Modular equations for congruence subgroups of genus zero (II)

We present a result that the modular equation of a Hauptmodul for a certain congruence subgroup Γ H ( N , t ) of genus zero satisfies Kronecker's congruence relation. This generalizes the author's previous result about Γ 1 ( m ) ⋂ Γ 0 ( m N ) . Furthermore we show that the similar result holds for a certain congruence subgroup Γ of genus zero with [ Γ : Γ H ( N , t ) ] = 2 . Finally we prove a conjecture of Lee and Park, asserting that the modular equation of the continued fraction of order six satisfies a certain form of Kronecker's congruence relation.