The Local-Global Principle for Integral Soddy Sphere Packings

Fix an integral Soddy sphere packing P. Let K be the set of all curvatures in P. A number n is called represented if n is in K, that is, if there is a sphere in P with curvature equal to n. A number n is called admissible if it is everywhere locally represented, meaning that n is in K(mod q) for all q. It is shown that every sufficiently large admissible number is represented.

(a) Four tangent spheres (b) Two more spheres (c) Reflection of (b) through a sphere centered at p Figure 1

Introduction
This paper is concerned with a 3-dimensional analogue of an Apollonian circle packing in the plane, constructed as follows. Given four mutually tangent spheres with disjoint points of tangency (Figure 1a), a generalization to spheres of Apollonius's theorem says that there are exactly two spheres (1.1) tangent to the given ones (Figure 1b). For a proof of (1.1), take a point p of tangency of two given spheres and reflect the configuration through a sphere centered at p. Thus p is sent to ∞, and the resulting configuration (Figure 1c) consists of two tangent spheres wedged between two parallel planes; whence the two solutions claimed in (1.1) are obvious. Returning to Figure 1b, one now has more configurations of tangent spheres, and can iteratively inscribe further spheres in the interstices (Figure 2a). Repeating this procedure ad infinitum, one obtains what we will call a Soddy sphere packing (Figure 2b).
The name refers to the radiochemist Frederick Soddy (1877Soddy ( -1956, who in 1936 wrote a Nature poem [Sod36] in which he rediscovered Descartes's Circle Theorem [Des01, pp. 37-50] and a generalization to spheres, see Theorem 2.3. The latter was known already in 1886 to Lachlan [Lac86], and appears in some form as early as 1798 in Japanese Sangaku problems [San]. We name the packings after Soddy because he was the first to observe that there are configurations of circle and sphere packings in which all curvatures 1 are integers [Sod37]; such a packing is called integral. The numbers illustrated in Figure 2b are some of the curvatures in that packing. By rescaling an integral packing, we may assume that the only integers dividing all of the curvatures are ±1; such a packing is called primitive. We focus our attention on bounded, integral, primitive Soddy sphere packings. In fact, all of the salient features persist if one restricts the discussion to the packing P 0 illustrated in Figure 2b; the reader may wish to do so henceforth. What numbers appear in Figure 2b? For a typical sphere S ∈ P, let κ(S) be its curvature, and let K = K (P) be the set of all curvatures in P, K := {n ∈ Z : ∃S ∈ P, κ(S) = n}. The bounding sphere is internally tangent to the others, so is given opposite orientation and negative curvature. The first few curvatures in P 0 are: that is, there are local obstructions. In analogy with Hilbert's 11th problem on representations of numbers by quadratic forms, we say that n is represented if n ∈ K. Let A = A (P) be the set of admissible numbers, that is, numbers n that are everywhere locally represented in the sense that n ∈ K (mod q) for all q. (1.4) In our example, A is the set of all numbers satisfying (1.3). The set of admissible numbers for any packing P satisfies either (1.3) or ≡ 0 or 2 (mod 3), (1.5) see Lemma 2.11. The number of spheres in P with curvature at most N (counted with multiplicity) is asymptotically equal to a constant times N δ , where δ is the Hausdorff dimension of the closure of the packing [Kim11]. Soddy packings are rigid (one can be mapped to any other by a conformal transformation), and so δ is a universal constant; it is approximately (see [Boy73a,BdPP94]) equal to δ ≈ 2.4739 . . . .
Hence one expects, on grounds of randomness, that the multiplicity of a given admissible curvature up to N is roughly N δ−1 , which should be quite large. In particular, every sufficiently large admissible should be represented. The main purpose of this paper is to confirm this claim.
Theorem 1.6. Integral Soddy sphere packings satisfy a local-to-global principle: there is an effectively computable N 0 = N 0 (P) so that if n > N 0 and n ∈ A , then n ∈ K .
This theorem is the analogue to Soddy sphere packings of the localglobal conjecture for integral Apollonian circle packings [GLM + 03, FS11, BF11, BK12]. Being in higher dimension puts more variables into play, making the problem much easier.
For the proof, we study a certain infinite index subgroup Γ of the integral orthogonal group G preserving a particular quadratic form of signature (4, 1). This group, Γ, which we call the Soddy group, is isomorphic to the group of symmetries of P; extended to act on hyperbolic 4-space, the quotient is an infinite volume hyperbolic 4-fold. Passing to the spin double cover of the orientation preserving subgroup of G, we find that Γ contains an arithmetic subgroup Ξ acting with finite co-volume on hyperbolic 3-space. A consequence is that the set K of curvatures contains the "primitive" values of a shifted quaternary quadratic form (in fact an infinite family of such). Then using Kloosterman's method, the local-global theorem follows. This generalizes to sphere packings the following related result in 2-dimensions due to Sarnak [Sar07]: the curvatures in an integral, primitive Apollonian circle packing contain the primitive values of a shifted binary quadratic form. It is in this sense that we have more variables: instead of binary forms, sphere packings contain values of quaternary forms.
Binary forms represent very few numbers, so despite some recent advances [BF11,BK12], the analogous problem in circle packings is wide open.
In dimension n ≥ 4, one can start with a configuration of n tangent hyperspheres, repeating the above-described generating procedure. Unfortunately this does not give rise to a packing, as the hyperspheres eventually overlap [Boy73b]. Moreover there are no longer any such configurations in which all curvatures are integral (they can be S-integral, with the set S of localized primes depending on the dimension n); this follows from Gossett's [Gos37] generalization (also in verse) of Soddy's Theorem 2.3 to n-space.

