INTEGER POINTS ON SPHERES AND THEIR ORTHOGONAL GRIDS
MENNY AKA, MANFRED EINSIEDLER, AND URI SHAPIRA
Abstract. The set of primitive vectors on large spheres in the eu- clidean space of dimension d ≥ 3 equidistribute when projected on the unit sphere. We consider here a refinement of this problem concern- ing the direction of the vector together with the shape of the lattice in its orthogonal complement. Using unipotent dynamics we obtained the desired equidistribution result in dimension d ≥ 6 and in dimension d = 4, 5 under a mild congruence condition on the square of the radius. The case of d = 3 is considered in a separate paper.
1. Introduction Let d 3 be a fixed integer. Let Zd be the set of primitive vectors in ≥ prim Zd. Set d 1 def d 2 d d 1 S − (D) = v Z : v = D = Z √DS − , ∈ prim k k2 prim ∩ n o def where Sd 1 = x Rd : x = 1 . We would like to discuss the simultane- − ∈ k k2 ous equidistribution of the direction v Sd 1 of the elements in Sd 1(D) √D ∈ − − and the shape [Λv] of the orthogonal lattice d Λ = Z v⊥. v ∩ d 1 def d 1 d To make this more precise fix a copy of R − = R − 0 in R and d 1 × { } choose for every v S − (D) a rotation kv SOd(R) with kvv = √Ded so ∈ d 1 ∈ that kvΛv becomes a lattice in R − . Note that (1.1) [Zd : (Zv Λ )] = D ⊕ v since primitivity of v implies that the homomorphism Zd Z defined by → u (u, v) is surjective and Zv Λv is the preimage of DZ. 7→ ⊕ d 1 Therefore, kvΛv is a lattice in R − of covolume √D. In order to nor- malize this covolume, we further multiply by the diagonal matrix av = −1 −1 1 diag(D 2(d−1) , , D 2(d−1) , D 2 ). Note that the set of possible choice of k · · · v is precisely SOd 1(R)kv and that av commutes with SOd 1(R). Recall − − M.A. acknowledges the support of ISEF, Advanced Research Grant 228304 from the ERC, and SNF Grant 200021-152819. M.E. acknowledges the support of the SNF Grant 200021-127145 and 200021-152819. U.S. acknowledges the support of the Chaya fellowship and ISF grant 357/13. 1 2 M.AKA,M.EINSIEDLER,ANDU.SHAPIRA that SLd 1(R)/ SLd 1(Z) is identified with the space of unimodular lattices d 1 − − in R − so that we obtain an element def [Λv] = SOd 1(R)avkvΛv d 1 = SOd 1(R) SLd 1(R)/ SLd 1(Z), − ∈X − − \ − − which we refer to as the shape of the lattice Λv. It is possible to obtain still a bit more geometric information from the primitive vector v as follows. Given v Sd 1(D) choose w Zd with ∈ − ∈ (w, v)=1. If now v1, . . . , vd 1 is a Z-basis of Λv we see that v1, . . . , vd 1, w d − − is a Z-basis of Z and we may assume that det(v1, , vd 1, w) = 1. Let gv · · · − ∈ SLd(Z) denote the matrix whose columns are v1, , vd 1, w. Set ASLd 1 = g · · · − − ∗ g SLd 1 . The set of possible choices for gv is the coset n0 1 | ∈ − o d 1 d 1 gvASLd 1(Z). We define a grid in R − to be a unimodular lattice Λ in R − − d 1 together with a marked point on the (d 1)-dimensional torus R − /Λ. The − d 1 space ASLd 1(R)/ASLd 1(Z) is the moduli space of grids in R − . Thus, − − kvgvASLd 1(Z) represent the grid consisting of the rotated image of Λv to d 1 − d 1 R − together with the rotated image of w orthogonally projected into R − , and the well-defined double coset def [∆v] = SOd 1(R)avkvgvASLd 1(Z) − − d 1 represent this grid up-to rotations of the hyperplane R − . Thus we obtain the element [∆v] of the space
def d 1 = SOd 1(R) ASLd 1(R)/ASLd 1(Z). Y − − \ − − One should think about [∆v] as the shape of the orthogonal lattice Λv to- gether with a point on the corresponding (d 1)-dimensional torus which marks the position of orthogonal projection of−w to the hyperplane contain- ing Λv. Letν ˜D denote the normalized counting measure on the set
v d 1 d 1 , [∆v] : v S − (D) S − d 1. v ∈ ⊂ ×Y − k k We are interested to find A N for which ⊂ weak∗ (1.2)ν ˜D mSd−1 m d−1 as D with D A −→ ⊗ Y →∞ ∈ where mSd−1 m d−1 is the product of the natural uniform measures on d 1 ⊗ Y S − and d 1. We propose the following conjecture as a generalization of Linnik’s ProblemY − on spheres: Conjecture 1.1. The convergence in (1.2) holds for the subset A = N if d> 4, holds for A = N (8N) if d = 4, and for the subset \ A = D 1 D is not congruent to 0, 4, 7 modulo 8 ≥ | if d = 3. INTEGER POINTS AND THEIR ORTHOGONAL GRIDS 3
By a theorem of Legendre the restriction to the proper subset of N as in the above conjecture for d = 3 is equivalent to S2(D) being nonempty, and hence necessary. A similar statement holds for d = 4. In a separate paper [AES14] we obtain for the case d = 3 some partial re- sults towards this conjecture. However, for d> 3 we can give much stronger results using the techniques presented here. For an odd rational prime p let D(p)= D : p ∤ D . The main result of this paper is the following: { } Theorem 1.2. Conjecture 1.1 is true for d> 5. For d = 5 (resp. d = 4), the convergence in (1.2) holds for the subset A = D(p) (resp. A = D(p) (8N)) where p is any fixed odd prime. \ Theorem 1.2 will be proven using the theorem of Mozes and Shah [MS95] concerning limits of algebraic measures with unipotent flows acting ergodi- cally. More precisely we will need a p-adic analogue of this result, which has been given more recently by Gorodnik and Oh [GO11]. In particular we note that Theorem 1.2 should therefore be considered a corollary of the measure classification theorems for unipotent flows on S-arithmetic quotients (see [Ra95] and [MT94]). As explained in Lemma 3.7, the congruence condition D D(p) is a splitting condition which enables us to use the existing theory∈ of unipo- tent dynamics. It is possible to remove this splitting condition for d = 4, 5 by giving effective dynamical arguments in the spirit of [EMMV14] (see also [EMV09]), but this result is not general enough for that purpose. In [ERW14] Ren´eR¨uhr, Philipp Wirth, and the second named author use the methods of [EMMV14] to remove the congruence condition for d = 4, 5 in Theorem 1.2. We note however, that the case d = 3 remains open (apart from the partial results in [AES14] that concerns itself only with the problem 2 on S 2 and involves some stronger congruence conditions). Our×X interest in this problem arose through the work of W. Schmidt [Sch98], J. Marklof [Mar10] (see also [EMSS]). However, as Peter Sarnak and Ruixi- ang Zhang pointed out to us, Maass [Maa56] already asked similar questions in 1956 (see also [Maa59]). More precisely, the above question is the common refinement of Linnik’s problem and the question of Maass.
