Explicit subconvexity savings for sup-norms of cusp forms on ${\mathrm{PGL}}_n(\mathbb{R})$

Abstract

Blomer and Maga recently proved that, if $F$ is an $L^2$-normalized Hecke Maass cusp form for ${\mathrm{SL}}_n(\mathbb{Z})$, and $\Omega$ is a compact subset of ${\mathrm{PGL}}_n(\mathbb{R})/{\mathrm{PO}}_n(\mathbb{R})$, then we have $\Vert F{|}_{\Omega }{\Vert }_{\infty }{\ll }_{\Omega }{\lambda }_F^{n(n-1)/8-{\delta }_n}$ for some ${\delta }_n>0$, where ${\lambda }_F$ is the Laplacian eigenvalue of $F$. In the present paper, we prove an explicit version of their result.

13 pages, minor edits, to appear in Journal of Number Theory