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.
Citation
Bumkyu Cho. Nam Min Kim. Yoon Kyung Park. "On the coefficients of certain family of modular equations." Osaka J. Math. 46 (2) 479 - 502, June 2009.
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