2. Notation and organization of the paper
A sequence of probability measures µn on a metric space X is said to equidistribute to a probability measure µ if µn converge to µ in the weak∗ topology on the space of probability measures on X. A probability measure µ is called a weak∗ limit of a sequence of measures µn if there exists a subse- quence (n ) such that µ equidistribute to µ as k . For a probability k nk →∞ measure µ and a measurable set A of positive measure, the restriction µ A of µ to A is defined by µ (B)= 1 µ(A B) for any measurable set B.| |A µ(A) ∩ Recall that a discrete subgroup Λ < L is called a lattice if L/Λ admits an L-invariant probability measure. Given a locally compact group L and a 4 M.AKA,M.EINSIEDLER,ANDU.SHAPIRA subgroup M
3. Equidistribution of joined S-adic orbits Fix some finite set S of valuations of Q. We define the algebraic ∋ ∞ groups G1 = SOd, G2 = ASLd 1, G2 = SLd 1, G = G1 G2, and G = − − S def × S S def G1 G2. Consider the homogeneous spaces = GS/G(Z ) and = × S S S S Y X GS/G(Z ). We write π : for the map induced by the natural Y → X S S projection ρSLd−1 : ASLd 1 SLd 1. Finally, let i = Gi,S/Gi(Z ). − → − Y d def For v Zprim we set Hv = StabG1 (v) under the natural action. The ∈ d group Hv is defined over Z Q as v Z . Let us mention at this point that we will prove Theorem 1.2⊂ by studying∈ the dynamics and the orbits of the stabilizer Hv of v which in particular will allows us to conclude that there are many primitive vectors in S( v 2) if D = v 2 is sufficiently large. k k k k INTEGER POINTS AND THEIR ORTHOGONAL GRIDS 5
1 We set SOd 1(R) = StabG1 (ed)(R) and note that kv− StabG1 (ed)(R)kv = 1 − kv− SOd 1(R)kv = Hv(R). Consider the diagonally embedded algebraic − group Lv defined by def 1 (3.1) L (R) = h, g− hg : h H (R) v v v ∈ v for any ring R. As gv SLd(Z), the group Lv is also defined over Z Q. We also define ∈ ⊂ def 1 L (R) = h, ρ (g− hg ) : h H (R) . v SLd−1 v v ∈ v d 1 ¯ For v S − (D) let θv = avkvgv, θv = ρSLd−1 (avkvgv) and consider the orbits ∈ (3.2) O def= (k , e ,θ , e )L+ G(ZS) v,S v f v f v,S ⊂YS and + O def= (k , e , θ¯ , e )L G(ZS) . v,S v f v f v,S ⊂XS As Lv is Q-anisotropic (e.g. because Lv(R) is compact), the Borel Harish- Chandra Theorem (see e.g. [Mar91, Theorem I.3.2.4]) implies that these are compact orbits. Let µv,S (resp. µv,S ) be the Haar measure on these orbits def def and define µ = µ + + S and µ = µ + + . S G /G G(Z ) S G /G G(ZS ) S S ∩ S S ∩ S S Remark 3.1. We have π (µv,S)= µv,S and π (µS)= µS. ∗ ∗ Theorem 3.2. Let p be a fixed odd prime and set S = ,p . For d > 5 and for any sequence v Zd with v as n {∞ }we have that { n}⊂ prim k nk→∞ →∞ µvn converge in the weak∗ topology to µS. The same conclusion holds for d = 4 or 5 when vn is a sequence of prim- { 2} itive vectors with vn as n and vn 2 D(p) for any n N, where p is a fixedk oddk→∞ prime number.→ ∞ k k ∈ ∈ The proof of Theorem 3.2 is divided into three steps. In the first we prove some preliminaries on quadratic forms and recall a theorem by Gorodnik and Oh. In the second we will establish the analog of Theorem 3.2 for the measures µv,S. In the last step, we will deduce the desired statement for the measures µv,S.
