Visualizing the Arithmetic of Imaginary Quadratic Fields. (26th February 2017) - Stange, Katherine E
Abstract: We study the orbit of R under the Bianchi group PSL_2(O_K), where K is an imaginary quadratic field. The orbit, called a Schmidt arrangement 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 -Δ and describe the curvatures of tangent circles in terms of the norm form of O_K . Second, we show that the circles themselves are in bijection with certain ideal classes in orders of 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 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.