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Apollonian Circle Packings: Number Theory II. Spherical and Hyperbolic Packings

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Apollonian Circle Packings: Number Theory II. Spherical and Hyperbolic Packings Apollonian Circle Packings: Number Theory II. Spherical and Hyperbolic Packings Nicholas Eriksson University of California at Berkeley Berkeley, CA 94720 Jeffrey C. Lagarias University of Michigan Ann Arbor, MI 48109 (February 1, 2005) ABSTRACT Apollonian circle packings arise by repeatedly filling the interstices between mutually tan- gent 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 Apol- lonian circle packing. There are infinitely many different integral packings; these were studied in the paper [8]. Integral circle packings also exist in spherical and hyperbolic space, provided a suitable definition of curvature is used (see [10]) and again there are an infinite number of dif- ferent integral packings. This paper studies number-theoretic properties of such packings. This amounts to studying the orbits of a particular subgroup of the group of integral automorphs A of the indefinite quaternary quadratic form Q (w,x,y,z) = 2(w2+x2+y2+z2) (w+x+y+z)2. D − This subgroup, called the Apollonian group, acts on integer solutions QD(w,x,y,z)= k. This paper gives a reduction theory for orbits of acting on integer solutions to Q (w,x,y,z)= k A D valid for all integer k. It also classifies orbits for all k 0 (mod 4) in terms of an extra param- ≡ eter n and an auxiliary class group (depending on n and k), and studies congruence conditions on integers in a given orbit. Keywords: Circle packings, Apollonian circles, Diophantine equations, Lorentz group AMS Subject Classification: 11H55 1. Introduction A Descartes configuration is a set of four mutually touching circles with distinct tangents. A (Euclidean) Apollonian circle packing in the plane is constructed starting from a Descartes configuration by recursively adding circles tangent to three of the circles already constructed in the packing; see [10], [5], [6], and Figure 1. Apollonian packings can be completely described by the set of Descartes configurations they contain. The curvatures (a, b, c, d) of the circles in a Descartes configuration satisfy an algebraic relation called the Descartes circle theorem. Using the Descartes quadratic form Q (a,b,c,d) := 2(a2 + b2 + c2 + d2) (a + b + c + d)2, (1.1) D − 182 182 114 146 146 114 86 86 131 62 62 131 186 42 42 186 150 75 75 150 26 143 143 26 171 171 90 90 87 87 194 194 35 182 182 35 107 66 66 107 158 158 111 14 14 111 122 122 131 131 47 47 147 147 98 98 98 98 167 167 170 59 59 170 191 191 -1 195 30 30 195 74 74 138 138 95 95 194 194 194 194 6 119 119 6 62 62 179 179 170 170 186 186 110 110 110 110 174 174 54 54 150 150 131 167 11 39 83 143 179 111 59 23 23 59 111 179 143 83 39 11 167 131 162 3 162 98 98 183 183 147 50 50 147 182 87 87 182 134 134 134 134 63 191 191 63 179 179 18 155 155 18 86 86 95 95 27 167 167 27 126 114 114 126 71 71 38 135 38 110 110 38 135 38 174 174 183 183 51 107 107 51 66 15 66 83 83 102 102 102 102 123 123 146 35 146 171 171 198 198 198 198 63 99 143 195 195 143 99 63 198 198 198 198 171 171 146 146 123 35 123 102 102 102 102 83 83 66 2 2 66 51 51 183 107 107 183 174 15 174 38 135 38 110 110 38 135 38 71 71 126 114 114 126 167 167 27 95 95 27 86 86 155 155 179 179 191 191 63 134 134 134 134 63 18 182 87 87 182 18 147 147 183 50 50 183 98 162 162 98 167 179 179 167 131 83 143 111 59 150 150 59 111 143 83 131 54 39 39 54 174 23 23 174 110 110 110 110 186 186 170 170 179 179 11 62 62 11 119 119 194 194 194 194 95 95 138 138 74 74 195 30 30 195 191 191 170 170 59 167 167 59 98 98 98 98 147 6 6 147 47 47 131 131 122 122 111 111 158 158 107 3 107 66 66 182 182 194 35 35 194 87 87 90 14 14 90 171 171 143 143 150 75 26 26 75 150 186 42 42 186 131 62 62 131 86 114 114 86 146 182 182 146 Figure 1: The Euclidean Apollonian circle packing ( 1,2,2,3) − the Descartes circle theorem states that QD(a,b,c,d) = 0. (1.2) Descartes originally stated this relation in another algebraic form; see [10]. In order to have this formula work in all cases, the curvatures of the circles in a Descartes configuration must be given proper signs, as specified in [10] and [5]. The form QD is an integral