Abstract
Apollonian circle packings arise by repeatedly filling the interstices between mutually tangent circles with further tangent circles. In Euclidean space it is possible for every circle in such a packing to have integer radius of curvature, and we call such a packing an integral Apollonian circle packing. There are infinitely many different integral packings; these were studied in Part I (J. Number Theory 100, 1–45, 2003). Integral circle packings also exist in spherical and hyperbolic space, provided a suitable definition of curvature is used and again there are an infinite number of different integral packings. This paper studies number-theoretic properties of such packings. This amounts to studying the orbits of a particular subgroup ${\mathcal{A}}$ of the group of integral automorphs of the indefinite quaternary quadratic form $Q_{{\mathcal{D}}}(w,x,y,z)=2(w^{2}+x^{2}+y^{2}+z^{2})-(w+x+y+z)^{2}$ . This subgroup, called the Apollonian group, acts on integer solutions $Q_{{\mathcal{D}}}(w,x,y,z)=k$ . This paper gives a reduction theory for orbits of ${\mathcal{A}}$ acting on integer solutions to $Q_{{\mathcal{D}}}(w,x,y,z)=k$ valid for all integer k. It also classifies orbits for all k≡0 (mod 4) in terms of an extra parameter n and an auxiliary class group (depending on n and k), and studies congruence conditions on integers in a given orbit.
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