Estimates of Picard modular cusp forms

Abstract

In this article, for $n\geq 2$, we compute asymptotic, qualitative, and quantitative estimates of the Bergman kernel of Picard modular cusp forms associated to torsion-free, cocompact subgroups of $\mathrm{SU}\big((n,1),\mathbb{C}\big)$. The main result of the article is the following result. Let $\Gamma\subset \mathrm{SU}\big((2,1),\mathcal{O}_{K}\big)$ be a torsion-free subgroup of finite index, where $K$ is a totally imaginary field. Let $\mathcal{B}_{\Gamma}^{k}$ denote the Bergman kernel associated to the $\mathcal{S}_{k}(\Gamma)$, complex vector space of weight-$k$ cusp forms with respect to $\Gamma$. Let $\mathbb{B}^{2}$ denote the $2$-dimensional complex ball endowed with the hyperbolic metric, and let $X_{\Gamma}:=\Gamma\backslash \mathbb{B}^{2}$ denote the quotient space, which is a noncompact complex manifold of dimension $2$. Let $\big|\cdot\big|_{\mathrm{pet}}$ denote the point-wise Petersson norm on $\mathcal{S}_{k}(\Gamma)$. Then, for $k\geq 6$, we have the following estimate \begin{equation*} \sup_{z\in X_{\Gamma}}\big|\mathcal{B}_{\Gamma}^{k}(z)\big|_{\mathrm{pet}}=O_{\Gamma}\big(k^{\frac{5}{2}}\big), \end{equation*} where the implied constant depends only on $\Gamma$.