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 $\{f_1,\dots ,f_d\}$ 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.