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 $\{f_{1},\ldots,f_{d_{2k}}\}$ 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 $S_{2k}^{\Gamma}(z):=\sum_{j=1}^{d_{2k}}\vert f_{j}(z)\vert^{2}\,\mathrm{Im}(z)^{2k}$ as $z$ ranges through $\mathbb{H}$.