Microlocal lifts and quantum unique ergodicity on GL2(ℚp)

Nelson, Paul

doi:10.2140/ant.2018.12.2033

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

Paul D. Nelson

Vol. 12 (2018), No. 9, 2033–2064

DOI: 10.2140/ant.2018.12.2033

Abstract

We prove that arithmetic quantum unique ergodicity holds on compact arithmetic quotients of ${GL}_{2} (ℚ_{p})$ 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.

Keywords

arithmetic quantum unique ergodicity, microlocal lifts, representation theory

Mathematical Subject Classification 2010

Primary: 58J51

Secondary: 22E50, 37A45

Milestones

Received: 26 January 2017

Revised: 9 April 2018

Accepted: 15 July 2018

Published: 21 December 2018

Authors

Paul D. Nelson
Departement Mathematik ETH Zurich Switzerland

Vol. 12, No. 9, 2018