quadratic form in four variables, and the equation (1.2) has many integer solutions. If the initial Descartes configuration generating a (Euclidean) Apollonian packing has all four circles with integral curvatures, then it turns out that all the circles in the packing have integer curvatures. This integer structure was studied in the first paper in this series [8] using the action of a particular discrete subgroup of automorphisms of the Descartes quadratic form, which was there termed the Apollonian group . It is a group of integer A matrices, with generators explicitly given in 3. The Descartes configurations in a (Euclidean) § Apollonian packing are described by an orbit of the Apollonian group. That is, the Apollonian group acts on the column vector of curvatures of the initial Descartes configuration, and generates the curvature vectors of all Descartes configurations in the packing. This fact explains the preservation of integrality of the curvatures. The Apollonian group and its relevance for Apollonian packings was already noted in 1992 by S¨oderberg [19]. The first paper in this series [8] studied Diophantine properties of the curvatures in Eu- clidean integer Apollonian circle packings, based on the Apollonian group action. In [10] it was observed that there exist notions of integer Apollonian packings in spherical and hyperbolic geometry. These are based on analogues of the Descartes equation valid in these geometries, as 2 follows. In hyperbolic space, if the “curvature” of a circle with hyperbolic radius r is taken to be coth(r), then the modified Descartes equation for the “curvatures” of circles in a Descartes configuration is QD(a,b,c,d) = 4. (1.3) Similarly, in spherical space, if the “curvature” of a circle with spherical radius r is defined to be cot(r), then the modified Descartes equation is Q (a,b,c,d)= 4. (1.4) D − There exist integral Apollonian packings in both these geometries, in the sense that the (spher- ical or hyperbolic) “curvatures” of all circles in the packing are integral; see the examples in Figures 2 and 3 below. Again, the vectors of “curvatures” of all Descartes configurations in such a packing form an orbit under the action of the Apollonian group. The purpose of this paper is to study Diophantine properties of the orbits of the Apollonian group acting on the integer solutions to the Diophantine equation A QD(a,b,c,d)= k, (1.5) where k is a fixed integer. Integer solutions exist for k 0 or 3 (mod 4), and our results ≡ on reduction theory in 3 are proved in this generality. Our other results are established § under the more restrictive condition k 0 (mod 4), where we will always write k = 4m. ≡ These results cover the motivating cases k = 4 corresponding to spherical and hyperbolic ± Apollonian packings, as well as the Euclidean case k = 0. We note that the Apollonian group is a subgroup of infinite index in the group Aut(Q , Z) of all integer automorphs of the A D Descartes quadratic form QD. As a consequence, the integral solutions of QD(a,b,c,d) = k (for k 0 or 3 ( mod 4)) fall in an infinite number of orbits under the action of . In contrast, ≡ A these same integral points fall in a finite number of orbits under the full Aut(QD, Z)-action. In 2, we study the distribution of integer solutions to the equation Q (a,b,c,d) = 4m, § D determining asymptotics for the number of such solutions to these equations of size below a given bound. These asymptotics are obtained via a bijective correspondence between integral representations of integers 4m by the Descartes form and representations of integers 2m by the Lorentzian form Q (W,X,Y,Z) = W 2 + X2 + Y 2 + Z2, and then applying results of L − Ratcliffe and Tschantz [17] on integer solutions to the Lorentzian form. In 3, we describe the Apollonian group, and apply it to give a reduction algorithm for § quadruples v = (a,b,c,d) satisfying QD(v) = k in a fixed orbit of the Apollonian group, starting from an initial integer quadruple satisfying QD(v) = k. This reduction algorithm differs in appearance from that given in part I [8, Sect. 3], which treated the case k = 0; however we show its steps coincide with those of the earlier algorithm when k = 0. A quadruple obtained at the end of this algorithm is called a reduced quadruple.
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