We study general properties of the dessins d'enfants associated with the Hecke congruence subgroups $\Gamma_0(N)$ of the modular group $\mathrm{PSL}_2(\mathbb{R})$. The definition of the $\Gamma_0(N)$ as the stabilisers of couples of projective lattices in a two-dimensional vector space gives an interpretation of the quotient set $\Gamma_0(N)\backslash\mathrm{PSL}_2(\mathbb{R})$ as the projective lattices $N$-hyperdistant from a reference one, and hence as the projective line over the ring $\mathbb{Z}/N\mathbb{Z}$. The natural action of $\mathrm{PSL}_2(\mathbb{R})$ on the lattices defines a dessin d'enfant structure, allowing for a combinatorial approach to features of the classical modular curves, such as the torsion points and the cusps. We tabulate the dessins d'enfants associated with the $15$ Hecke congruence subgroups of genus zero, which arise in Moonshine for the Monster sporadic group.

中文翻译：

---

尖点、同余群和怪物结构

我们研究了与模群 $\mathrm{PSL}_2(\mathbb{R})$ 的 Hecke 同余子群 $\Gamma_0(N)$ 相关的 dessins d'enfants 的一般性质。$\Gamma_0(N)$ 作为二维向量空间中射影格对的稳定器的定义给出了商集 $\Gamma_0(N)\backslash\mathrm{PSL}_2(\mathbb {R})$ 作为距参考点的射影格 $N$-超远距离，因此作为环 $\mathbb{Z}/N\mathbb{Z}$ 上的射影线。$\mathrm{PSL}_2(\mathbb{R})$ 在晶格上的自然作用定义了 dessin d'enfant 结构，允许对经典模曲线的特征进行组合方法，例如扭点和尖。我们列出了dessins d'