Asymptotic distribution of values of isotropic quadratic forms at $ S ...

arXiv:1605.02490v1 [math.DS] 9 May 2016

ASYMPTOTIC DISTRIBUTION OF VALUES OF ISOTROPIC QUADRATIC FORMS AT S-INTEGRAL POINTS JIYOUNG HAN, SEONHEE LIM, KEIVAN MALLAHI-KARAI

Abstract. We prove a generalization of a theorem of Eskin-Margulis-Mozes [6] in an S-arithmetic setup: suppose that we are given a finite set of places S over Q containing the archimedean place, an irrational isotropic form q of rank n ≥ 4 on QS , a product of p-adic intervals Ip , and a product Ω of star-shaped sets. We show that if the real part of q is not of signature (2, 1) or (2, 2), then the number of S-integral vectors v ∈ TΩ satisfying simultaneously q(v) ∈ Ip for p ∈ S is asymptotically λ(q, Ω)|I| · kTkn−2 , as T goes to infinity, where |I| is the product of Haar measures of the p-adic intervals Ip .

Contents 1. Introduction 1.1. The Quantitative Oppenheim Conjecture 1.2. The S-arithmetic Oppenheim Conjecture 1.3. Statements of results 2. Preliminaries 2.1. Norms on exterior products 2.2. Quadratic forms 2.3. Orthogonal groups 3. The S-arithmetic Geometry of Numbers 3.1. S-lattices and a generalization of the α function 3.2. Integration on subvarieties 4. Passage to homogeneous dynamics 5. Counting Results 6. Equidistribution of translated orbits 7. Establishing upper bounds for the certain integrals 7.1. Coarsely superharmonic functions 8. Counterexamples for ternary forms Appendix A. A version of Witt theorem References

1 2 3 3 7 7 7 7 8 8 12 12 16 20 22 28 32 34 36

1. Introduction The Oppenheim conjecture, settled by Margulis in 1986 [14], states that for any non-degenerate indefinite quadratic form q over Rn , n ≥ 3, the set q(Zn ) of values of integral vectors is a dense subset of R if q is irrational, i.e. if q is not proportional to a form with rational coefficients. Both indefiniteness and irrationality conditions are easily seen to be necessary. Margulis’ proof 1

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JIYOUNG HAN, SEONHEE LIM, KEIVAN MALLAHI-KARAI

is based on a study of orbits of certain unipotent one-parameter groups on the space of lattices in R3 . The idea of using unipotent flows dates back in an explicit form to Raghunathan and in implicit forms to Cassels and Swinnerton-Dyers. More precisely, Raghunathan observed that Oppenheim conjecture follows from the assertion that every bounded orbit of the orthogonal group SO(2, 1) on SL3 (R)/SL3 (Z) is compact. Raghunathan further conjectured in mid-seventies that if G is a connected Lie group, Γ a lattice in G, and U a unipotent subgroup of G, then the closure of any orbit U x, for x ∈ G/Γ is itself an orbit Lx, where L is a closed connected subgroup of G containing U . Note that the subgroup L depends on x. Raghunathan’s conjecture was later proved by Ratner. 1.1. The Quantitative Oppenheim Conjecture. Quantitative Oppenheim conjecture, a refinement of Oppenheim conjecture, is about the asymptotic distribution of the values of q(v), where v runs over integral vectors in a large ball. To be more precise, let Ω be a radial set defined by Ω = {x ∈ Rn : ||x|| < ρ(x/||x||)}, where ρ is a positive continuous function on the unit sphere and T Ω is the dilation of Ω by a factor T > 0. Let us also denote by N(a,b),q,Ω (T ) the number of vectors x ∈ T Ω such that q(x) ∈ (a, b). In [3], Dani and Margulis obtained the asymptotic exact lower bound: for q an irrational indefinite quadratic form on Rn with n ≥ 3, for a given interval (a, b) and a set Ω, we have lim inf T →∞

N(a,b),q,Ω (T ) ≥ 1. λq,Ω (b − a)T n−2

Note that the denominator is the asymptotic volume of the set {v ∈ T Ω : q(v) ∈ (a, b)}. The proof of the above theorem is based on studying the distribution of the shifted orbits of the form ut Kx, where ut is a certain one-parameter unipotent subgroup, K is a maximal compact subgroup of the orthogonal group SO(q) associated to q, and x ∈ SLn (R)/SLn (Z). A suitable choice of a function f on Rn allows one to approximate the number of the integral vectors v in T Ω satisfying q(v) ∈ (a, b) by an integral of the form Z fe(ut kx)dµ(x), K

where dµ is the invariant probability measure on SLn (R)/SLn (Z), and fe : SLn (R)/SLn (Z) is the P Siegel transform of f defined by fe(gΓ) = v∈gZn f (v). Although fe is an unbounded function to which the uniform version of Ratner’s theorem proved in [3] does not apply, it is yet possible to establish asymptotically sharp lower bounds by approximating fe from below by bounded functions. The question of the upper bound, which turned out to be substantially subtler, was taken up in the groundbreaking work [6]: if q is a form of signature (r, s) with r ≥ 3 and s ≥ 1, then N(a,b),q (T ) = 1. T →∞ λq,Ω (b − a)T n−2 lim

For quadratic forms of signature (2, 1) and (2, 2), certain quadratic forms were constructed in [6] so that by choosing Ω to be the unit ball, for any ǫ > 0, N(a,b),qi (Tj ) ≫ Tji (log Tj )1−ǫ for an infinite sequence Tj → ∞. These counterexamples are very well approximable by rational forms. When q has signature (2, 2), it is proven in [7] that if forms of a class called EWAS are excluded, then a similar asymptotic statement continues to hold.

