Zeros of weakly holomorphic modular forms of level 4

TL;DR

This paper studies the zeros of weakly holomorphic modular forms of level 4, establishing a basis with zeros mostly on the lower boundary and revealing a duality in their Fourier coefficients.

Contribution

It introduces a canonical basis for these modular forms and demonstrates the zeros' distribution and a novel duality property of Fourier coefficients.

Findings

Most zeros lie on the lower boundary of the fundamental domain.

Fourier coefficients exhibit a duality property.

Basis provides new insights into modular form zeros.

Abstract

Let $M_k^\sharp(4)$ be the space of weakly holomorphic modular forms of weight $k$ and level $4$ that are holomorphic away from the cusp at $\infty$. We define a canonical basis for this space and show that for almost all of the basis elements, the majority of their zeros in a fundamental domain for $\Gamma_0(4)$ lie on the lower boundary of the fundamental domain. Additionally, we show that the Fourier coefficients of the basis elements satisfy an interesting duality property.