Visualizing the Arithmetic of Imaginary Quadratic Fields

Abstract

We study the orbit of |$\widehat{{\mathbb{R}}}$| 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 realization, as an intricate circle packing, of the arithmetic of |$K$|⁠. This article 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 generalizing Apollonian groups, in a companion paper.

Locations

• International Mathematics Research Notices - View - PDF