3.1. Properties of the quadratic forms. Let Q0 denote the quadratic d 2 d 1 form i=1 xi . Fix a vector v S − (D) and a rational matrix γ SLd 1(Q). ∈ ∈ − Fix aP choice of gv and consider the following quadratic map γ d 1 φ : Q − Q, u (Q g γ)(u). v → 7→ 0 ◦ v ◦ As before we identify Qd 1 with Qd 1 0 Qd so that g (γ(u)) Qd − − × { } ⊂ v ∈ def e is well-defined. We set φv = φv. For a quadratic map φ let Bφ be the associated bilinear form 1 B (u , u )= (φ(u + u ) φ(u ) φ(u )) . φ 1 2 2 1 2 − 1 − 2 6 M.AKA,M.EINSIEDLER,ANDU.SHAPIRA
Finally, the determinant of φ with respect to b1, , bd 1 is det Mφ where · · · − d 1 d 1 Mφ = (Bφ(bi, bj)) . When b1, , bd 1 is a basis for Z − Q − 1 i,j d 1 · · · − ⊂ and φ(b ) Z for all≤ i,≤ the− determinant is a well-defined integer which does i ∈ not depend of the choice of the basis. This is the case for φv with the standard basis and the choice of gv merely changes the basis, so does not influence the value of the determinant. d 1 Lemma 3.3. For any v S − (D) we have det(φv)= D. Moreover, there exist u , u Zd 1 such that∈ B (u , u ) = 1. 1 2 ∈ − φv 1 2 Proof. Using Equation (1.1) we see that the determinant of Q0 with respect 2 v1, , vd 1, v is D . But MQ0 with respect this basis, is a block matrix · · · − having a d 1-block whose determinant is det(φv) and a 1-block whose value is (v, v−)= D. Thus the first assertion follows. Let v1, , vd 1, w be a basis chosen as in the introduction. Since d 3 we can assume· · · without− loss of generality that v is in v w . It is enough≥ to 1 ⊥∩ ⊥ show the second assertion while considering the map φv with this choice of a basis as the columns of the matrix g . As v is primitive we can find u Zd v 1 ∈ with (u, v1) = 1. As (w, v1) = 0 we can add to u multiples of w, so we can d d 1 assume (as v1, , vd 1, w is a basis for Z ) that u = i=1− aivi, ai Z. This ·d · · 1 − ∈ implies that i=1− ai(vi, v1) = 1 showing Bφv (e1, (a1P, , ad 1)) = 1. · · · − P Define Hφ = SO(φ) < SLd 1 by T SLd 1 : φ T = φ . Recall that − { ∈ − ◦ } ρSLd−1 : ASLd 1 SLd 1 denotes the natural projection. Following the definitions, we− have→ that−
γ 1 1 1 (3.3) Hφv = γ− Hφv γ = γ− ρSLd−1 (gv− Hvgv)γ. Lemma 3.4. Let φ, φ : Qd 1 Q be two quadratic maps and assume that ′ − → Hφ = Hφ′ as Q-algebraic subgroups of SLd 1. Then there exists r Q× such − ∈ that φ = rφ′. Proof. It is enough to prove this statement over C. Thus, we can assume that Mφ is the identity matrix. We need to show that Mφ′ is a scalar ij matrix. Fix 1 i < j d 1 and let Mφ be the 2 by 2 matrix whose ≤ ≤ − ij entries are the ii, ij, ji, jj entries of Mφ and similarly for Mφ′ . Acting on t ij ij matrices with A.M = A MA, Mφ is preserved by SO2(C), and so is Mφ′ ij by our assumption. A direct calculation show that this implies that Mφ′ = diag(r, r) for some r = 0. Applying this argument for all possible i = j implies the claim. 6 6 d Proposition 3.5. γv γu If Hφv = Hφu for some v, u Zprim and some γv, γu 2 ∈ ∈ S v S SL (Z ), then k k (Z ) . d 1 u 2 × − k k ∈ γv γu Proof. By Lemma 3.4 φv = rφu for some r Q. Using the standard basis d 1 t t ∈ of Z − , it follows that γvMφv γu = r γuMφu γu. By Lemma 3.3 there exist d 1 t · S w1, w2 Z − with w1Mφu w2 = 1. Since γv, γu SLd 1(Z ) it follows ∈ ∈ − INTEGER POINTS AND THEIR ORTHOGONAL GRIDS 7
S 1 S that r Z . Switching the roles of v and u we see that r− Z and so ∈ 2 ∈ r (ZS) . By Lemma 3.3, det M = v and similarly for u. Noting that ∈ × φv k k det γ = det γ = 1, we get v 2 = rd 1 u 2 and the claim follows. v u k k − k k Lemma 3.6. If P is the orthogonal group of a definite quadratic form then P+ = P(R). {∞} Proof. See [Pro07, 5.1]. § d 1 Lemma 3.7. Let v S − (D) and recall that p = 2. Then the Lie algebra ∈ 1 6 of Hv (resp. ρSLd−1 (gv− Hvgv)) is a maximal semisimple Lie sub-algebra of G1 (resp. G2). If d > 5 these groups are Qp-isotropic and the same holds for d = 4 or 5 whenever D D(p). ∈ Proof. Maximality goes back to a classification made in 1952 by Dynkin [Dyn52] whose english translation may be found at [Dyn00]. As being isotropic is preserved by conjugation by gv, it is enough to prove the second statement for Hv. The group Hv is naturally the orthogonal group of the d 1 dimensional quadratic lattice Q def= (Q, Λ ), so it is enough to show − v v that Qv is isotropic. This follows from [Kit99, Theorem 3.5.1] using the congruence condition D D(p) when d = 4 or 5. Indeed, for d 1 5, ∈ − ≥ Qv is automatically isotropic over Qp. We have seen in Lemma 3.3 that Qv has discriminant D. Denote the Hasse invariant of Qv by S(Qv). Note that for p = 2 the congruence condition p ∤ D implies that S(Q ) = 1 (use 6 v [Kit99, Theorem 3.3.1(d)]). This, in turn, implies that Qv is isotropic (use [Kit99, Theorem 3.5.1]).
