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arXiv · 1602.00589

Zeros of modular forms of half integral weight

Amanda Folsom·Paul Jenkins
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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.

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Amanda Folsom, Paul Jenkins. 2016-02-01. Zeros of modular forms of half integral weight. https://arxiv.org/abs/1602.00589

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