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

TL;DR

This paper employs the heat kernel method to establish uniform bounds on the sup-norms of cusp forms of integral and half-integral weight, extending results to cofinite groups and providing bounds independent of the group.

Contribution

It introduces a heat kernel approach to derive sup-norm bounds for cusp forms of various weights, including the extension to cofinite groups, which is a novel application.

Findings

Sup-norms of cusp forms grow at most linearly with weight k.

Bounds are uniform and independent of the Fuchsian subgroup.

Extension of results to cofinite groups using existing theorems.

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 $\lbrace f_{1},\ldots,f_{j_{k}} \rbrace$ denote an orthonormal basis of $S^{k}(\Gamma)$. In this article, we show that as $k\rightarrow \infty,$ 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.