$L^4$-norms of automorphic forms in the depth aspect

Abstract

Let $p$ be an odd prime, and suppose $f$ is an $L^2$-normalised newform for ${\Gamma }_0(p^n)$ with bounded spectral parameters and trivial central character. We prove the optimal $L^4$-norm bound $\Vert f{\Vert }_4{\ll }_{p,\epsilon }(p^n{)}^{\epsilon }$ for all $\epsilon >0$ as $n\to \infty$.

arXiv comment51 pages