Estimates of Picard modular cusp forms

Abstract

In this article, for $n\ge 2$, we compute asymptotic, qualitative, and quantitative estimates of the Bergman kernel of Picard modular cusp forms associated to torsion-free, cocompact subgroups of $\mathrm{SU}((n,1),\mathbb{C})$. The main result of the article is the following result. Let $\Gamma \subset \mathrm{SU}((2,1),{\mathcal{O}}_K)$ 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 $|\cdot {|}_{\mathrm{pet}}$ denote the point-wise Petersson norm on ${\mathcal{S}}_k(\Gamma )$. Then, for $k\ge 6$, we have the following estimate \begin{equation*} \sup_{z\in X_Γ}\big|\mathcal{B}_Γ^{k}(z)\big|_{\mathrm{pet}}=O_Γ\big(k^{\frac{5}{2}}\big), \end{equation*} where the implied constant depends only on $\Gamma$.

This is the first draft, and any comments, suggestions, and remarks are most welcome