Heat kernel approach for sup-norm bounds for cusp forms of integral and half integral weight

Abstract

In this article, using the heat kernel approach from \cite{bouche}, we derive sup-norm bounds for cusp forms of integral and half integral weight. Let $\Gamma \subset {\mathrm{PSL}}_2(\mathbb{R})$ be a cocompact Fuchsian subgroup of first kind. For $k\in \frac{1}{2}\mathbb{Z}$ (or $k\in 2\mathbb{Z}$), let $S^k(\Gamma )$ denote the complex vector space of weight-$k$ cusp forms. Let $\{f_1,\dots ,f_{j_k}\}$ denote an orthonormal basis of $S^k(\Gamma )$. In this article, we show that as $k\to \infty ,$ the sup-norm for ${\sum }_{i=1}^{j_k}{y}^k|f_i(z){|}^2$ is bounded by $O(k)$, where the implied constant is independent on $\Gamma$. Furthermore, using results from \cite{berman}, we extend these results to the case when $\Gamma$ is cofinite.

arXiv comment6 pages