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

TL;DR

This work advances the level-aspect sup-norm problem for Hecke-Maaß newforms on spaces $X_n(N)$ with prime level by developing a higher-rank reduction theory that uses a generalized Fricke involution to define a bulk region $\Omega_N$ where most of the volume concentrates. The authors employ an amplified pre-trace formula, translate the analytic bound into a counting problem for integral matrices in $\mathcal{M}_n(\mathbb{Z},N)$, and develop a lattice-theoretic framework to control the geometry via Iwasawa coordinates and exterior powers. A key innovation is the analysis of sparse determinant sequences and the unique- Fricke involution symmetry (when $n>2$), together with a detailed reduction of the domain and an iterative lattice-counting strategy that yields sub-baseline bounds: $\|\phi|_{\Omega_N}\|_\infty \ll N^{-1/(4n^2)+\varepsilon}$ for prime $n$, and under GRH the bound extends to all $n\ge 2$ with refined exponents. The results illuminate the level-aspect behavior in higher rank and provide a blueprint for further improvements via sharpened amplification and deeper determinant-analysis, potentially impacting subconvexity and equidistribution questions for automorphic forms on $\mathrm{PGL}(n)$.

Abstract

Let $N$ be a prime and $φ$ be a Hecke-Maass cuspidal newform for the Hecke congruence subgroup $Γ_0(N)$ in $\operatorname{SL}_n(\mathbb{R})$. Let $Ω$ be an adelic compactum and let $Ω_N$ be its projection to $Γ_0(N) \backslash \operatorname{SL}_n(\mathbb{R}) / \operatorname{SO}(n)$. For any prime $n$, we prove sub-baseline bounds for the sup-norm of $φ$ restricted to $Ω_N$. Conditionally on GRH, we generalise this result to all $n \geq 2$. The methods involve a new reduction theory with level structure, based on generalisations of Atkin-Lehner operators.

Theorems & Definitions (64)

• Theorem 1
• Theorem 2
• Definition
• Theorem 3
• Lemma 2.1
• proof
• Lemma 2.2
• Lemma 2.3
• Remark 2.4
• Remark 2.5