The sup-norm problem for automorphic cusp forms of $\mathrm{PGL}(n,\mathbb{Z}[i])$

TL;DR

This paper establishes a new bound on the maximum size of automorphic cusp forms on complex groups, improving understanding of their growth and distribution in the context of the sup-norm problem.

Contribution

It provides the first non-trivial subconvex bound for the sup-norm of Hecke--Maass cusp forms on $ ext{PGL}_n(b{Z}[i])$, advancing the analytic theory of automorphic forms.

Findings

Proves a bound $ orm{|_o{Omega}}_e{infty} r lambda_^{n(n-1)/4-e{ ext{delta}}}$ for cusp forms.

Shows the bound depends only on the eigenvalue and a compact subset, with $e{ ext{delta}}>0}$.

Extends sup-norm bounds to higher rank groups over complex integers.

Abstract

Let $\phi$ be an $L^2$-normalized Hecke--Maa{\ss} cusp form for $\mathrm{PGL}_n(\mathbb{Z}[i])$ on the locally symmetric space $X:=\mathrm{PGL}_n(\mathbb{Z}[i])\backslash \mathrm{PGL}_n(\mathbb{C}) / \mathrm{PU}_n$. If $\Omega$ is a compact subset of $X$, then we prove the bound $\|\phi|_{\Omega}\|_{\infty}\ll_{\Omega} \lambda_{\phi}^{n(n-1)/4-\delta}$ for some $\delta>0$ depending only on $n$, where $\lambda_{\phi}$ is the Laplace eigenvalue of $\phi$.