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 $\mathbb {H}$.