The sup-norm problem for PGL(4)

TL;DR

This paper establishes a power-saving bound on the maximum size of Hecke-Maass cusp forms for SL(4, Z) with respect to their eigenvalues, assuming Ramanujan at infinity, improving understanding of their growth behavior.

Contribution

It provides the first power-saving sup-norm bound for PGL(4) cusp forms under the Ramanujan conjecture at infinity.

Findings

Achieved a power-saving bound relative to the generic bound

Bound applies to forms restricted to compact sets

Assumes Ramanujan conjecture at infinity

Abstract

Let F be a Hecke-Maass cusp form for the group SL(4, Z) with Laplace eigenvalue lambda. Assume that F satisfies the Ramanujan conjecture at infinity (this is satisfied by almost all cusp forms). We show a power-saving sup-norm bound in terms of lambda (relative to the generic bound) for F restricted to a compact set.