Divisibility properties for weakly holomorphic modular forms with sign vectors

Abstract

In this paper, we prove some divisibility results for the Fourier coefficients of reduced modular forms with sign vectors. More precisely, we generalize a divisibility result of Siegel on constant terms when the weight [Formula: see text], which is related to the weight of Borcherds lifts when [Formula: see text]. In particular, we see that such divisibility of the weight of Borcherds lifts only exists for [Formula: see text]. By considering Hecke operators for the spaces of weakly holomorphic modular forms with sign vectors, we obtain divisibility results in an “orthogonal” direction on reduced modular forms.