ASYMPTOTIC DISTRIBUTION OF VALUES OF ISOTROPIC QUADRATIC FORMS AT S-INTEGRAL POINTS 3

1.2. The S-arithmetic Oppenheim Conjecture. The generalization we study in this article involves considering various places. Let Sf be a finite set {p1 , . . . , ps } of odd prime numbers and S = {∞} ∪ Sf . Each element of S (Sf , respectively) is called a place (a finite place, respectively). For p ∈ Sf , the p-adic norm on Q is defined by |x|p = p−vp (x) , where vp (x) is the p-adic valuation of x. The completion of Q with respect to | · |p is the field of p-adic numbers and is denoted by Qp . We will sometimes write Q∞ for R. When p ∈ Sf , Qp contains a maximal compact subring Zp consisting of elements with norm at most 1. We also write Up = Zp − pZp for the set of multiplicative units in Zp . When p = ∞, we set Up = {±1}. We will use the notation Unp = Znp − p(Znp ) for p ∈ Sf and Un∞ = Sn−1 , the set of vectors of length 1 in Rn . λp stands for the Haar measure on Qp normalized so that λp (Zp ) = 1. The Haar measure on Qnp and QnS are denoted by λnp , λnS , respectively. Q We denote by QS = p∈S Qp the direct product of Qp , p ∈ S and write ZS = {x ∈ Q : |x|p ≤ 1, ∀p ∈ Sf } for the ring of S-adic integers. We also denote by λS the product measure ⊗p∈S λp . A quadratic form q on QnS is an S-tuple (qp )p∈S , where qp is a quadratic form on Qp . The form q is called isotropic, if each qp is isotropic, that is, if there exists a non-zero vector v ∈ Qnp such that qp (v) = 0. A quadratic form over R is isotropic if and only if it is indefinite. q is called non-degenerate if each qp is non-degenerate. Finally, we call q rational if q = λq0 for a form q0 on QnS and an invertible element λ ∈ QS . Otherwise q is called irrational. Borel and Prasad proved the S-arithmetic Oppenheim conjecture over any number field k. In the case when k = Q, the theorem says that for a non-degenerate irrational isotropic quadratic form q on QnS (n ≥ 3) and any given ǫ > 0, there exists a non-zero vector x ∈ ZnS with |qp (x)| < ǫ for all p ∈ S. 1.3. Statements of results. The goal of this paper is to prove a quantitative version of the theorem of Borel and Prasad mentioned above. Note that due to the presence of |S| valuations, each one of these objects will depend on |S| parameters. Let T = (Tp )p∈S , be an S-tuple of positive real numbers with the components Tp ∈ pZ for p ∈ Sf . Such an S-tuple T willQbe called an S-time. The set of all S-time vectors is denoted by TS . For T ∈ TS , write kTk = p∈S Tp ., and m(T) = minp∈S Tp . For T = (Tp ), T′ = (Tp′ ) ∈ TS , define T T′ if Tp ≥ Tp′ for all p ∈ S. Similarly, Ti = (Tp,i )p∈S → ∞ means that Tp,i → ∞ for each p ∈ S. Since T is of the form (et∞ , pt11 , · · · , ptss ), let us denote t = (t∞ , t1 , · · · , ts ) and consider S-parameter groups Y Y Z+ . Z, T+ = R+ × T=R× p∈Sf

p∈Sf

t′

which we will use later. The notions t and t → ∞ are defined accordingly. For p ∈ S, let ρp : Unp → R be a positive continuous function. For p ∈ Sf , we will assume throughout the paper that ρ satisfies the following condition: (Iρ )

ρ(ux) = ρ(x),

Define (1)

Ω=

∀x ∈ Unp , Y

∀u ∈ Up .

Ωp ⊆ QnS ,

p∈S

where Ωp is the set of vectors vp whose norm is bounded by the value of ρp in the direction of vp . Condition (Iρ ) is indeed very mild and is satisfied by many sets of interest (e.g., the unit ball, which is defined by the constant function). Having fixed an S-time T, we denote by

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TΩ = {(zp vp )p∈S : zp ∈ Qp , |zp | ≤ Tp , vp ∈ Ωp } the dilation of Ω by T. For a vector v ∈ QnS , we denote kvk = (kvp kp )p∈S . A p-adic interval of length p−b is a set of the form Ip = a + pb Zp , where a ∈ Qp and Q b ∈ Z. An S-adic interval I is a product of p-adic interval Ip , with p ∈ S. Let us denote |I| = λp (Ip ).