3.2. Limits of algebraic measures. Let G SLk (for some integer k) be a connected semisimple Q-group, S a finite set⊂ of valuations containing all the valuations for which G(Qv) is compact, GS = G(QS) and Γ a finite-index subgroup of G(ZS) = G(Q ) SL (ZS). Let X def= G /Γ and let (X ) S ∩ k S S P S denote the space of probability measures on XS. Mozes and Shah showed in [MS95] that limits of algebraic probability measure are again algebraic if some unipotent flows act ergodically for each of the measures in the sequence. We are going to use the following analogue for S-arithmetic quotients obtained by Gorodnik and Oh, which we include here in a slightly simplified version.
Theorem 3.8 ([GO11, Theorem 4.6]). Let Li be a sequence of connected semisimple subgroups of G and assume that there exists p S such that for ∈ any Li and any Qp-normal subgroup N < Li, N(Qp) is non-compact. Let gi def be a sequence of elements of GS and set νi = µ L+ . If the centralizers of gi i,SΓ all L are Q-anisotropic, then ν ,ν ,... is relatively compact in (X ). i { 1 2 } P S Assume that νi weakly converge to ν in (XS ), then the following statements hold: P (1) There exists a Zariski connected algebraic group M defined over Q such that ν = µgMΓ where M is a closed finite-index subgroup of MS 8 M.AKA,M.EINSIEDLER,ANDU.SHAPIRA
and g GS. If the centralizers of all Li are Q-anisotropic, then M is semi-simple.∈ (2) There exists a sequence γi Γ such that for all i sufficiently large 1 ⊂ we have γiLiγi− M. ⊂ + 1 (3) There exists a sequence hi Li,S such that gihiγi− converges to g as i . ∈ →∞ 3.3. Step I: Proof of Theorem 3.2 for orthogonal lattices. Let C =
vn : n N and S = ,p . We apply Theorem 3.8 with Ln = Lvn and { ∈ } ¯ {∞ } gn = (kvn , ep, θv, ep) GS. By Lemma 3.7 and the congruence assumption in Theorem 3.2 when∈ d = 4 or 5 the main assumption to Theorem 3.8 is satisfied.
Let ν be a weak∗ limit of µv,S v C which is the limit of a subsequence ∈ µ for C1 C. We wish to show that ν = µ . By Lemma 3.7 the v,S v C1 S ∈ ⊂ centralizer of Hv is finite hence Q-anisotropic so ν is a probability measure by Theorem 3.8. Applying Theorem 3.8.(1)–(2), we find a semisimple algebraic Q-group M < G and C C such that C C < and for all v C we have 2 ⊂ 1 | 1 \ 2| ∞ ∈ 2 1 (3.4) γvLvγv− < M for some γ G(ZS). v ∈ Assume, for a moment, that M = G. We will again use Theorem 3.8 to conclude the desired equidistribution. First we note that all the dy- + namics takes place in the orbit of GS . Indeed, by Lemma 3.6 the element + + + g = (k , e, θ¯ , e) belongs to G for any n N and L G . Using The- n vn v S ∈ vn,S ⊂ S orem 3.8.(1), we see that ν = µ S for a subgroup M of finite-index gM0G(Z ) 0 + in G and some g G . By [BT73, 6.7] M+ = G is a minimal finite- S ∈ S § S S index subgroup of GS and therefore contained in M0. Since ν is supported + S + inside GS G(Z ), it follows that M0 = GS and that ν = µ + = µv,S. GS G(Z) Therefore, the proof of this step will be concluded once we show: Claim. M = G.
Proof of the Claim . Let π1 : G1 G2 G1 and π2 : G1 G2 G2 denote × def→ × → the natural projection and define Mi = πi(M). Since M is semisimple and G1 and G2 have non-isomorphic simple Lie factors, it is enough to show that M1 = G1 and M2 = G2. Case I: v 2 is not eventually supported on a geometric progression k k v C2 k∈ of the form D0p k N. We begin with M1: we know that M1 contains ∈ 1 S subgroups of the form γv− Hvγv for any v C2 where γv G1(Z ). By 1 ∈ ∈ Lemma 3.7, each γv− Hvγv is a maximal semisimple subgroup. Thus, if 1 1 M1 = G1 then for all v, u C2 we have γv− Hvγv = γu− Huγu, which says that 6 ∈ 1 1 H −1 = H −1 . This in turn implies γ v = αγ u for some α = 0. As v, u γv v γu u v− u− d 1 1 6 S d 1 d are primitive vectors in Z , γv− v, γv− u are primitive vectors in (Z ) = Z[ p ] INTEGER POINTS AND THEIR ORTHOGONAL GRIDS 9
(considered as a ZS-module). Thus α (ZS) = pn : n Z . Noting ∈ × {± ∈ } that for all v C we have γ 1v 2 = v 2, we obtain a contradiction ∈ 2 v− k k under the assumption of Case I.
Assume now that M2 = G2. It follows from (3.4) and from (3.3) that M2 6 S contains subgroups of the form H γv for all v C where γ G (Z ). By φv ∈ 2 v ∈ 2 γ γu γv Lemma 3.7 Hφv is always maximal, so we have that Hφu = Hφv for all v, u C2. It follows from Proposition 3.5, that the assumption of Case I is not satisfied.∈ This concludes the proof of Case I. Note that this is the only relevant case when d = 4 or 5 (as in that case we assume that p ∤ v 2 for all v C). k k ∈ Case II: v 2 is eventually supported on a geometric progression k k v C2 ∈ k def def of the form D0p k N. Consider Rp,q = µv, ,p,q and Rq = {∞ } v C2 ∈ ∈ µv, ,q for some odd prime q = p. For Rq we are again in Case {∞ } v C2 6 ∈ ,p,q I, so by the above, Rq converges to µ ,q . We consider the space {∞ } {∞ } X with the natural projections
,p,q Xr{∞ ▲▲} rrr ▲▲▲ rrr ▲▲▲ ry rr ▲▲% ,p ,q X {∞ } X {∞ } and apply Theorem 3.8 to R . For any converging subsequence R R p,q p,q′ ⊂ p,q we obtain a subgroup M(Rp,q′ ) < G. As the push-forward of Rp,q′ under the ,p,q ,q natural projection {∞ } {∞ } equidistribute to µ ,q we have that X →X {∞ } M(Rp,q′ )= G. As above, this implies that Rp,q converges to µ ,p,q and in {∞ } turn, that µv, ,p equidistribute to µ ,p (and in particular that {∞ } v C2 {∞ } ∈ M = G) as we needed to show.
