Abstract
We examine the distribution of zeroes of half-integral weight Hecke cusp forms on the manifold Γ0(4)\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 Re(s) = - 1/2 and Re(s) = 0 as the weight tends to infinity. We show that, for ≫εK2/(logK)3/2+ε 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.