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

Abstract

For $n\ge 2$, let $\hat{I}“\subset \mathrm{SU}((n,1),{\mathcal{O}}_K)$ 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 $X_{\hat{I}}“:=\hat{I}“\backslash {\mathbb{B}}^n$ denote the quotient space. Furthermore, let $\hat{I}{¼}_{\mathrm{hyp}}^{\mathrm{vol}}$ denote the volume form associated to the hyperbolic metric. Let $\hat{I}›:=\hat{I}{\text{c◯}}_{{\overline{X}}_{\hat{I}}“}^n$ denote the line bundle, where ${\overline{X}}_{\hat{I}}“:=X_{\hat{I}}“\cup \{\infty \}$. For any $k\ge 1$, let $\hat{I}{»}^k:=\hat{I}{›}^{\otimes k}\otimes O_{{\overline{X}}_{\hat{I}}“}((k-1)\infty )$. For any $k\ge 1$, the hyperbolic metric induces a point-wise metric on $H^0({\overline{X}}_{\hat{I}}“,\hat{I}{»}^k)$. For any $k\ge 1$, let ${\mathcal{B}}_{X_{\hat{I}}“}^{\hat{I}{»}^k}$ denote the Bergman kernel associated $H^0({\overline{X}}_{\hat{I}}“,\hat{I}{»}^k)$. Then, for $k\gg 1$, the first main result of the article, is the following estimate $$\underset{z\in {\overline{X}}_{\hat{I}}“}{\sup }|{\mathcal{B}}_{X_{\hat{I}}“}^{\hat{I}{»}^k}(z,z){|}_{\mathrm{hyp}}=O_{X_{\hat{I}}“}(k^{n+1/2}).$$ For any $k\ge 1$, and $z\in X_{\hat{I}}“$, let $\hat{I}{¼}_{\mathrm{Ber},k}(z)$ denote the Bergman metric associated to the line bundle $\hat{I}{»}^k$, and let $\hat{I}{¼}_{\mathrm{ber},k}^{\mathrm{vol}}$ denote the associated volume form. Then, for $k\gg 1$, the second main result of the article is the following estimate $$\underset{z\in {\overline{X}}_{\hat{I}}“}{\sup }\left|\frac{\hat{I}{¼}_{\mathrm{Ber},k}^{\mathrm{vol}}(z)}{\hat{I}{¼}_{\mathrm{hyp}}^{\mathrm{vol}}(z)}\right|=O_{X_{\hat{I}}“}\left(k^{2(n-1)(n+1)+n+3}\right).$$ Our estimate for the Bergman metric completes our arguments and corrects our estimate from arXiv:2305.11609, for $n=1$.

This is the first draft, and any comments, suggestions, and remarks are most welcome. arXiv admin note: text overlap with arXiv:2301.11160