3.4. Upgrading from orthogonal lattices to orthogonal grids. Let ν be a weak limit of (µ ) which is the limit of a subsequence (µ ) ∗ v,S v C v,S v C1 ∈ S S ∈ S for C1 C. We need to show that ν = µS. First notice that π : has compact⊂ fibers, which together with Remark 3.1 and 3.3 givesY →X that S § π ν = µS. In particular, ν is also a probability measure. ∗ We will use the same type of argument as in 3.3: assuming that ν = µS we will use the algebraic information furnished by§ Theorem 3.8 to deduce6 a contradiction to the fact that the primitive vectors in C1 have their length going to infinity. As before, we may assume that v 2 are not even- k k v C1 ∈ n tually varying along a geometric progression of the type (D0p )n N. In fact, if this is not the case, a similar argument as in Case II above can∈ be applied, and we will not repeat it. 10 M.AKA,M.EINSIEDLER,ANDU.SHAPIRA
More precisely, we will apply Theorem 3.8 within the quotient (G SL ) / (G SL ) (ZS). 1 × d S 1 × d To simplify the notation we set G = G SL and recall that ′ 1 × d G = G1 G2 = SOd ASLd 1 < G′. × × − S We note that the orbit GSG′(Z ) is isomorphic to (and will be identified S with) S = GS/G(Z ). Recall that this implies that the finite volume Y S S orbit GSG′(Z ) is a closed subset of GS′ /G′(Z ), equivalently a closed subset of G or even of the homogeneous space W = G G . S′ S\ S′ From Theorem 3.8.(1)–(2) we find an algebraic Q-subgroup M < G′ and C C such that C C < and for all v C 2 ⊂ 1 | 1 \ 2| ∞ ∈ 2 1 (3.5) γvLvγv− < M S S for some γv G1 SLd(Z ). Moreover, ν = µgM0G1 SL (Z ) for some finite- ∈ × × d index subgroup M0 < MS and g GS′ . By construction all orbits in our S∈ sequence are contained in GSG′(Z ), which implies that the support of ν is S S also contained in this set and in particular that gG′(Z ) GSG′(Z ). This implies that we may change M by a conjugate γMγ 1 for∈ some γ G (ZS) − ∈ ′ and assume that g GS. If m M , we obtain∈ that the element gmG (ZS) belongs to the sup- ∈ 0 ′ port of ν. Therefore, GSm W belongs to the closed (and discrete) set G G (ZS). If m M is∈ sufficiently close to the identity this im- S ′ ∈ S plies m GS. As M is Zariski connected we conclude that M < G. Using the same∈ argument it also follows from Theorem 3.8.(3) that (γ ) G v v C3 ∈ ⊂ for some subset C3 C2 with C2 C3 < . ⊂ | \ | S ∞ By the previous step we know that π ν = µS, which implies that either ∗ M = G or M = G1 M2 where M2 is a Q-subgroup which is Q-isomorphic to × q q a fixed copy of SLd 1. Such subgroups are of the form SLd 1 where SLd 1 − − − SLd−1 0 Id−1 q is the conjugation of ι(SLd 1) = by cq = for − 0 1 0 1 d 1 some fixed q Q − . As above, we will be done once we show that M = G. ∈ q Assume therefore that we are in the second case and M2 = SLd 1. Using − the definition of Lv in (3.1) and projecting (3.5) using the canonical map G G , we get that for all v C that → 2 ∈ 3 1 1 1 cq− γv,−2 gv− Hvgvγv,2cq ι(SLd 1), ⊂ − where γ genotes the projection of γ to G . Let N N be such that v,2 v 2 ∈ Nq Zd 1 and set u def= g γ c (Ne ). Note that Ne is a simultaneous ∈ − v v q d d eigenvector of the right hand side. It follows that for each v C3, we have S d ∈ that u (Z ) and that u is a simultaneous eigenvector for Hv. Therefore ∈ S u = αvv for some αv Z 0 . Using that γv G2, and the definition of ∈ \ {d }1 ∈S g , we see that u = Nw + − a v with a Z . Recall that (w, v) = 1. v i=1 i i i ∈ P INTEGER POINTS AND THEIR ORTHOGONAL GRIDS 11
Taking the inner product of u with v we get α v 2 = (u, v) = (Nw, v)= N v k k for all v C . This gives a contradiction under the assumption that ∈ 3 v 2 is not eventually varying along a geometric progression of the k k v C1 ∈ n type (D0p )n N. ∈ 4. An equivalence relation d 1 Let Gi = Gi(R), Γi = Gi(Z) for i = 1, 2, K = Hed (R) and fix v S − (D) d 1 ∈ throughout this section. We identify K G1 = S − via the action of G1 d 1 1 \ ∼ on S − defined by w k− w using the initial point ed. We will also 1 7→ d 1 write w.k = k− w for this right action of k G1 on w S − (R). Recall def ∈ ∈ that this action is transitive with K = SOd 1(R) = StabG(ed). As in the − introduction, the group K can also be embedded into G2. We denote the def diagonal embedding of K by ΘK = (k, k) : k K G1 G2. d 1 d 1 d 1 { d 1 ∈ }⊂ × Let S − = S − /Γ1 and S − (D) = S − (D)/Γ1. Set v = v.Γ1 and [∆v]=[∆v]. The latter is well-defined as [∆γv]=[∆v] for all γ Γ1. Note d 1 v d 1 ∈ also that the projection v S (D) S − is well-defined. It ∈ − 7→ v ∈ follows that the following double coset k k (4.1) K K (k ,θ )Γ Γ × v v 1 × 2 represents the pair
