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 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 √ −∆ 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 generalising Apollonian groups, in a companion paper.

The Apollonian structure of Bianchi groups

“…The orbit of the extended real line R under PSL 2 (O K ) forms a delicately intertwined collection of circles, some examples of which are shown in Figures 1, 4 and 7. This is the Schmidt arrangement of K, denoted S K , named, in [41], in honour of the work of Asmus Schmidt generalizing continued fractions to the complex setting [32,33,34,35,36]. The individual images of R are called K-Bianchi circles.…”

Quaternion orders and sphere packings

“…In contrast, for circle packings constructed as orbits of R under the action of a Bianchi group SL(2, O), Stange proved that all intersections are rational and furthermore are tangential if and only if the action of the unit group preserves R [Sta17]. Moreover, in that context, even if the unit group does not preserve R, it is nevertheless true that there exist two spheres that intersect at angle θ if and only if there exists a unit u ∈ O × such that…”

“…Apollonian circle packing with initial cluster (−10, 18, 23, 27). In [Sta17], Stange showed that the superpacking of the Apollonian circle packing is simply described as the orbit of R under the action of SL(2, Z[i]). In analogy, she also considered the orbit of R under the action of other Bianchi groups SL(2, O)-here O is the ring of integers of some imaginary quadratic field.…”

Local-global principles in circle packings

“…Note that x 0 , y 0 , r are rational numbers which may be written with denominator b, the curvature of C T (formulae for these integers in terms of the entries of T are given in [27,Proposition 3.7]). Then the intersection points of γT −1 · P 1 (R) with the imaginary axis are of the form √ −ds where, in the case that C T is a line, √ −ds is the height of the line, and if C T is a circle, s = y 0 ± r, and r √ d is the radius of T −1 · P 1 (R).…”