The sup-norm problem for newforms of large level on $\mathrm{PGL}(n)$

Abstract

Let $N$ be a prime and $\varphi$ be a Hecke-Maass cuspidal newform for the Hecke congruence subgroup ${\Gamma }_0(N)$ in ${\mathrm{SL}}_n(\mathbb{R})$. Let $\Omega$ be an adelic compactum and let ${\Omega }_N$ be its projection to ${\Gamma }_0(N)\backslash {\mathrm{SL}}_n(\mathbb{R})/\mathrm{SO}(n)$. For any prime $n$, we prove sub-baseline bounds for the sup-norm of $\varphi$ restricted to ${\Omega }_N$. Conditionally on GRH, we generalise this result to all $n\ge 2$. The methods involve a new reduction theory with level structure, based on generalisations of Atkin-Lehner operators.

47 pages; new version contains corrections and clarifications