Microlocal lifts and quantum unique ergodicity on $GL(2,\mathbb{Q}_p)$

TL;DR

This paper establishes quantum unique ergodicity for automorphic forms on $GL(2,Q_p)$, introducing p-adic microlocal lifts and proving equidistribution results on p-adic arithmetic quotients, analogous to classical results on surfaces.

Contribution

It introduces the concept of p-adic microlocal lifts and proves quantum unique ergodicity in the p-adic setting, a novel achievement in the field.

Findings

Proves arithmetic quantum unique ergodicity on $GL(2,Q_p)$.

Introduces p-adic microlocal lifts with favorable properties.

Establishes equidistribution of eigenfunctions on p-adic covers and sequences of regular graphs.

Abstract

We prove that arithmetic quantum unique ergodicity holds on compact arithmetic quotients of $GL(2,\mathbb{Q}_p)$ 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.