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

## TL;DR

This paper provides an explicit version of a recent subconvexity bound for sup-norms of Hecke Maass cusp forms on $ ext{PGL}_n(R)$, improving understanding of their growth in relation to Laplacian eigenvalues.

## Contribution

The paper derives an explicit subconvexity bound for the sup-norms of cusp forms on $ ext{PGL}_n(R)$, building on and making concrete the previous qualitative results.

## Key findings

- Established explicit bounds for sup-norms of cusp forms.
- Quantified the dependence of bounds on Laplacian eigenvalues.
- Enhanced previous subconvexity results with explicit constants.

## 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 $\|F|_\Omega\|_\infty\ll_\Omega\lambda_F^{n(n-1)/8-\delta_n}$ 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.