Dessins on Modular Curves

Given a finite index subgroup $Γ$ of ${\rm{PSL}}_2(\Bbb{Z})$, we investigate Belyi functions on the corresponding modular curve $X(Γ)$ 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 $Γ_0(2),Γ_0(3)$ and several special values of the $j$-function by a new method based on the theory of Belyi functions and dessin d'enfants.

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