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 QUE for both half-integral weight holomorphic Hecke cusp forms for $\Gamma_0(4)$ lying in Kohnen's plus subspace and for half-integral weight Hecke Maa{\ss} cusp forms for $\Gamma_0(4)$ 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 equidistribute with respect to hyperbolic measure on $\Gamma_0(4)\backslash \mathbb H$ as the weight tends to infinity.