On the real zeroes of half-integral weight Hecke cusp forms

We examine the distribution of zeroes of half-integral weight Hecke cusp forms on the manifold ${\Gamma }_0\left(4\right)\backslash H$ near a cusp at infinity. In analogue of the Ghosh-Sarnak conjecture for classical holomorphic Hecke cusp forms, one expects that almost all of the zeroes sufficiently close to this cusp lie on two vertical geodesics $\text{Re}\left(s\right)=-1/2$ and $\text{Re}\left(s\right)=0$ as the weight tends to infinity. We show that, for ${\gg }_{\epsilon }{K}^2/{(logK)}^{3/2+\epsilon }$ of the half-integral weight Hecke cusp forms in the Kohnen plus subspaces with weight bounded by a large parameter K, the number of such "real" zeroes grows almost at the expected rate. We also obtain a weaker lower bound for the number of real zeroes that holds for a positive proportion of forms. One of the key ingredients is the estimation of averaged first and second moments of quadratic twists of modular L-functions.