Some applications of modular units

TL;DR

This paper explores expressing weakly holomorphic modular functions as sums of higher-level modular units and characterizes when Siegel modular functions are free of zeros and poles on specific domains.

Contribution

It introduces a method to decompose certain modular functions into modular units and provides a criterion for the absence of zeros and poles in Siegel modular functions.

Findings

Weakly holomorphic modular functions can be expressed as sums of higher-level modular units.

A necessary and sufficient condition is established for Siegel modular functions to have no zeros or poles on specific subsets.

The results deepen understanding of the structure of modular functions and their behavior on Siegel upper half-spaces.

Abstract

We show that a weakly holomorphic modular function can be written as a sum of modular units of higher level. We further find a necessary and sufficient condition for a Siegel modular function of degree $g$ to have neither zero nor pole on the domain when restricted to certain subset of the Siegel upper half-space $\mathbb{H}_g$.