Estimates of automorphic forms on $\mathrm{SU}(n,1)$

Abstract

For $n\geq 2$, let $Γ\subset \mathrm{SU}((n,1),\mathcal{O}_{K})$ be a torsion-free, finite-index subgroup, where $\mathcal{O}_K$ denotes the ring of integers of a totally imaginary number field $K$ of degree $2$. Let $\mathbb{B}^n$ denote the $n$-dimensional complex ball endowed with the hyperbolic metric, and let $X_Γ:=Γ\backslash \mathbb{B}^n$ denote the quotient space. Furthermore, let $μ_{\mathrm{hyp}}^{\mathrm{vol}}$ denote the volume form associated to the hyperbolic metric. Let $Λ:=Ω_{\overline{X}_Γ}^{n}$ denote the line bundle, where $\overline{X}_Γ:=X_Γ\cup\lbrace \infty\rbrace$. For any $k\geq 1$, let $λ^{k}:=Λ^{\otimes k}\otimes O_{\overline{X}_Γ}((k-1)\infty)$. For any $k\geq 1$, the hyperbolic metric induces a point-wise metric on $H^{0}(\overline{X}_Γ,λ^{k})$. For any $k\geq 1$, let $\mathcal{B}_{X_Γ}^{λ^{k}}$ denote the Bergman kernel associated $H^{0}(\overline{X}_Γ,λ^{k})$. Then, for $k\gg1$, the first main result of the article, is the following estimate $$ \sup_{z\in \overline{X}_Γ}\big|\mathcal{B}_{X_Γ}^{λ^{k}}(z,z)\big|_{\mathrm{hyp}}=O_{X_Γ}(k^{n+1/2}).$$ For any $k\geq 1$, and $z\in X_Γ$, let $μ_{\mathrm{Ber},k}(z)$ denote the Bergman metric associated to the line bundle $λ^{ k}$, and let $μ_{\mathrm{ber},k}^{\mathrm{vol}}$ denote the associated volume form. Then, for $k\gg1$, the second main result of the article is the following estimate $$ \sup_{z\in \overline{X}_Γ}\bigg|\frac{μ_{\mathrm{Ber},k}^{\mathrm{vol}}(z)}{μ_{\mathrm{hyp}}^{\mathrm{vol}}(z)}\bigg|=O_{X_Γ}\big(k^{2(n-1)(n+1)+n+3} \big).$$ Our estimate for the Bergman metric completes our arguments and corrects our estimate from arXiv:2305.11609, for $n=1$.