Recurrence Relations Satisfied by the Traces of Singular Moduli for Gamma(0)(N)

Abstract

We compute the divisor of the modular equation on the modular curve Gamma(0)(N)\H* and then find recurrence relations satisfied by the modular traces of the Hauptmodul for any congruence subgroup Gamma(0)(N) of genus zero. We also introduce the notions and properties of Gamma-equivalence and Gamma-reduced forms about binary quadratic forms. Using these, we can explicitly compute the recurrence relations for N = 2, 3, 4, 5.