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 -> infinity. The growth of these coefficients was first obtained by Cohen ([5]). And, recently Cais and Conrad ([1], 7) 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 -> infinity, and observed that 60 is the group index [<(Gamma(1))over bar> : <(Gamma(5))over bar>]. In this paper we prove that their observation is true for Hauptmoduln of somewhat general Fuchsian groups of the first kind with genus zero.