Visualising the arithmetic of imaginary quadratic fields

Abstract

We study the orbit of $\mathbb{R}$ under the Bianchi group $\operatorname{PSL}_2(\mathcal{O}_K)$, where $K$ is an imaginary quadratic field. The orbit, called a Schmidt arrangement $\mathcal{S}_K$, is a geometric realisation, as an intricate circle packing, of the arithmetic of $K$. This paper presents several examples of this phenomenon. First, we show that the curvatures of the circles are integer multiples of $\sqrt{-\Delta}$ and describe the curvatures of tangent circles in terms of the norm form of $\mathcal{O}_K$. Second, we show that the circles themselves are in bijection with certain ideal classes in orders of $\mathcal{O}_K$, the conductor being a certain multiple of the curvature. This allows us to count circles with class numbers. Third, we show that the arrangement of circles is connected if and only if $\mathcal{O}_K$ is Euclidean. These results are meant as foundational for a study of a new class of thin groups generalising Apollonian groups, in a companion paper.