Zeros of modular forms of half integral weight

TL;DR

This paper investigates zeros of modular forms of half-integer weight, revealing that most have zeros on a boundary arc, and explores connections between generating functions, mock modular forms, and Fourier coefficients.

Contribution

It introduces canonical bases for weakly holomorphic modular forms of level 4 and half-integer weights, analyzing zeros and relations to mock modular Poincaré series.

Findings

Most modular forms have zeros on a boundary arc of the fundamental domain.

The generating function for Hurwitz class numbers equals a mock modular Poincaré series at many points.

Certain Fourier coefficients of cusp forms cannot all vanish simultaneously.

Abstract

We study canonical bases for spaces of weakly holomorphic modular forms of level 4 and weights in $\mathbb{Z}+\frac{1}{2}$ and show that almost all modular forms in these bases have the property that many of their zeros in a fundamental domain for $\Gamma_0(4)$ lie on a lower boundary arc of the fundamental domain. Additionally, we show that at many places on this arc, the generating function for Hurwitz class numbers is equal to a particular mock modular Poincar\'{e} series, and show that for positive weights, a particular set of Fourier coefficients of cusp forms in this canonical basis cannot simultaneously vanish.