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

Abstract An effective formula for the Bergman kernel on $${\mathbb {H}}_{\gamma } = \{|z_1|^\gamma< |z_2| < 1 \}$$ H γ = { | z 1 | γ < | z 2 | < 1 } is obtained for rational $$\gamma = \frac{m}{n} >1$$ γ = m n > 1 . The formula depends on arithmetic properties of $$\gamma $$ γ , 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 $$\gamma _j \searrow 1$$ γ j ↘ 1 , where each $${\mathbb {H}}_{\gamma _j}$$ H γ j has a Bergman kernel with zeros (while $${\mathbb {H}}_1$$ H 1 is known to have a zero-free kernel), resolving an open question on this domain class.