Uniform Sup-Norm Bounds on Average for Cusp Forms of Higher Weights

Abstract

Let $\Gamma \subset \mathrm{ PSL}_{2}(\mathbb{R})$ be a Fuchsian subgroup of the first kind acting by fractional linear transformations on the upper half-plane $\mathbb{H}$. Consider the d-dimensional space of cusp forms $\mathcal{S}_{2k}^{\Gamma }$ of weight 2k for $\Gamma$, and let {f ₁, …, f _d } be an orthonormal basis of $\mathcal{S}_{2k}^{\Gamma }$ with respect to the Petersson inner product. In this paper we show that the sup-norm of the quantity $S_{2k}^{\Gamma }(z):=\sum _{ j=1}^{d}\vert f_{j}(z)\vert ^{2}\,\mathrm{Im}(z)^{2k}$ 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.