Zeros of replicable functions

Following the work of Asai, Kaneko, and Ninomiya for Faber polynomials associated to ${\mathrm{P}\mathrm{S}\mathrm{L}}_2(\mathbb{Z})$, and Bannai, Kojima, and Miezaki’s partial proof for the case of ${\mathrm{\Gamma }}_0^{\ast }(2)$, we show that the zeros of certain modular functions associated to some low-level genus zero groups are all located on the boundary of certain natural fundamental domains for $\mathrm{\Gamma }$. The groups considered are ${\mathrm{\Gamma }}_0^{\ast }(2)$, ${\mathrm{\Gamma }}_0^{\ast }(3)$, ${\mathrm{\Gamma }}_0(2\mathrm{\parallel }2)$, ${\mathrm{\Gamma }}_0^{\ast }(5)$, ${\mathrm{\Gamma }}_0(6)+$, ${\mathrm{\Gamma }}_0^{\ast }(7)$, ${\mathrm{\Gamma }}_0(4\mathrm{\parallel }2)+$, ${\mathrm{\Gamma }}_0(3\mathrm{\parallel }3)$, and ${\mathrm{\Gamma }}_0(10)+$.