2016/01/31 by Paul D. Nelson, Paul Nelson · 6 citations
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Automorphic form #Diagonal #Eigenfunction #Entropy (arrow of time) #Ergodicity #Limit (mathematics) #Microlocal analysis #Quantum #Quotient #math.DS #math.NT #math.RT #msc:22E50 #msc:37A45 #msc:58J51
paper · pdf · doi:10.2140/ant.2018.12.2033
published in Algebra & Number Theory 12(9), 2033-2064 (Mathematical Sciences Publishers) · 27 pages, rewritten introduction, minor edits
openalex created_date 2016/06/24 · arxiv created 2016/11/13 · openalex publication_date 2018/12/21 · arxiv updated 2019/01/02 · openalex updated_date 2026/08/05
We prove that arithmetic quantum unique ergodicity holds on compact arithmetic quotients of [math] for automorphic forms belonging to the principal series. We interpret this conclusion in terms of the equidistribution of eigenfunctions on covers of a fixed regular graph or along nested sequences of regular graphs.\n¶ Our results are the first of their kind on any [math] -adic arithmetic quotient. They may be understood as analogues of Lindenstrauss’s theorem on the equidistribution of Maass forms on a compact arithmetic surface. The new ingredients here include the introduction of a representation-theoretic notion of “ [math] -adic microlocal lifts” with favorable properties, such as diagonal invariance of limit measures; the proof of positive entropy of limit measures in a [math] -adic aspect, following the method of Bourgain–Lindenstrauss; and some analysis of local Rankin–Selberg integrals involving the microlocal lifts introduced here as well as classical newvectors. An important input is a measure-classification result of Einsiedler–Lindenstrauss.