Quantum Unique Ergodicity for half-integral weight forms

TL;DR

This paper proves the Quantum Unique Ergodicity conjecture for certain half-integral weight automorphic forms under GRH, showing equidistribution of mass and zeros as weight increases.

Contribution

It establishes QUE for half-integral weight automorphic forms under GRH, a significant extension of QUE results to this class of forms.

Findings

QUE holds for half-integral weight holomorphic Hecke cusp forms under GRH

Zeros of these cusp forms equidistribute with hyperbolic measure under GRH

Results extend QUE to half-integral weights in the Kohnen plus subspace

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.