Uniform sup-norm bounds on average for cusp forms of higher weights

Abstract

Let $\Gamma\subseteq\mathrm{PSL}_{2}(\mathbb{R})$ be a Fuchsian subgroup of the first kind acting on the upper half-plane $\mathbb{H}$. Consider the $d$-dimensional space of cusp forms $\mathcal{S}_{k}^{\Gamma}$ of weight $2k$ for $\Gamma$, and let $\{f_{1},\ldots,f_{d}\}$ be an orthonormal basis of $\mathcal{S}_{k}^{\Gamma}$ with respect to the Petersson inner product. In this paper we show that the sup-norm of the quantity $S_{k}^{\Gamma}(z):=\sum_{j=1}^{d}| f_{j}(z)|^{2}\,\mathrm{Im}(z)^{2k}$ is bounded as $O_{\Gamma}(k)$ in the cocompact setting, and as $O_{\Gamma}(k^{3/2})$ in the cofinite case, where the implied constants depend solely on $\Gamma$. We also show that the implied constants are uniform if $\Gamma$ is replaced by a subgroup of finite index.