Estimates of Picard modular cusp forms

In this article, for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>n</m:mi> <m:mo>≥</m:mo> <m:mn>2</m:mn> </m:mrow> </m:math> {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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>SU</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>n</m:mi> <m:mo>,</m:mo> <m:mn>1</m:mn> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>,</m:mo> <m:mi>ℂ</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {SU((n,1),C)} . The main result of the article is the following result. Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi mathvariant="normal">Γ</m:mi> <m:mo>⊂</m:mo> <m:mrow> <m:mi>SU</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mn>2</m:mn> <m:mo>,</m:mo> <m:mn>1</m:mn> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>,</m:mo> <m:msub> <m:mi mathvariant="script">𝒪</m:mi> <m:mi>K</m:mi> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> {Γ((2,1),OK)} be a torsion-free subgroup of finite index, where K is a totally imaginary field. Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msubsup> <m:mi mathvariant="script">ℬ</m:mi> <m:mi mathvariant="normal">Γ</m:mi> <m:mi>k</m:mi> </m:msubsup> </m:math> {{{BΓᵏ}}} denote the Bergman kernel associated to the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi mathvariant="script">𝒮</m:mi> <m:mi>k</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi mathvariant="normal">Γ</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {Sₖ(Γ)} , complex vector space of weight- k cusp forms with respect to Γ. Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>𝔹</m:mi> <m:mn>2</m:mn> </m:msup> </m:math> {B²} denote the 2-dimensional complex ball endowed with the hyperbolic metric, and let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>X</m:mi> <m:mi mathvariant="normal">Γ</m:mi> </m:msub> <m:mo>:=</m:mo> <m:mrow> <m:mi mathvariant="normal">Γ</m:mi> <m:mo>\</m:mo> <m:msup> <m:mi>𝔹</m:mi> <m:mn>2</m:mn> </m:msup> </m:mrow> </m:mrow> </m:math> {XΓ:=Γ²} denote the quotient space, which is a noncompact complex manifold of dimension 2. Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">|</m:mo> <m:mo>⋅</m:mo> <m:msub> <m:mo stretchy="false">|</m:mo> <m:mi>pet</m:mi> </m:msub> </m:mrow> </m:math> {|·|ₚₑₜ} denote the point-wise Petersson norm on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi mathvariant="script">𝒮</m:mi> <m:mi>k</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi mathvariant="normal">Γ</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {Sₖ(Γ)} . Then, for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>k</m:mi> <m:mo>≫</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> {k 1} , we have the following estimate: <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mrow> <m:munder> <m:mo movablelimits="false">sup</m:mo> <m:mrow> <m:mi>z</m:mi> <m:mo>∈</m:mo> <m:msub> <m:mi>X</m:mi> <m:mi mathvariant="normal">Γ</m:mi> </m:msub> </m:mrow> </m:munder> <m:mo>⁡</m:mo> <m:msub> <m:mrow> <m:mo stretchy="false">|</m:mo> <m:mrow> <m:msubsup> <m:mi mathvariant="script">ℬ</m:mi> <m:mi mathvariant="normal">Γ</m:mi> <m:mi>k</m:mi> </m:msubsup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>z</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:mi>pet</m:mi> </m:msub> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:msub> <m:mi>O</m:mi> <m:mi mathvariant="normal">Γ</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msup> <m:mi>k</m:mi> <m:mfrac> <m:mn>5</m:mn> <m:mn>2</m:mn> </m:mfrac> </m:msup> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> <m:mo>,</m:mo> </m:mrow> </m:math> _{z∈ XΓ}|{{BΓᵏ}}(z)|ₚₑₜ=O% Γ(k5/2), where the implied constant depends only on Γ.