On the coefficients of certain family of modular equations

Abstract

The $n$-th modular equation for the elliptic modular function $j(z)$ has large coefficients even for small $n$, and those coefficients grow rapidly as $n \to \infty$. The growth of these coefficients was first obtained by Cohen ([5]). And, recently Cais and Conrad ([1], \S7) considered this problem for the Hauptmodul $j_{5}(z)$ of the principal congruence group $\Gamma(5)$. They found that the ratio of logarithmic heights of $n$-th modular equations for $j(z)$ and $j_{5}(z)$ converges to 60 as $n \to \infty$, and observed that 60 is the group index $[\overline{\Gamma(1)} : \overline{\Gamma(5)}]$. In this paper we prove that their observation is true for Hauptmoduln of somewhat general Fuchsian groups of the first kind with genus zero.