Definition 1.1 (Counting and volume functions). Fixing Ω as above, the counting and volume functions are defined respectively by NI,q,Ω(T) = card{v ∈ ZnS ∩ TΩ : qp (vp ) ∈ Ip , ∀p ∈ S}, VI,q,Ω (T) = vol{v ∈ QnS ∩ TΩ : qp (vp ) ∈ Ip , ∀p ∈ S}, where card and vol denote the cardinality and the Haar measure λnS . We start with the following statement about the asymptotic volume of the set of vectors in Tp Ωp ⊆ Qnp with the condition q(v) ∈ I. This proposition will be proven in Section 4. Proposition 1.2. For a constant λ = λ(q, Ω), we have, VI,q,Ω (T) ∼ λ(q, Ω) · |I| · kTkn−2 , as T → ∞.

The asymptotic behavior of NI,q,Ω (T) is more intricate than the volume asymptotics. The issue for real forms of signature (2, 1) and (2, 2) persists in the S-arithmetic setup. However, p-adic forms of rank 4 behave more nicely than the corresponding real forms of signature (2, 2), which allow us to prove a partial result even in this case. Definition 1.3. Let q = (qp )p∈S be an isotropic quadratic form on QnS . We say that q is exceptional if either n = 3, or n = 4 and q∞ is of signature (2, 2). n Note that the set OQ S (n) of non-degenerate quadratic forms on QS can be identified with a Zariski open subset of p∈S symn (Qp ), where symn (Qp ) denotes the set of n × n matrices over Qnp . In this correspondence, a form q = (qp )p∈S is associated to its Gram matrix.

Theorem 1.4. Let D be a compact subset of non-exceptional quadratic forms over QnS of discriminant 1, I an S-interval, and Ω as in (1). Then there exists a constant C = C(D, I, Ω) such that for any q ∈ D and any sufficiently large S-time T we have NI,q,Ω (T) ≤ CkTkn−2 .

Note that this theorem is effective in the sense that the constant C can be explicitly given in terms of the data. The next theorem provides asymptotically sharp bounds for NI,q,Ω (T). For the sake of simplicity the theorem below is stated for an individual irrational form. Modifications of the proof, along the lines of the proofs in [6], can be made to establish a uniform version when q runs over a compact set of forms. Theorem 1.5. Let n ≥ 4 and q be an irrational isotropic form in O(n), I be an S-interval, and Ω be as in (1). If q is non-exceptional, then NI,q,Ω (T) = 1. T→∞ λ(q, Ω)|I| · kTkn−2 lim

Remark 1.6. From the proof of Theorem 1.5, we can also obtain a partial result in the case when q∞ has signature (2, 2). Assume that T → ∞ while being contained in the cone X log Tp log T∞ ≤ δ p∈Sf


where δ is the constant in Theorem 1.7. Then N′ I,q (T) ∼ 1, ˆ λ(q)|I| · kTkn−2 where N′ I,q (T) counts the number of integral solutions of kv∞ k ≤ T∞ ,

kvp kp = Tp ,

q(gv) ∈ I

ˆ and λ(q) is a constant depending on q. Note that the conditions above defines the intersection of a ball in the archimedean place with spheres in non-archimedean places. The proof of Theorem 1.5 rests upon a number of ingredients. We will first relate the counting question to a question about the asymptotic behavior of integrals of the form Z f (at kx)dm(k). K

Denote the space of unimodular lattices by LS (see Definition 3.1 for S-lattices).

Theorem 1.7. With the notation as above, assume that q is not exceptional. For 0 < s < 2 and for any S-lattice ∆ ∈ LS , Z sup α(at k∆)s dm(k) < ∞. t>1

K

Moreover, the bound is uniform as ∆ varies over a compact subset C of LS . When the signature of q∞ is (2, 2), the same inequality is valid if t is inside the cone X tp , t∞ < δ p∈Sf

where δ is a constant depending on S and quadratic form q.

We will deduce the main theorem from the following result: Theorem 1.8. Let φ : G/Γ → R be a continuous function, and ν be a positive continuous function Q n on p∈S Up . Assume that for some 0 < s < 2, we have |φ(∆)| < Cα(∆)s for all ∆ ∈ G/Γ. Let x0 ∈ G/Γ be such that Hx0 is not closed. Then Z Z Z φ(y)dµ(y) ν dm(k). φ(at kx0 )ν(k)dm(k) = lim t→∞ K

G/Γ

K

The same holds when the signature of q∞ is (2, 2), assuming that t is inside a the cone defined by X t∞ < δ tp , p∈Sf

where δ is the same constant as in Theorem 1.7.

As in [6], the proof of this theorem relies upon a uniform version of Ratner theorem. To explain this version, we will recall some definitions. Let G be a Q-algebraic Q group, and let Sf be a set of odd primes. Set S = Sf ∪ {∞} as before, and let G = G(QS ) = p∈S G(Qp ). Let Γ be an S-arithmetic subgroup of G, i.e. a group commensurable to G(ZS ). Note that G(Qp ) is naturally embedded in G. For each p ∈ S, let Up = {up (zp ) : zp ∈ Qp } be a one-parameter unipotent Qp -subgroup, which means that up : Qp → G(Qp ) is a non-trivial Qp -rational homomorphism. [17]. We will need a uniform version of Ratner’s theorem for the action of unipotent groups on G/Γ, in the special case that Γ is an S-arithmetic subgroup. The following class of groups introduced by Tomanov plays an important role in this theorem.

