Zeros of modular forms of half integral weight

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 $Γ_0(4)$ 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 Poincaré series, and show that for positive weights, a particular set of Fourier coefficients of cusp forms in this canonical basis cannot simultaneously vanish.

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