Apollonian Circle Packings: Number Theory II. Spherical and Hyperbolic Packings
| dc.creator | Eriksson, Nicholas | |
| dc.creator | Lagarias, Jeffrey C. | |
| dc.date | 2004-03-17 | |
| dc.date | 2005-03-09 | |
| dc.date.accessioned | 2026-07-07T12:09:52Z | |
| dc.date.available | 2026-07-07T12:09:52Z | |
| dc.description | 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. | |
| dc.description | 32 pages, 5 figures. To appear in the Ramanujan Journal. Proof of Thm 3.2 made more clear, otherwise small changes | |
| dc.identifier | https://arxiv.org/abs/math/0403296 | |
| dc.identifier | http://arxiv.org/abs/math/0403296 | |
| dc.identifier | Ramanujan Journal 14 (2007), no. 3, 437--469 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/209775 | |
| dc.subject | Number Theory | |
| dc.subject | 11H55 | |
| dc.title | Apollonian Circle Packings: Number Theory II. Spherical and Hyperbolic Packings | |
| dc.type | text |