Preliminaries
Let S = (S 1 , S 2 , S 3 , S 4 , S 5 ) be a configuration of five mutually tangent spheres, and let v 0 = v(S) = (κ 1 , κ 2 , κ 3 , κ 4 , κ 5 ) be the corresponding quintuple of curvatures, with κ j = κ(S j ). Any four tangent spheres, say S 1 , S 2 , S 3 , S 4 have six cospherical points of tangency, and determine a dual sphereS 5 passing through these points. Similarly, for j = 1, . . . , 4, letS j be the dual sphere orthogonal to all those in S except S j , and callS = (S 1 , . . . ,S 5 ) the dual configuration. Reflection throughS 5 fixes S 1 , S 2 , S 3 , S 4 , and sends S 5 to S 5 , the other sphere satisfying (1.1), see Figure 5a. The same holds for the other S j , and iteratively reflecting the original configuration through theS j ad infinitum yields the Soddy packing P = P(S) corresponding to S. Observe that unlike the Apollonian case, the dual spheres inS are not tangent, but intersect non-trivially, see Figure 3b.
Extend the reflections through dual spheres to hyperbolic 4-space, replacing the action of the dual sphereS j by a reflection through a hyper(hemi)sphere s j whose equator (at y = 0) isS j (with j = 1, . . . , 5).
We abuse notation, writing s j for both the hypersphere and the conformal map reflecting through s j . The group generated by these reflections acts discretely on H 4 . The A-orbit of any fixed base point in H 4 has a limit set in the boundary ∂H 4 ∼ = R 3 ∪ {∞}, which is the closure of the original sphere packing, a fundamental domain for this action being the exterior in H 4 of the five dual hyperspheres s j . Hence the quotient hyperbolic 4-fold A\H 4 is geometrically finite (with orbifold singularities corresponding to nontrivial intersections of the dual spheresS j ), and has infinite hyperbolic volume with respect to the hyperbolic measure in the coordinates (2.1). The group A is the symmetry group of all conformal transformations fixing P.
To spy out spherical affairs / An oscular surveyor / Might find the task laborious, / The sphere is much the gayer, / And now besides the pair of pairs / A fifth sphere in the kissing shares. / Yet, signs and zero as before, / For each to kiss the other four / The square of the sum of all five bends / Is thrice the sum of their squares.
If κ 1 , . . . , κ 4 are given, it then follows from (2.4) that the variable κ 5 satisfies a quadratic equation, and hence there are two solutions. This is an algebraic proof of (1.1). Writing κ 5 and κ 5 for the two solutions, it is elementary from (2.4) that In other words, if the quintuple (κ 1 , κ 2 , κ 3 , κ 4 , κ 5 ) is given, then one obtains the quintuple with κ 5 replaced by κ 5 via a linear action: This is an algebraic realization of the geometric action ofS 5 (or s 5 ) on a quintuple. Call the above 5 × 5 matrix M 5 . One can similarly replace other κ j by κ j keeping the four complementary curvatures fixed, via the matrices The inner product ·, · in (2.9) is the standard one on R 5 . This explains the integrality of all curvatures in Figure 2b: the group Γ has only integer matrices, so if the initial quintuple v 0 (or for that matter any curvatures of five mutually tangent spheres in P) is integral, then the curvatures in P are all integers (as first observed by Soddy [Sod37]). (2.10)

