Estimates of Picard modular cusp forms

TL;DR

This paper provides asymptotic estimates for the Bergman kernel of Picard modular cusp forms on complex hyperbolic spaces, revealing growth rates as the weight increases.

Contribution

It offers the first detailed asymptotic bounds for the Bergman kernel of Picard modular cusp forms on complex hyperbolic manifolds.

Findings

Supremum of Bergman kernel grows at most like k^{5/2} for large weight k.

Provides explicit estimates depending only on the subgroup mma.

Extends understanding of automorphic forms on complex hyperbolic spaces.

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 $\Gamma$.