发表机构
National Institute of Technology, Niihama College; Mathematical Inst. Tohoku Univ.（国立高等专门学校新居滨高专; 东北大学数学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对维曼六次曲线的光滑射影模型确定显式Belyi函数，结合非同余子群$\boldsymbol{\tilde{W}}$的模函数描述其复单值化，通过$\boldsymbol{F}_5$上的退化与Puiseux级数展开，给出权2下非同余子群无界分母猜想的不同证明。
AI 中文摘要
本文中，我们在维曼六次曲线W的唯一光滑射影模型$\tilde{W}$上显式确定了一个代数Belyi函数，并通过与$SL_2(\boldsymbol{Z})$的某个非同余子群$\boldsymbol{\tilde{W}}$相关的模函数描述其复单值化。作为应用，我们对$\boldsymbol{\tilde{W}}$证明了权2下的无界分母猜想。该猜想现已通过Calegari、Dimitrov和Tang的工作得到完全证明，此前已有Dong、Lin和Ng等学者的进展。不过，我们的证明利用了$\tilde{W}$在$\boldsymbol{F}_5$上的退化以及显式Puiseux级数展开，与他们采用的方法有显著差异。
英文摘要
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.
Comments20 pages, Appendix is added
