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

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.