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 ${\Gamma }_0(N)\backslash {\mathbb{H}}^{\ast }$ 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 $N=2,3,4,5$.

Accepted for publication in the Taiwanese Journal of Mathematics. arXiv admin note: text overlap with arXiv:1711.00230