Arithmetic properties and zeros of the Bergman kernel on a class of quotient domains

Abstract

An effective formula for the Bergman kernel on ${\mathbb{H}}_{\gamma }=\{|z_1{|}^{\gamma }<|z_2|<1\}$ is obtained for rational $\gamma =\frac{m}{n}>1$. The formula depends on arithmetic properties of $\gamma$, which uncovers new symmetries and clarifies previous results. The formulas are then used to study the Lu Qi-Keng problem. We produce sequences of rationals ${\gamma }_j\searrow 1$, where each ${\mathbb{H}}_{{\gamma }_j}$ has a Bergman kernel with zeros (while ${\mathbb{H}}_1$ is known to have a zero-free kernel), resolving an open question on this domain class.

24 pages, 2 figures. To appear in Math Z