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Definition 1.9. [23] A connected Q-algebraic subgroup P of G is a subgroup of class F relative to S if for each proper normal Q-algebraic subgroup Q of P, there exists p ∈ S such that (P/Q)(Qp ) contains a non-trivial unipotent element. Let Γ be an S-arithmetic lattice in G. If P is a subgroup of class F in G, then for any subgroup P ′ of finite index in P(QS ), P ′ ∩ Γ is an S-arithmetic lattice in P ′ . We will use the following S-arithmetic version of Dani-Margulis theorem: Theorem 1.10. [13] Let G be a Q-algebraic group, G = G(QS ), and Γ an S-arithmetic lattice in G, and µ denote the G-invariant probability measure on G/Γ. Let U = {(up (zp ))p∈S |zp ∈ Qp } be a one-parameter unipotent QS -subgroup of G, and let φ : G/Γ → R be a bounded continuous function. Let K be a compact subset of G/Γ, and let ǫ > 0. Then there exist finitely many proper subgroups P1 , . . . , Pk of G of class F, and compact subsets Ci ⊆ X(Pi , U ), where Pi = Pi (QS ), 1 ≤ i ≤ k, such that the following holds: given a compact subset F of K − ∪i Ci Γ/Γ, for all x ∈ F and all T with m(T) ≫ 0, the following holds Z Z 1 φdµ ≤ ǫ. φ(uz x)dλS (z) − λS (I(T)) I(T) G/Γ It is natural to inquire what happens in the case of exceptional forms. In this direction, we can prove the following theorem:

Theorem 1.11. Let I ⊆ QS be an interval and Ω be the product of unit balls in Q3p , for p ∈ S. Then, there exists an isotropic quadratic form on Q3S which is not proportional to a rational form, a constant c > 0, and a sequence Ti → ∞ such that Then NI,q (Tj ) > ckTj k(log kTj k)1−ǫ . Since scalar multiples of a single form q cover all possible equivalence classes of isotropic quadratic forms over Q3p (this can be easily seen from the proof of Proposition 2.1), the above theorem shows that the counterexamples can be constructed in any equivalence class of quadratic forms. This theorem, however, leaves out the case of forms with four variables. It would be interesting to study these forms in details in the future. Remark 1.12. In this paper, sans serif roman letters are reserved for S-adic objects. Algebraic groups will Q be denoted by bold letters. For instance, when G = SLn , then G(Qp ) = SLn (Qp ) and G = p∈S G(Qp ). Element of G, H, etc. are denoted by lower case g, h, etc.

Remark 1.13. Prime powers pn can be considered both real and p-adic numbers. In some occasions (see Lemma 4.1) we need to transform a real pn ∈ R to pn ∈ Qp , which will be denoted by: (pn )⋄ = p−n ∈ Qp . This notation also renders some formulae more familiar; for instant, the unit p-adic vector in the same direction as a vector v is v/kvk⋄ . When p = ∞, we set x⋄ = x. For an S-vector x = (xp )p∈S ∈ RS , we set x⋄ = (x⋄p )p∈S ∈ QS . Remark 1.14. For simplicity, we have dealt only with the field of rational numbers, but a generalization to the case of number fields is not too difficult.

Acknowledgement. We would like to thank G. Margulis, A. Mohammadi and A. Eskin for helpful discussions. We also acknowledge the support of Seoul National University, MSRI and Jacobs University during our visits. We were informed that Rex Chung has some result on the upper bound but we do not know the details.


2. Preliminaries In this section, we recall some definitions about Qp -vector spaces and the quadratic forms defined on them. 2.1. Norms on exterior products. Let 1 ≤ i ≤ n. One equips the ith exterior product with an inner product defined by

