbounded denominators conjecture for weight $2$ cusp forms with respect to ΓW̃. 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 W̃ over 𝔽5 together with explicit Puiseux series expansions and is substantially different from the methods employed in their work.
Related
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BibTeX
@misc{horie2026explicit,
title = {{An Explicit Belyi Map for the Wiman Sextic and Cusp Forms for a Noncongruence Subgroup}},
author = {Madoka Horie and Takuya Yamauchi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.30798},
eprint = {2608.30798},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.30798},
}
RIS
TY - UNPB
TI - An Explicit Belyi Map for the Wiman Sextic and Cusp Forms for a Noncongruence Subgroup
AU - Madoka Horie
AU - Takuya Yamauchi
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2608.30798
UR - https://arxiv.org/abs/2608.30798
AB - In this paper, we explicitly determine an algebraic Belyi function on a unique smooth projective model W̃ of the Wiman sextic curve W and describe its complex uniformization in terms of modular functions associated with a certain noncongruence subgroup Γ_W̃ ⊂ SL_2(ℤ). As an application, we give a direct proof of the unbounded denominators conjecture for weight $2$ cusp forms with respect to Γ_W̃. 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 W̃ over 𝔽_5 together with explicit Puiseux series expansions and is substantially different from the methods employed in their work.
ER -