Quantum unique ergodicity for half-integral weight automorphic forms

Abstract

We investigate the analogue of the quantum unique ergodicity (QUE) conjecture for half-integral weight automorphic forms. Assuming the generalized Riemann hypothesis (GRH), we establish both QUE for half-integral weight Hecke Maaß cusp forms for ${\Gamma }_0\left(4\right)$ lying in Kohnen’s plus subspace and mass equidistribution for half-integral weight holomorphic Hecke cusp forms for ${\Gamma }_0\left(4\right)$ lying in Kohnen’s plus subspace. By combining the former result along with an argument of Rudnick, it follows that under GRH the zeros of these holomorphic Hecke cusp forms equidistribute with respect to hyperbolic measure on ${\Gamma }_0\left(4\right)\backslash \mathbb{H}$ as the weight tends to infinity.