Local bounds for $L^p$ norms of Maass forms in the level aspect

TL;DR

This paper establishes new local bounds for the $L^p$ norms of Maass forms on quaternion division algebras, especially when the level is not squarefree, extending understanding of their behavior in the level aspect.

Contribution

It introduces novel local bounds for the sup norm of Maass forms and extends $L^p$ norm bounds analogous to Sogge's theorem to the level aspect.

Findings

New candidate for local sup norm bound when level is not squarefree

Bound for $L^p$ norms in the level aspect similar to Sogge's theorem

Application of harmonic analysis techniques to quaternion algebra Maass forms

Abstract

We apply techniques from harmonic analysis to study the $L^p$ norms of Maass forms of varying level on a quaternion division algebra. Our first result gives a candidate for the local bound for the sup norm in terms of the level, which is new when the level is not squarefree. The second result is a bound for $L^p$ norms in the level aspect that is analogous to Sogge's theorem on $L^p$ norms of Laplace eigenfunctions.