Vi

Rn

hx1 ∧ · · · ∧ xi , y1 ∧ · · · ∧ yi i = det (xk · yj )1≤k,j≤i . V For v ∈ i Rn , the induced norm is given by kvk = |hv, vi|1/2 . Denote by e1 , . . . , en the canonical basis for Rn . For J = {1 ≤ j1 < . . . < ji ≤ n}, denote eJ = ej1 ∧ · · · ∧ eji . As J runs over V all subsets of {1, 2, . . . , n} of i elements, the set {eJ } forms an orthonormal basis for i Rn and P P 2 1/2 . The inner product defined above is preserved by the action of hence k J aJ eJ k = J aJ n the orthogonal group O(n) on R . The most convenient norm to work with, for our purpose, is given by kvk = maxJ |aJ |p . The V action of SLn (Zp ) on i Qnp preserves this norm. Indeed, since the entries of k ∈ SLn (Zp ) are p-adic integers, we immediately have the inequality kkvkp ≤ kvkp . Equality follows by applying this inequality to k−1 instead of k. 2.2. Quadratic forms. In this subsection, we review the classification of quadratic forms over the p-adic fields Qp for odd prime p following [20]. Recall that a quadratic form q on Rn has signature (r, s) if there is g ∈ GLn (R) such that q(x) = q(gx), P Pr+s where q(x) = ri=1 x2i − i=r+1 x2j . Two quadratic forms over the field Qp are equivalent if and only if they have the same rank, the same discriminant, and the same Hasse invariant. Recall that the discriminant of a non-degenerate quadratic form over a field F is the image of the determinant of the Gram matrix G in the group F ∗ /(F ∗ )2 . This group contains 4 elements when F = Qp , and p is an odd prime, representing the parity of vp (det G) and whether the image of det G in the residue field is a quadratic residue or not. Hasse invariant can take two values and is invariant under scalar multiplication. It follows that there are at most 8 different quadratic forms over Qp of any given dimension. An isotropic quadratic form q over Qp is called standard if it is of the form q(x) = x1 xn + a2 x22 + · · · + an−1 x2n−1 , where ai ∈ {1, p, u, pu}. Here u ∈ Up is a fixed element such that its image in Zp /pZp is a non-residue. 2.3. Orthogonal groups. Q For a quadratic form q = (qp )p∈S over QS , we define the orthogonal group of q by SO(q) = p∈S SO(qp ), where SO(qp ) is the p-adic analytic group consisting of matrices of determinant 1 preserving the quadratic form qp . Let K∞ be the maximal compact subgroup of SO(q∞ ) which is isomorphic to SO(r) × SO(s), where (r, s) is the signature of q∞ . For p ∈ Sf , we define Kp = SLn (ZQ p ) ∩ SO(qp ). It is easy to see that Kp is a maximal subgroup of SO(qp ). Finally, we set K = p∈S Kp and let mK , mKp be the Haar measures on K, Kp , respectively. Being a subgroups of SLn (Zp ), the linear action of Kp on Qnp preserves the p-adic max norm on Qnp . We will also need the following one-parameter subgroups. For each p ∈ Sf ,

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define (2)

apt

( diag(pt , 1, . . . , 1, p−t ) : = diag(e−t , 1, . . . , 1, et ) :

t∈Z t∈R

p ∈ Sf p = ∞.

We denote by Ap the group consisting of all apt with t ∈ Z when p ∈ Sf and t ∈ R, when p = ∞. Note that these subgroups are isomorphic to Z when p ∈ Sf and isomorphic to R, otherwise. Then we can define subgroups of SO(q) for a standard quadratic form q as Y p Y A= Ap = {at = a∞ atp } = {at : t ∈ T}, A+ = {at : t ∈ T+ }. t∞ · p∈Sf

The following simple fact about ternary forms will be used later. Proposition 2.1. Let q be a non-degenerate isotropic ternary quadratic form over Qp . Then SO(q) is isomorphic to PSL2 (Qp ). Proof. This is well-known for R, so assume p ∈ Sf . The conjugation action of SL2 (Qp ) on the space of 2 × 2 matrices of trace zero over Qp preserves the determinant, which can be viewed as a non-degenerate isotropic ternary quadratic form q0 . The kernel is ±I, hence the orthogonal group of q0 is isomorphic to PSL2 (Qp ). For an arbitrary non-degenerate isotropic ternary quadratic form q and c ∈ Qp \{0}, the form cq has the same orthogonal group as q, and for an appropriate value of c has the same discriminant as q0 . The proof of the following fact is deferred to the appendix: Proposition 2.2. For given c1 ∈ Qp and c2 ∈ pZ , Kp acts transitively on {vp ∈ Qnp : q(vp ) = c1 and kvp kp = c2 }. 3. The S-arithmetic Geometry of Numbers In this section, we prove some basic results on the S-arithmetic geometry of numbers. Due to lack of reference in S-arithmetic case, we provide proofs. 3.1. S-lattices and a generalization of the α function. The proof of Oppenheim conjecture is based on the dynamics of unipotent flow on the space of lattices, thus we need the analogous S-arithmetic notion. Definition 3.1. Let V be a free QS -module of rank n. A ZS -module ∆ is an S-lattice in V if there exist x1 , . . . , xn ∈ V such that ∆ = ZS x1 ⊕ · · · ⊕ ZS xn and V is generated by x1 , . . . , xn as a QS -module. Note that an S-lattice is a discrete and cocompact subgroup of QnS . The base lattice ∆0 is the S-lattice generated by the canonical basis e1 , . . . , en . Note that SLn (QS ) acts on the set of lattices in QnS and the orbit of ∆0 under this action can be identified with LS := SLn (QS )/SLn (ZS ). For an S-lattice ∆, we say that a subspace L ⊆ QnS is ∆-rational if the intersection L ∩ ∆ is an Slattice in L, that is, L is a subspace of QnS generated by finite number of elements in ∆. Suppose that L is i-dimensional and L ∩ ∆ = ZS v1 ⊕ · · · ⊕ ZS vi . Define d∆ (L) or simply d(L) by Y (3) d(L) = d∆ (L) = kv1 ∧ · · · ∧ vi kp . p∈S


