Visualizing the arithmetic of imaginary quadratic fields

Abstract

We study the orbit of R under the Bianchi group PSL2(OK), where K is an imaginary quadratic field. The orbit, called a Schmidt arrangement SK, 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 − and describe the curvatures of tangent circles in terms of the norm form of OK. Second, we show that the circles themselves are in bijection with certain ideal classes in orders of OK, 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 OK 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.