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

Abstract We examine the distribution of zeroes of half-integral weight Hecke cusp forms on the manifold $$\Gamma _0(4)\backslash \mathbb H$$ Γ 0 ( 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 $$\text {Re}(s)=-1/2$$ Re ( s ) = - 1 / 2 and $$\text {Re}(s)=0$$ Re ( s ) = 0 as the weight tends to infinity. We show that, for $$\gg _\varepsilon K^2/(\log K)^{3/2+\varepsilon }$$ ≫ ε K 2 / ( log K ) 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.