On the coefficients of certain family of modular equations

Bumkyu Cho, Nam Min Kim, Yoon Kyung Park

Research output: Contribution to journalArticlepeer-review

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 → ∞. The growth of these coefficients was first obtained by Cohen ([5]). And, recently Cais and Conrad ([1], §7) considered this problem for the Hauptmodul j5(z) of the principal congruence group (5). They found that the ratio of logarithmic heights of n-th modular equations for j(z) and j5(z) converges to 60 as n → ∞, and observed that 60 is the group index [(1):(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.

Original languageEnglish
Pages (from-to)479-502
Number of pages24
JournalOsaka Journal of Mathematics
Volume46
Issue number2
StatePublished - Jun 2009

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