## Algebra & Number Theory

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

Paul D. Nelson

#### Abstract

We prove that arithmetic quantum unique ergodicity holds on compact arithmetic quotients of $GL2(ℚp)$ 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

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

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

https://projecteuclid.org/euclid.ant/1546657274

Digital Object Identifier
doi:10.2140/ant.2018.12.2033

Mathematical Reviews number (MathSciNet)
MR3894428

#### Citation

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. https://projecteuclid.org/euclid.ant/1546657274

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