v d 1 , [∆v] S − d 1. v ∈ ×Y − k k Note that all the measures appearing in equation (1.2) are Γ1-invariant so d 1 if we consider their projection νD ofν ˜D to S − d 1 we have that the convergence (1.2) is equivalent to ×Y − weak∗ (4.2) νD mSd−1 m as D with D A −→ ⊗ Yd−1 →∞ ∈ 4.1. Definition of Pv and the measure νv. From now on we will use only S 1 one odd prime p so we fix S = ,p and then QS = R Qp and Z = Z[ p ]. d 1 {∞ } d × Fix v S − (D). We say w v for w Z if there exist gp G1(Zp) and ∈ 1 ∼ ∈ 1 + ∈ γp G1(Z[ p ]) such that gpw = v, γpw = v and gpγp− Hv, p . ∈ ∈ { } Lemma 4.1. The relation is an equivalence relation. ∼ d + Proof. The key fact we will use is that for any v Zprim, Hv, p is a minimal ∈ { } finite index subgroup of Hv, p (see [BT73, 6.7]). Reflexivity is immediate. { } § For symmetry, assume w v with gp, γp as above. Taking the inverse 1 + ∼ 1 we have γpgp− Hv, p . It follows from gp− Hvgp = Hw and the key fact 1 ∈+ { } + 1 + above that gp− Hv, p gp = Hw, p showing that gp− γp Hw, p establishing { } { } ∈ { } symmetry. For transitivity, let v Sd 1(D) be given and let w , w be two ∈ − 1 2 vectors satisfying v w . We will show that w w . Let γ G (Z[ 1 ]), g ∼ i 2 ∼ 1 i ∈ 1 p i ∈ 12 M.AKA,M.EINSIEDLER,ANDU.SHAPIRA
G1(Zp), be the elements arising from the definition of v wi. Then, w2 = 1 1 ∼ γ1− γ2w1 = g1− g2w1. Transitivity follows as 1 1 1 1 1 1 1 + + g− g (γ− γ )− = g− g γ− γ g− g g− H g = H 2 1 2 1 2 1 1 2 2 2 2 v, p 2 w2, p ∈ { } { } where the latter equality again follows from the key fact.
The equivalence relation satisfies that if w v and γ Γ1 then γw ∼ ∼ d 1 ∈ def∼ γv, and so it descends to an equivalence relation on S − (D). We set Pv = w w : w v and Rv = w , [∆w] : w Pv . We finally define νv = { ∼ } k k ∈ ν . In the next sectionn we will relate ν to theo measure µ . D|Rv v v,S 5. Deducing Theorem 1.2 from Theorem 3.2. def 5.1. Restriction to the principal genus. Consider the open orbit ′ = G(R Z )G(Z[ 1 ]) . The set is also closed by [PR94, TheoremU × p p ⊂ YS U ′ 5.1]. By [PR94, 8.2] G+ < G is a clopen finite-index subgroup. Finally, § S S we consider the clopen set = G+G(Z[ 1 ]) . Thus, with v as in U U ′ ∩ S p n Theorem 3.2, we have that def weak∗ def (5.1) ηvn = µvn,S η = µ as n |U −→ |U →∞ We have a projection map π from , considered as a subset of S, to U ′ Y ∞ = G(R)/G(Z) obtained by dividing from the left by e G(Zp). Noting Y + { }× that G = G(R) (this follows from Lemma 3.6 and the fact that SLd 1 − is simply-connected),{∞} it follows that π (η) is a probability measure which is invariant under G(R), that is, it is∗ the uniform probability measure on G(R)/G(Z). Therefore, under the assumptions of Theorem 3.2 we have weak∗ (5.2) π (ηvn ) mG(R)/G(Z). ∗ −→ In addition, we have the projection map: (5.3) ρ : G G /Γ Γ K K G G /Γ Γ 1 × 2 1 × 2 → × \ 1 × 2 1 × 2 Below we will show that the measures (ρ π) ηv and νQv are closely related. ◦ ∗ 5.2. Description of η as union of orbits. Fix v Sd 1(D) and note v ∈ − that H (Z ) = H (Q ) G (Z ). Let π : S S def= G /G (ZS) be v p v p ∩ 1 p 1 Y → Y1 1,S 1 the natural projection. For h H+ the double coset H (Z )hH (Z[ 1 ]) ∈ v, p v p v p is either contained in π ( ), in which{ } case we set s(h) = 0, or it is dis- 1 U joint from π1( ), in which case we set s(h) = other. We will not need this, but wishU to note that the symbol 0 corresponds here to our quadratic 2 2 form Q0(x1,...,xd)= x1 + +xd and ‘other’ for the other quadratic forms in the spin genus of Q . For· · ·s 0, other choose a set M satisfying the 0 ∈ { } s following equality (as subsets of Hv(Qp)): 1 1 Hv(Zp)hHv(Z[ p ]) = Hv(Zp)hHv(Z[ p ]) h H+ [,s(h)=s hGMs ∈ v,{p} ∈ INTEGER POINTS AND THEIR ORTHOGONAL GRIDS 13
