Divisibility properties for weakly holomorphic modular forms with sign vectors

TL;DR

This paper establishes new divisibility properties for Fourier coefficients of weakly holomorphic modular forms with sign vectors, generalizing classical results and exploring Hecke operators' effects.

Contribution

It introduces generalized divisibility results for modular forms with sign vectors and analyzes Hecke operators' influence on these forms.

Findings

Generalized Siegel's divisibility result for non-positive weights.

Established divisibility properties via Hecke operators.

Connected divisibility results to Borcherds lift weights.

Abstract

In this paper, we prove some divisibility results for the Fourier coefficients of reduced modular forms of sign vectors. More precisely, we generalize a divisibility result of Siegel on constant terms when the weight is non-positive, which is related to the weight of Borcherds lifts when the weight is zero. By considering Hecke operators for the spaces of weakly holomorphic modular forms with sign vectors, and obtain divisibility results in an "orthogonal" direction on reduced modular forms.