Dessins on Modular Curves

Given a finite index subgroup $\mathrm{\Gamma }$ of ${\mathrm{P}\mathrm{S}\mathrm{L}}_2(\mathbb{Z})$, we investigate Belyi functions on the corresponding modular curve $X(\mathrm{\Gamma })$ 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 ${\mathrm{\Gamma }}_0(2),{\mathrm{\Gamma }}_0(3)$ and several special values of the $j$-function by a new method based on the theory of Belyi functions and dessin d’enfants.