The definition of d(L) is independent of the choice of basis. This is an analogue of the volume of the quotient space L/(L ∩ ∆) for lattices in Rn . If L = {0}, we write d(L) = 1. We now show that there is indeed a more intimate connection between the LS and the space L of lattices (of covolume 1) in Rn . More precisely, we show that the LS fibers over L in such a way that many natural functions on LS factor through the projection map. Definition 3.2 (The projection map). Let ∆ ∈ LS . Define π(∆) to be the set of vectors v ∈ Rn for which there exists w ∈ ∆ with w∞ = v and wp ∈ Znp for all p ∈ Sf . Remark 3.3. For any vector v ∈ π(∆) there is exactly one vector w ∈ ∆ satisfying w∞ = v. Indeed, if both w1 and w2 satisfy this property, Q then w = w1 − w2 ∈ ∆ is a non-zero vector with real component zero. This implies that ( p∈Sf pm )w → 0 as m → ∞, which contradicts the discreteness of ∆. Proposition 3.4. π(∆) is a lattice of covolume 1 in Rn . Proof. Since the p-adic components of the elements of π(∆) belong to compact sets SLn (Zp ), for p ∈ Sf , the vectors w = v∞ cannot have an accumulation point, since then by passing to a subsequence, we can find an accumulation point for a sequence of vectors in ∆. Start with a ZS -basis v1 , . . . , vn for ∆. By replacing the vectors vi with certain ZS linear combinations of them, we may assume that the determinant of the matrix Ap with columns vp1 , . . . , vpn is a unit in Zp , for every p ∈ Sf . Note that this is equivalent to the condition (4)

kvp1 ∧ · · · ∧ vpn kp = 1

1 , . . . , vn form a Z-basis for π(∆). If w ∈ π(∆), then for every p ∈ Sf . We now claim that v∞ ∞ P n w = v∞ for some v ∈ ∆ with vp ∈ Zp . Write v = nk=1 ck vk , for ck ∈ ZS , and let c be the n column vector with entries c1 , . . . , cn . Since Ap c = vp , A−1 p ∈ SLn (Zp ), and vp ∈ Zp , we deduce n n that c ∈ Zp for all p ∈ Sf . This implies that c ∈ Z , which proves our claim. Since ∆ is 1 ∧ · · · ∧ vn k unimodular, (4) implies that kv∞ ∞ ∞ = 1, showing that π(∆) has covolume 1.

Definition 3.5. For an S-lattice ∆, define 1 : L is a ∆-rational subspace of dimension i , αSi (∆) = sup d(L) where 1 ≤ i ≤ n and αS (∆) = max {αi (∆) : 0 ≤ i ≤ n} . The function αS is not only an analogue of the original function α, but is in fact intimately related to it. Proposition 3.6. For any ∆ ∈ LS , we have αS (∆) = α(π(∆)). Proof. Assume that αSi (∆) = 1/d(L) where L is the ZS -span of v1 , . . . , vi . Similar to the proof of Proposition 3.4, after possibly replacing v1 , . . . , vi by another basis, we can assume that 1 ∧ · · · ∧ vi k . As before, kvp1 ∧ · · · ∧ vpi kp = 1 for all p ∈ Sf . This implies that d(L) = kv∞ ∞ ∞ i 1 v∞ , . . . , v∞ form a Z-basis for π(L), implying that αi (π(∆)) ≥ αSi (∆). The converse can be proven in a similar way.

10


Since S is fixed throughout the discussion, we will henceforth drop the superscript S in αSi and αS . We now define an S-adic Siegel transform. Let f : QnS → R be a bounded function vanishing outside a compact P set. The S-adic Siegel transform of f is defined as follows: for g ∈ SLn (QS ), let f˜(g) = v∈Zn f (gv). More generally, f˜ is defined on the space of S-lattices; for S such a lattice ∆, define X f˜(∆) = f (v). v∈∆

Since the diagonal embedding of ZnS in QS is discrete, the defining sum involves only finitely many non-zero terms. The following lemma is a natural extension of Schmidt’s theorem in geometry of numbers.

Lemma 3.7. (S-adic Schmidt lemma) Let f : QnS → R be a bounded function vanishing outside a compact subset. Then there exists a positive constant c = c(f ) such that f˜(∆) < cαS (∆) for any S-lattice ∆ in QnS . Proof. Without loss of generality, we can assume that f is the characteristic function of B = Q Bp , where Bp is a ball of radius pnp , for p ∈ S and a ball of radius R for p = ∞. Set p∈S Q D = p∈Sf pnp and consider the scalar map δ : ∆ → ∆ defined by δ(x) = D · x. Note that if x ∈ ∆ ∩ B, then kxkp ≤ 1 for all p ∈ Sf and kxk∞ ≤ RD. In particular, π(δ(x)) will be inside a ball B ′ of radius RD. Note that Schmidt’s lemma states that card(∆′ ∩ B ′ ) = O(α(∆′ )), holds for any lattice ∆′ in Rn , where the implies constant depends on B ′ . Using this fact and Remark 3.3 we obtain card(∆ ∩ B) ≤ card(π(∆) ∩ B ′ ) = O(α(π(∆)) = O(αS (∆)). Lemma 3.8. Let ∆ be an S-lattice in we have that

QnS

and let L and M be two ∆-rational subspaces. Then

d(L)d(M ) ≥ d(L ∩ M )d(L + M ),

(5) where d = d∆ is defined in (3).

