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

TL;DR

This paper derives an explicit formula for the Bergman kernel on certain quotient domains, revealing arithmetic-dependent symmetries and addressing the Lu Qi-Keng problem by constructing domains with kernels that have zeros.

Contribution

It provides a new explicit formula for the Bergman kernel on $\

Findings

Derived an effective formula for the Bergman kernel on $\

Identified sequences of domains where the Bergman kernel has zeros, resolving an open question.

Uncovered new symmetries related to the arithmetic properties of $\

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.