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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 establishvarious pointwise upper bounds for $\phi$ in terms of its Laplace eigenvalue$\lambda_\phi$. These imply, for $\phi$ arithmetically normalized and temperedat the archimedean place, the bound $\|\phi\|_\infty\ll_\epsilon\lambda_{\phi}^{39/40+\epsilon}$ for the global sup-norm (without restrictionto a compact subset). On the way, we derive a new uniform upper bound for the$\mathrm{GL}_3$ Jacquet-Whittaker function.

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