On Differentials of the First Kind and Theta Constants for Certain Congruence Subgroups

Abstract

Let $\Gamma (8)$ denote the principal congruence subgroup of level 8 and let $\Gamma (16,32)$ denote the subgroup of $\Gamma (16)$ satisfying $cd \equiv ab \equiv 0\;(\bmod 32)$. We are dealing only with the elliptic modular case. Consider the spaces of cusp forms of weight 2 (differentials of the first kind) with respect to these groups. It is proved that these spaces are generated by certain monomials of theta constants of degree 4.