ON THE COEFFICIENTS OF CERTAIN FAMILY OF MODULAR EQUATIONS

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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.
Publisher
OSAKA JOURNAL OF MATHEMATICS
Issue Date
2009
Language
English
Article Type
Article
Citation

OSAKA JOURNAL OF MATHEMATICS, v.46, no.2, pp.479 - 502

ISSN
0030-6126
URI
http://hdl.handle.net/10203/93571
Appears in Collection
RIMS Journal Papers
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