def where the later denotes a disjoint union. The sets M0, Mother, Mfull = M0 M are finite by [PR94, Theorem 5.1]. Using that H+ G+ and∪ other v, p ⊂ 1, p the definition of we have { } { } U + + 1 h Hv, p : s(h) = 0 = Hv, p G1(Zp)G1(Z[ p ]). n ∈ { } o { } ∩ + By Lemma 3.6, Lv, = Lv(R). Recall θv = avkvgv and note that {∞} (kv,θv) Lv(R) =ΘK (kv,θv). Using this we can express the orbit Ov,S 1 from (3.2) in a different form: set l(h) = gv− hgv and let us allow to re- order entries of products as needed so that + + (5.4) ΘK Lv, p = (k,h,k,l(h)) : k K, h Hv, p GS. × { } n ∈ ∈ { }o ⊂ In this notation we get + 1 Ov,S =ΘK Lv, p (kv, ep,θv, ep)G(Z[ p ]). × { } + + We also set L (Zp)= Lv, p G(Zp) and obtain { } ∩ O = Θ L+(Z )(k ,h,θ , l(h))G(Z[ 1 ]), v K × p v v p h GM ∈ full where we used the same identification as in (5.4). Thus, the restricted measure η (see (5.1)) is a Θ L+(Z )-invariant probability measure on v K × p (5.5) O = Θ L+(Z )(k ,h,θ , l(h))G(Z[ 1 ]). v ∩U K × p v v p hGM0 ∈ We note that the last equality could also have been used as the definition of + the finite set M0 Hv, p of representatives. ⊂ { } 5.3. The support of the measure π (ηv). By definition each h M0 belongs to G (Z )G (Z[ 1 ]). So we can write∗ h = c (h)γ (h) 1 where c ∈(h) 1 p 1 p 1 1 − 1 ∈ G (Z ) and γ (h) G (Z[ 1 ]). Using the fact that G has class number 1, we 1 p 1 ∈ 1 p 2 can write l(h)= c (h)γ (h) 1 where c (h) G (Z ) and γ (h) G (Z[ 1 ]). 2 2 − 2 ∈ 2 p 2 ∈ 2 p Proposition 5.1. The measure π (ηv) is a ΘK-invariant probability mea- sure on ∗ def (5.6) = Θ (k γ (h), a k g γ (h))Γ. Oh K v 1 v v v 2 hGM0 hGM0 ∈ ∈ Proof. The proposition follows immediately by plugging the decompositions 1 1 h = c1(h)γ1(h)− and l(h) = c2(h)γ2(h)− into (5.5) while recalling two facts. The first is that the map π is dividing by e G(Z ) from the left. { }× p The second fact is that (γ (h), γ (h), γ (h), γ (h)) G(Z[ 1 ]). 1 1 2 2 ∈ p Let us note that ΘK(kvγ1(h), avkvgvγ2(h))Γ does not depend on the choice + 1 of the representative of the double coset Hv, p (Zp)hH(Z[ p ]) and also not on the choice of the above decompositions.{ In} fact, let us first assume 1 1 that h = c1γ1− = c1′ (γ1′ )− are two decompositions as above. This gives that 14 M.AKA,M.EINSIEDLER,ANDU.SHAPIRA
1 1 1 c1− c1′ = γ1− γ1′ belongs to G1(Zp) G1(Z[ p ]) = G1(Z)=Γ1, which implies ∩ 1 the first half of the claimed independence in the G1-factor. If now h = c1γ1− 1 1 as above, g H (Z ), and β H (Z[ ]), then g hβ = (g c )(γ− β) and p ∈ v p ∈ v p p p 1 1 we associate to this point the double coset Kk β 1γ Γ . Using k β 1k 1 v − 1 1 v − v− ∈ K the latter equals Kkvγ1Γ1, which is the claimed independence for the components in G1. The proof for the component in G2 is similar. We will now relate the set appearing in the above proposition with the set P introduced in 4.1. v § Proposition 5.2. For h M set ϕ(h) = Kk γ (h)Γ . Then ϕ is a ∈ 0 v 1 1 bijection from M0 to KkuΓ1 : u Pv . Noting that ϕ(h) corresponds to 1 { ∈ } u = γ1(h)− v we further claim that Kavkvgvγ2(h)Γ2 = [∆u].
Proof. Fix h M0 and recall that h stabilizes v. We first need to show that u = γ (h) 1v∈ Zd 1: indeed, we have 1 − ∈ − 1 d 1 1 1 d Z[ ] γ (h)− v = c (h)− hv = c (h)− v Z p ∋ 1 1 1 ∈ p so u Zd as Z[ 1 ] Z = Z. Now, the elements c (h), γ (h) satisfy ∈ p ∩ p 1 1 1 c1(h)u = c1(h)γ1(h)− v = hv = v and γ1(h)u = v. 1 + As c1(h)γ1(h)− = h Hv, p this shows that u v and therefore Kkvγ1(h)Γ1 ∈ { } ∼ belongs to KkuΓ1 : u Pv . { ∈ } 1 + To see that ϕ is onto, fix u v and let hu = gpγp− Hv, p arising from ∼ ∈ { } the definition of in 4.1. Then γpu = v and s(hu) = 0. Let h¯ M0 be ∼ § 1 ¯ 1 ∈ such that Hv(Zp)huHv(Z[ p ]) = Hv(Zp)hHv(Z[ p ]). We have explained above that KkuΓ1 = KkvγpΓ1 = Kkvγ1(h¯)Γ1. For injectivity, let h , h M and set α = γ (h ), k = c (h ), i = 1, 2. 1 2 ∈ 0 i 1 i i 1 i Assuming ϕ(h1)= ϕ(h2), there exist a γ Γ1 such that Kkvα1γ = Kkvα2. 1 1 ∈ 1 1 Thus α1γα2− stabilize v so α1γα2− Hv(R) G(Z[ p ]) = Hv(Z[ p ]). Also 1 1 1 1 1 ∈ ∩ 1 1 (k2γ− k1− )v = (h2α2γ− α−1 h1− )v = v so k2γ− k1− Hv(Qp) G(Zp) = 1 1 1 ∈ ∩ Hv(Zp). As (k2γ− k1− )h1(α1γα2− )= h2 we see that 1 1 Hv(Zp)h1Hv(Z[ p ]) = Hv(Zp)h2Hv(Z[ p ]). For the second assertion, fix h M and let u = γ (h) 1v. We will use the ∈ 0 1 − abbreviations γi = γi(h), ci = ci(h) for i = 1, 2 which satisfy by definition 1 1 1 that h = c1γ1− and l(h)= gv− hgv = c2γ2− . We need to show that ? (5.7) Kavkvgvγ2Γ2 = [∆u]= KaukuguΓ2.