Let Γ, isomorphic to
The orbit under Γ, reduced mod 3 is then elementarily computed. In general we have the following Lemma 2.11. For K the set of curvatures of an integral, primitive Soddy packing P, there is always a local obstruction mod 3, either of the form (1.3) or (1.5). In particular, there is an ε = ε(P) ∈ {1, 2} so that, for any quintuple v in the cone (2.4) over Z, two entries are ≡ 0(mod 3) and three entries are ≡ ε(mod 3).
Note that we are not (yet) claiming that these are the only local obstructions; this will follow from a proof of the local-to-global theorem.
Proof. One may first attempt to understand the cone (2.4) over Z/3Z, but the form Q in (2.5) reduced mod 3 is highly degenerate. So instead consider the cone over Z/9Z. Disregarding the origin (since the packing is assumed to be primitive), there are 140 vectors mod 9, not counting permutations. Reducing these mod 3 leaves only the two vectors (0, 0, ε, ε, ε), ε ∈ {1, 2}, and their permutations. The action of Γ(mod 3) on these is trivial: each vector is fixed. This is all verified by direct computation. In this subsection, we prove that the Soddy group Γ, while being infinite index in O Q ∼ = O(4, 1), contains a subgroup which is arithmetic in SO(3, 1), or alternatively, in its spin double cover SL 2 (C). The method is a generalization of Sarnak's in [Sar07].
Recall the configuration S = (S 1 , . . . , S 5 ) of five mutually tangent spheres and the group A in (2.2) of reflections through spheres in the configurationS dual to S. Let A 1 = s 2 , ..., s 5 be the subgroup of A which fixes the sphere S 1 in S. It acts discontinuously on the interior of S 1 , which we now consider as the ball model for hyperbolic 3-space H 3 . A fundamental domain for the quotient A 1 \H 3 is the curvilinear tetrahedron interior to S 1 and exterior to the dual spheresS 2 , . . . ,S 5 , see Figure 4. In particular, it has finite volume.
To realize this geometric action algebraically, let be the corresponding subgroup of Γ, where the M j are given in (2.6).
We immediately pass to its orientation preserving subgroup, setting Then Ξ is generated by Then for j = 1, 2, 3, the conjugates are given bỹ Proof. Of course this can be verified by direct computation. But we elucidate the role of J as follows.
The matrix J is then simply the change of variables matrix from v to a.
The convenience of this conjugation is made apparent in the following Lemma 3.10. The quadratic form F in (3.9) has signature (3, 1). Its special orthogonal group SO F has spin double cover isomorphic to SL 2 (C). There is a homomorphism ρ : SL 2 (C) → SO F given explicitly (for our purposes embedded in GL 5 ) by mapping (3.12) The preimages under ρ of the matricesξ 1 ,ξ 2 ,ξ 3 in (3.6) are ±t 1 , ±t 2 , ±t 3 , respectively, where: (3.13) Here Proof. The signature of F is computed directly, and its spin group being SL 2 (C) is a general fact in the theory of quadratic forms, see e.g. [Cas78]. We construct ρ explicitly as follows. Observe that the matrix is Hermitian and has determinant −F (a). Then for g ∈ SL 2 (C), and then drawing another Dirichlet domain using these shows the equality of fundamental domains. Hence the groups are also equal. In hindsight, one can now directly verify that words in the generators (3.15) of Λ give the matrices in (3.19).
The point is that Λ is now seen to be an arithmetic group, so its elements can be parametrized, giving an injection of affine space into the otherwise intractable thin Soddy group Γ.
Then n 1 := n + κ 1 also has (n 1 , κ 1 ) = 1, and n 1 = n + κ 1 ≡ 0(mod 3). That is, n 1 is coprime to the discriminant (3.25). By (3.28), n 1 moreover satisfies the local condition (3.27). Kloosterman's method for quaternary quadratic forms (see e.g. [IK04,Theorem 20.9]) shows that every sufficiently large (with an effective bound) number which is locally represented by the form f v and coprime to its discriminant is also globally represented by f v . Hence if n 1 is sufficiently large, it is represented by f v . But then n = n 1 − κ 1 is represented by F v in (3.23), and thus appears in K by Corollary 3.22.
Since there is nothing special about κ 1 , Theorem 1.6 follows immediately from Proposition 3.26 by primitivity.