Proof. We first verify that if L and M are ∆-rational, then L ∩ M and L + M are also ∆rational. Since a subspace of QnS is ∆-rational if and only if it is generated by elements of ∆, L + M is clearly a ∆-rational subspace of QnS . On the other hand, consider a projection map π : QnS → QnS /∆. Note that the projection image π(H) of a subspace H in QnS is closed if and only if H ∩ ∆ is a lattice subgroup in H, that is, H is ∆-rational. Since π is proper, if we set H = π −1 (π(L ∩ M )) = π −1 (π(L) ∩ π(M )), we can easily check that H is ∆-rational and H = L ∩ M. Let p : QnS → QnS /(L∩M ). Since d∆ (H) = dp(∆) (p(H))d∆ (L∩M ) for any ∆-rational subspace H, the inequality (5) is equivalent to dp(∆) (L)dp(∆) (M ) ≥ dp(∆) (L + M ). Let {v1 , . . . , vℓ } and {w1 , . . . , wm } be bases of p(L) and p(M ) consisting of elements in p(∆), respectively. Then since (p(L) ∩ p(∆)) + (p(M ) ∩ p(∆)) ⊆ p(L + M ) ∩ p(∆), dp(∆) (L)dp(∆) (M ) = kv1 ∧ · · · ∧ vℓ kkw1 ∧ · · · ∧ wm k ≥ kv1 ∧ · · · ∧ vℓ ∧ w1 ∧ · · · ∧ wm k ≥ dp(∆) (L + M ).

ASYMPTOTIC DISTRIBUTION OF VALUES OF ISOTROPIC QUADRATIC FORMS AT S-INTEGRAL POINTS11

The following Lemma is the S-arithmetic version of Lemma 3.10 in [6]. Lemma 3.9. The function α belongs to Lr (SLn (QS )/SLn (ZS )) for all values of 1 ≤ r < n. Proof. Consider the map π : LS → L. Denote the invariant probability measures on LS and L by µS and µ, respectively. We claim that π is measure-preserving, that is, for any measurable A ⊆ LR , we have µS (π −1 (A)) = µ(A). To see this, define ν(A) = µS (π −1 (A)). One can easily see that ν is an SLn (R)-invariant probability measure on L, implying that ν = µ. The result will immediately follows from Proposition 3.6. We can now prove the following generalization of the classical Siegel’s formula in geometry of numbers. Proposition 3.10. (S-arithmetic Siegel integral formula) Let f be a continuous function with compact support on QnS , G = SLn (QS ) and Γ = SLn (ZS ). Then Z Z e f dλnS + f (0). f dµ = G/Γ

Qn S

The proof is based on the following lemma. For each p ∈ S, we denote by λnp the Haar measure on Qnp normalized such that λnp (Znp ) = 1. Denote by δp the delta measure at the zero point of Qnp . Lemma 3.11. Let ν be an SLn (QSN )-invariant Radon measure on QnS . Then ν is a linear combination of the measures of the form p∈S νp , where each νp is either the Haar measure λnp or the delta measure δp . Proof. Observe that if ν is a positive Radon measure on a locally compact space X invariant under the action of a group H and aH has Pa finite number of orbits, namely X1 , . . . , Xm , then ν naturally decomposes into the sum ν = m j=1 νi , where νi (A) = ν(A ∩ Xi ) is H-invariant and supported on Xi , for 1 ≤ i ≤ m. Let us now consider the case at hand, namely, the action of SLn (QS ) on QnS . Since the natural action of GQ= SLn (Qp ) on Qnp has two orbits on Qnp , the G-orbits in the action on QnS are of the form p∈S Xp , where Xp is either the zero point or Qnp − {0}. Each orbit is a homogenous space G/L, where L is the stabilizer of an arbitrary point in the respective orbit. The invariant measure on the homogeneous space G/L, if exists, is unique up to a factor. The result follows. Proof of Proposition 3.10. Note that by Lemmas 3.7 and 3.9, fe is integrable. This implies that R the map f 7→ G/Γ fedµ defines a positive linear functional on the space of compactly supported continuous functions on QnS , hence defines a Radon measure ν on QnS . Since µ isN G-invariant, so is ν. Using Lemma 3.11, ν splits as a linear combination of measures of the form p∈S νp , where N each νp is either λnp or the delta measure δp . The coefficient of p∈S λnp in this linear combination is one. Indeed, taking an increasing sequence of compactly supported non-negative continuous functions fn approximating the characteristic functions of increasingly larger balls in R normalized n n f QS , one obtains fn → 1, and Qn fn dλS → 1. Similarly, by choosing a family fn of non-negative S continuous functions with support in a ball of radius 1/n with |fn | ≤ 1 and fn (0) = 1, and applying the Lebesgue’s dominant convergence theorem, we have Z ff lim n dµ = 1, n→∞ G/Γ