Note first that av = au and that kvγ1 is a legitimate choice of ku. With these choices (and using the identity of K on both sides), (5.7) will follow once we 1 1 1 1 show gu− γ1− gvγ2 Γ2. The element gu− γ1− gvγ2 is certainly a determinant ∈ d 1 1 element which map R − to itself. Furthermore, the last entry of its last column is positive by the orientation requirement in the definition of gv and INTEGER POINTS AND THEIR ORTHOGONAL GRIDS 15
d gu. Therefore, it will be enough to show that this element maps Z to itself. We use again that Z[ 1 ] Z = Z to obtain p ∩ p 1 d 1 1 d 1 1 1 1 d Z[ ] g− γ− g γ Z = g− c− c γ− g γ c− c Z = p ⊃ u 1 v 2 u 1 1 1 v 2 2 2 1 1 1 1 d 1 1 d d = g− c− hg g− h− g c Z = g− c− g c Z Z . u 1 v v v 2 u 1 v 2 ⊂ p
5.4. Weights of (ρ π) ηv and νv. Fix a sequence (vn) of vectors satisfy- ◦ ∗ def ing the conditions of Theorem 3.2 and set µn = (ρ π) ηvn (with π as in def ◦ ∗ (5.2) and ρ as in (5.3)) and νn = νvn . It follows from Propositions 5.1–5.2 that Rvn = Supp(νn) = Supp(µn). Let λn denote the normalised counting measure on Rvn . In this section we show n n (5.8) µ λ →∞ 0 and ν λ →∞ 0, n − n −→ n − n −→ That is, the measures µn and νn are equal to λn up-to a negligible error. For d 1 ˜ u S − let S(u)= StabΓ1 (u) for some u u and E = E d 1 where ∈ | | ∈ ×Y − def d 1 d 1 E˜ = u S − : for u u, S(u) > 1 S − . n ∈ ∈ o ⊂ The convergences in (5.8) follow from the following two lemmata:
Lemma 5.3. Fix n N and let v = vn. Set Mn = maxx Rv µn(x) and ∈ Mn ∈ Nn = maxx Rv νn(x) and a = Γ1 . For every x Rv, µn(x) Mn ∈ a Nn | | ∈ ≤ ≤ and a νn(x) Nn. Furthermore, equality holds in the right hand side of both inequalities≤ ≤ when x R E. ∈ v \ Lemma 5.4. We have that n (5.9) R E / R →∞ 0. | vn ∩ | | vn | −→ u def u Proof of Lemma 5.3. Let x(u) = u , [∆u] and S(x(u)) = S( u ). By k k k k Γ1 the definition of ν , we have for x(u) R that ν (x(u)) = | | . So n vn n Rv S(x(u)) ∈ | n | the lemma follows for νn. For µn first note that, using (5.5) we have µ (x(u)) = η (Θ L+(Z )(k ,h,θ , l(h))G(Z[ 1 ])) n vn K × p v v p where h = h(x(u)) is the unique (by Prop. 5.2) element corresponding to x(u) in M0. Therefore, we will be done once we show that the stabilizer of the above orbit, namely,
+ 1 1 (5.10) ΘK L (Zp) αhG(Z[ p ]))αh− × ∩ def is bounded by S(x(u)), where αh = (kv,h,θv, l(h)). To this end, notice that + as ΘK L (Zp) embeds diagonally into the product G1 G2, the third and the fourth× coordinate of an element in this stabilizer are× determined by the first and the second. As we are only interested in getting an upper bound it 16 M.AKA,M.EINSIEDLER,ANDU.SHAPIRA is enough to consider the stabilizer in G1. Using that Kkv = kvHv(R) and + def + H (Zp) = Hv, p G1(Zp) G1(Zp) it is enough to bound { } ∩ ⊂ 1 1 (Hv(R) G1(Zp)) (e, h)G1(Z[ p ])(e, h− ) . × ∩ def 1 Using the decomposition h = cγ = c(h)γ(h)− and that c G1(Zp) the latter is bounded by ∈ 1 1 (5.11) (Hv(R) γG1(Zp)γ− ) G1(Z[ p ]) . × ∩ 1 1 1 1 As γ G1(Z[ ]) we have γG1(Zp)γ− G1(Z[ ]) = γG 1(Z)γ− . Therefore ∈ p ∩ p 1 1 H (R) γG (Z)γ− = γ− H (R)γ G (Z) = S(x(u)) v ∩ 1 v ∩ 1 bounds (5.11). n Proof of Lemma 5.4 . We have that µn(E)= µn(E˜ d 1) →∞ 0 since by ˜ ˜ ×Y − −→ d 1 (5.2) we have lim supn (π1) µn(E) mSd−1 (E) = 0. Here π1 : S − d 1 →∞ ∗ ≤ × d 1 S − is the projection map. Using Lemma 5.3 we have Y − → Mn µ (E R ) E Rv 1 E R (5.12) µ (E)= n ∩ vn a | ∩ n | | ∩ vn | n µ (R ) ≥ M R ≥ a R n vn n| vn | | vn | which gives (5.9).
This shows (5.8) and thus that
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Departement Mathematik, ETH Zurich,¨ Ramistrasse¨ 101, 8092 Zurich, Switzer- land E-mail address: [email protected]
Departement Mathematik, ETH Zurich,¨ Ramistrasse¨ 101, 8092 Zurich, Switzer- land E-mail address: [email protected]
Department of Mathematics, Technion, Haifa, Israel E-mail address: [email protected]