Quantum ergodicity for shrinking balls in arithmetic hyperbolic manifolds

Abstract

We study a refinement of the quantum unique ergodicity conjecture for shrinking balls on arithmetic hyperbolic manifolds, with a focus on dimensions $2$ and $3$. For the Eisenstein series for the modular surface ${\mathrm{PSL}}_2(\mathbb{Z})\backslash {\mathbb{H}}^2$ we prove failure of quantum unique ergodicity close to the Planck-scale and an improved bound for its quantum variance. For arithmetic $3$-manifolds we show that quantum unique ergodicity of Hecke-Maaß forms fails on shrinking balls centered on an arithmetic point and radius $R\asymp t_j^{-\delta }$ with $\delta >3/4$. For ${\mathrm{PSL}}_2({\mathcal{O}}_K)\setminus {\mathbb{H}}^3$ with ${\mathcal{O}}_K$ being the ring of integers of an imaginary quadratic number field of class number one, we prove, conditionally on the generalized Lindelöf hypothesis, that equidistribution holds for Hecke-Maa{ss} forms if $\delta <2/5$. Furthermore, we prove that equidistribution holds unconditionally for the Eisenstein series if $\delta <(1-2\theta )/(34+4\theta )$ where $\theta$ is the exponent towards the Ramanujan-Petersson conjecture. For ${\mathrm{PSL}}_2(\mathbb{Z}[i])$ we improve the last exponent to $\delta <(1-2\theta )/(27+2\theta )$. Studying mean Lindelöf estimates for $L$-functions of Hecke-Maaß forms we improve the last exponent on average to $\delta <2/5$. Finally, we study massive irregularities for Laplace eigenfunctions on $n$-dimensional compact arithmetic hyperbolic manifolds for $n\ge 4$. We observe that quantum unique ergodicity fails on shrinking balls of radii $R\asymp t^{-{\delta }_n+\epsilon }$ away from the Planck-scale, with ${\delta }_n=5/(n+1)$ for $n\ge 5$.

42 pages