Generators of the ring of weakly holomorphic modular functions for Gamma(1)(N)

Abstract

For a positive integer N divisible by 4, 5, 6,7 or 9, let O-1,O-N(Q) be the ring of weakly holomorphic modular functions for the congruence subgroup Gamma(1)(N) with rational Fourier coefficients. We present explicit generators of the ring O-1,O-N(Q) over Q by making use of modular units which have infinite product expansions.