{
  "id": "math/0403296",
  "version": "v2",
  "published": "2004-03-17T23:22:29.000Z",
  "updated": "2005-03-09T07:47:06.000Z",
  "title": "Apollonian Circle Packings: Number Theory II. Spherical and Hyperbolic Packings",
  "authors": [
    "Nicholas Eriksson",
    "Jeffrey C. Lagarias"
  ],
  "comment": "32 pages, 5 figures. To appear in the Ramanujan Journal. Proof of Thm 3.2 made more clear, otherwise small changes",
  "journal": "Ramanujan Journal 14 (2007), no. 3, 437--469",
  "categories": [
    "math.NT"
  ],
  "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 the paper \\cite{GLMWY21}. Integral circle packings also exist in spherical and hyperbolic space, provided a suitable definition of curvature is used (see \\cite{LMW02}) 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 $\\sA$ of the group of integral automorphs of the indefinite quaternary quadratic form $Q_{\\sD}(w, x, y, z)= 2(w^2+x^2 +y^2 + z^2) - (w+x+y+z)^2$. This subgroup, called the Apollonian group, acts on integer solutions $Q_{\\sD}(w, x, y, z)=k$. This paper gives a reduction theory for orbits of $\\sA$ acting on integer solutions to $Q_{\\sD}(w, x, y, z)=k$ valid for all integer $k$. It also classifies orbits for all $k \\equiv 0 \\pmod{4}$ 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.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2005-03-09T07:47:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11H55"
    ],
    "keywords": [
      "number theory",
      "hyperbolic packings",
      "integral packings",
      "integer solutions",
      "apollonian circle packings arise"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math......3296E"
    }
  }
}