On the global sup-norm of GL(3) cusp forms

Abstract

Let $\phi$ be a spherical Hecke-Maass cusp form on the non-compact space $\mathrm{PGL}_3(\mathbb{Z})\backslash\mathrm{PGL}_3(\mathbb{R})$. We establish various pointwise upper bounds for $\phi$ in terms of its Laplace eigenvalue $\lambda_\phi$. These imply, for $\phi$ arithmetically normalized and tempered at the archimedean place, the bound $\|\phi\|_\infty\ll_\epsilon \lambda_{\phi}^{39/40+\epsilon}$ for the global sup-norm (without restriction to a compact subset). On the way, we derive a new uniform upper bound for the $\mathrm{GL}_3$ Jacquet-Whittaker function.