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 Bouche (Asymptotic results for Hermitian line bundles over complex manifolds: The heat kernel approach, Higher-dimensional complex varieties, pp 67–81, de Gruyter, Berlin, 1996), 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 ${S^{k}_{\nu}(\Gamma)}$ denote the complex vector space of cusp forms of weight-k and nebentypus ${\nu^{2k}}$ (${\nu^{k\slash 2}}$, if ${k \in 2\mathbb{Z}}$) with respect to ${\Gamma}$, where ${\nu}$ is a unitary character. Let ${\lbrace f_{1},\ldots,f_{j_{k}} \rbrace}$ denote an orthonormal basis of ${S^{k}_{\nu}(\Gamma)}$. 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 of ${\Gamma}$. Furthermore, using results from Berman (Math. Z. 248:325–344, 2004), we extend these results to the case when ${\Gamma}$ is cofinite.