Quantum variance on quaternion algebras, III
Abstract
We determine the asymptotic quantum variance of microlocal lifts of Hecke--Maass cusp forms on the arithmetic compact hyperbolic surfaces attached to maximal orders in quaternion algebras. Our result extends those of Luo--Sarnak--Zhao concerning the non-compact modular surface. The results of this article's prequel (which involved the theta correspondence, Rallis inner product formula and equidistribution of translates of elementary theta functions) reduce the present task to some local problems over the reals involving the construction and analysis of microlocal lifts via integral operators on the group. We address these here using an analytic incarnation of the method of coadjoint orbits.
Cite
@article{arxiv.1903.08686,
title = {Quantum variance on quaternion algebras, III},
author = {Paul D. Nelson},
journal= {arXiv preprint arXiv:1903.08686},
year = {2025}
}
Comments
71 pages, final version, to appear in Annals of Math. Following the suggestions of referees, v2 now contains what was Part 1 of paper II in the series; paper II will be shortened and revised accordingly. v3: updated references