Estimates of Picard modular cusp forms

Abstract

In this article, for n ≥ 2 , we compute asymptotic, qualitative, and quantitative estimates of the Bergman kernel of Picard modular cusp forms associated to torsion-free, cocompact subgroups of SU ⁢ ( ( n , 1 ) , ℂ ) . The main result of the article is the following result. Let Γ ⊂ SU ⁢ ( ( 2 , 1 ) , 𝒪 K ) be a torsion-free subgroup of finite index, where K is a totally imaginary field. Let ℬ Γ k denote the Bergman kernel associated to the 𝒮 k ⁢ ( Γ ) , complex vector space of weight-k cusp forms with respect to Γ. Let 𝔹 2 denote the 2-dimensional complex ball endowed with the hyperbolic metric, and let X Γ := Γ \ 𝔹 2 denote the quotient space, which is a noncompact complex manifold of dimension 2. Let | ⋅ | pet denote the point-wise Petersson norm on 𝒮 k ⁢ ( Γ ) . Then, for k ≫ 1 , we have the following estimate:

where the implied constant depends only on Γ.