Algebra & Number Theory

Microlocal lifts and quantum unique ergodicity on $GL_2(\mathbb{Q}_p)$

Paul D. Nelson

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We prove that arithmetic quantum unique ergodicity holds on compact arithmetic quotients of GL2(p) 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.

Our results are the first of their kind on any p-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 “p-adic microlocal lifts” with favorable properties, such as diagonal invariance of limit measures; the proof of positive entropy of limit measures in a p-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.

Article information

Algebra Number Theory, Volume 12, Number 9 (2018), 2033-2064.

Received: 26 January 2017
Revised: 9 April 2018
Accepted: 15 July 2018
First available in Project Euclid: 5 January 2019

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Primary: 58J51: Relations between spectral theory and ergodic theory, e.g. quantum unique ergodicity
Secondary: 22E50: Representations of Lie and linear algebraic groups over local fields [See also 20G05] 37A45: Relations with number theory and harmonic analysis [See also 11Kxx]

arithmetic quantum unique ergodicity microlocal lifts representation theory


Nelson, Paul D. Microlocal lifts and quantum unique ergodicity on $GL_2(\mathbb{Q}_p)$. Algebra Number Theory 12 (2018), no. 9, 2033--2064. doi:10.2140/ant.2018.12.2033.

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