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 Gamma subset of SU(( 2, 1), O-K) be a torsion-free subgroup of finite index, where K is a totally imaginary field. Let B-Gamma(k) denote the Bergman kernel associated to the S-k(Gamma), complex vector space of weight-k cusp forms with respect to Gamma. Let B-2 denote the 2-dimensional complex ball endowed with the hyperbolic metric, and let X-Gamma := Gamma\B-2 denote the quotient space, which is a noncompact complex manifold of dimension 2. Let | center dot |(pet) denote the point-wise Petersson norm on S-k(Gamma). Then, for k >> 1, we have the following estimate: