Quantum variance on quaternion algebras, III
Paul D. Nelson
TL;DR
This work extends the quantum variance program to compact arithmetic surfaces arising from non-split quaternion algebras, building a complete theta-function framework that parallels the non-compact SL_2 setting. By translating variance sums into integrals of theta kernels and applying the Rallis inner product formula, the authors obtain precise asymptotics for microlocal lifts and holomorphic analogues, with a robust local–global adelic approach. The main contributions include a quadratic trace formula, a thickening operator that links variance to theta constructions, and a detailed archimedean microlocal-lift calculus that unifies general variance sums with microlocal lift statistics. The results illuminate the arithmetic variance landscape, provide explicit leading terms in terms of L-values and diagonal-H-integrals, and establish a powerful methodology for analyzing variances on compact arithmetic quotients with broad potential impact in quantum chaos and automorphic forms.
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.