On two families of modular subgroups

Abstract: In this work, we study various properties of two families of normal subgroups of the Hecke group $\Gamma_{0}(3)$ and their associated modular curves and function fields. As consequences, we obtain the stabilizer subgroups of the $N$-th roots of a uniformizer for $X(\Gamma_{0}(3))$, and realize such $N$-th roots as uniformizers for the associated modular curves to their stabilizer groups.