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

TL;DR

This paper establishes effective bounds on the supremum of the sum of squared cusp form values, weighted by the imaginary part, for a given Fuchsian subgroup, advancing understanding of their growth behavior.

Contribution

It provides explicit upper and lower bounds for the supremum of the sum of cusp form squares, improving previous asymptotic estimates with effective constants.

Findings

Derived explicit bounds for the supremum of cusp form sums.

Enhanced understanding of cusp form growth on the upper half-plane.

Results applicable to Fuchsian groups of the first kind.

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 $\mathbb{H}$.