Effective sup-norm bounds on average for cusp forms of even weight

Abstract

Let $\Gamma \subset {\mathrm{PSL}}_2(\mathbb{R})$ be a Fuchsian subgroup of the first kind acting on the upper half-plane $\mathbb{H}$. Consider the $d_{2k}$-dimensional space of cusp forms ${\mathcal{S}}_{2k}^{\Gamma }$ of weight $2k$ for $\Gamma$, and let $\{f_1,\dots ,f_{d_{2k}}\}$ be an orthonormal basis of ${\mathcal{S}}_{2k}^{\Gamma }$ with respect to the Petersson inner product. In this paper we will give effective upper and lower bounds for the supremum of the quantity $S_{2k}^{\Gamma }(z):={\sum }_{j=1}^{d_{2k}}|f_j(z){|}^2\mathrm{Im}(z{)}^{2k}$ as $z$ ranges through $\mathbb{H}$.