2020/07/22 by Chatzakos, Dimitrios, Frot, Robin, Raulf, Nicole · 1 citation
#FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2007.11473
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 PSL2( \mathbb Z) \backslash ℍ2 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 tj-δ with δ> 3/4 . For PSL2(OK) ∖ ℍ3 with OK 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-Maass forms if δ< 2/5 . Furthermore, we prove that equidistribution holds unconditionally for the Eisenstein series if δ< (1-2θ)/(34+4θ) where θ is the exponent towards the Ramanujan-Petersson conjecture. For PSL2(ℤ[i]) we improve the last exponent to δ< (1-2θ)/(27+2θ) . Studying mean Lindelöf estimates for L -functions of Hecke-Maaß forms we improve the last exponent on average to δ< 2/5. Finally, we study massive irregularities for Laplace eigenfunctions on n -dimensional compact arithmetic hyperbolic manifolds for n ≥ 4 . We observe that quantum unique ergodicity fails on shrinking balls of radii R \asymp t-δn+ε away from the Planck-scale, with δn = 5/(n+1) for n ≥ 5 .