Dessins on Modular Curves

Abstract

Given a finite index subgroup $\Gamma$ of ${\mathrm{PSL}}_2(\mathbb{Z})$, we investigate Belyi functions on the corresponding modular curve $X(\Gamma )$ by introducing two methods for constructing such functions. Numerous examples have been worked out completely and as an application, we have derived modular equations for ${\Gamma }_0(2),{\Gamma }_0(3)$ and several special values of the $j$-function by a new method based on the theory of Belyi functions and dessin d'enfants.

29 pages, 10 figures. Minor corrections