12


which implies that the coefficient of f (0) is 1. It remains to be shown that the rest of the coefficients are zero. This can be obtained in a similar way from the fact that a lattice ∆ in QnS cannot contain a non-zero point (vp )p∈S with vp0 = 0 for some p0 ∈ S. Otherwise, one can find an appropriate sequence ni of S-integers, such that |ni |q → 0 for all q 6= p0 and hence ni v → 0 contradicting the discreteness of ∆. From here, and by choosing appropriate sequences of continuous functions as above, we can deduce that the rest of the coefficients are zero. 3.2. Integration on subvarieties. Now let us introduce volume forms of submanifolds in QnS inherited from the volume form dλnS of QnS in order to measure the volumes of orbits of maximal compact subgroups of orthogonal groups in QnS . Since such maximal compact subgroups are products of maximal compact subgroups at each place, it suffices to deal with volume forms of p-adic submanifolds in Qnp . Let us introduce two equivalent definitions of volume form on submanifolds. Recall that any orbit of the p-adic maximal compact subgroup Kp = SLn (Zp ) ∩ SO(qp ) of SO(qp ) for a given quadratic form qp on Qnp is of the form {w ∈ Qnp : ||w||p = pc1 , qp (w) = c2 }, where c1 ∈ Z and c2 ∈ Qp are some constants. Note that the above Kp -orbit is an open subset of the (n − 1)-dimensional p-adic variety in Qnp given by {w ∈ Qnp : qp (w) = c2 }. Hence from now on, we will consider Y as a compact open subset of a d-dimensional p-adic variety in Qnp or a parallelepiped Zp v1 ⊕ · · · ⊕ Zp vd where v1 , . . . , vd ∈ Qnp . Then Y is contained in p−r Znp for some r ∈ N. Consider the projection πℓ : p−r Znp → p−r Znp /pℓ Znp and let Yℓ = πℓ (Y ). Then the volume form νd defined over Y is defined by #Yℓ νd (Y ) = lim dℓ . ℓ→∞ p As in the real case, we can define the volume form of a p-adic submanifold by describing the volume form of its tangent space. Let Y be a parallelepiped Zp v1 ⊕ · · · ⊕ Zp vd , where v1 , . . . , vd are linearly independent in Qnp . Then the volume form νd′ of the parallelepiped is defined as νd′ (Zp v1 ⊕ · · · ⊕ Zp vd ) = ||v1 ∧ · · · ∧ vd ||p , V where || · || is the maximum norm of d (Qnp ). We will use the fact (see [22, Theorem 9]) that the normalized measures νd and νd′ are equal. 4. Passage to homogeneous dynamics In this section, we define a p-adic analogue of Jf function introduced in Section 3 in [6] for the real case. This function is used to reduce the counting problem to the asymptotic formula of certain integrals on the space of S-lattices. Fix a prime p ∈ Sf , and let q be a standard quadratic form on Qnp . Let π1 : Qnp → Qp be the projection to the first coordinate and Qnp,+ be the set {x ∈ Qnp : π1 (x) 6= 0}. We denote by pZ the subset of Qp consisting of p-powers. Lemma 4.1. Let f be a continuous function of compact support on Qnp,+ which satisfies the invariance property (If )

f (ux1 , x2 , . . . , xn−1 , u−1 xn ) = f (x1 , x2 , . . . , xn )


for all units u ∈ Up . Let ν be a non-negative continuous function on the unit sphere Unp such that (Iν )

ν(ux) = ν(x),

for all u ∈ Up . Let Jf be the real-valued function on pZ × Qp ⊂ Q2p defined by Z 1 −r (6) Jf (p , ζ) = r(n−2) f (p−r , x2 , . . . , xn−1 , xn ) dx2 · · · dxn−1 , n−2 p Qp

where in the integral, xn = pr (ζ − q(0, x2 , . . . , xn−1 , 0)). Then for any ǫ > 0, if t and kvk are sufficiently large, we have Z v t ⋄ t(n−2) −1 (7) f (at kv)ν(k e1 )dm(k) − Jf p kvk , ζ ν( ) < ǫ, c(Kp )p kvk⋄ Kp

where c(Kp ) = vol(Kp .e1 )/(1 − 1/p) and m is the normalized Haar measure on Kp .

Proof. Let πi denote the projection onto the ith coordinate in Qnp , and at := apt = diag pt , 1, . . . , 1, p−t as before. Since f has a compact support, for t ≫ 1, if f (at w) 6= 0, we have kwkp = |π1 (w)|p . This implies that on the support of the function k 7→ f (at kv), for t large enough we have (8) at kv = kvk⋄p pt · u(kv), π2 (kv), . . . , πn−1 (kv), vn u(kv)−1 . Using (If ) we have

f (at kv) = f kvk⋄p pt , π2 (kv), . . . , πn−1 (kv), vn . where vn is determined by q kvk⋄p pt , π2 (kv), . . . , πn−1 (kv), vn = q(v). Since Kp preserves the norm, given δ > 0, for t > t0 (δ), if f (at kv) 6= 0, we have kv/kvk⋄p − u(π1 (kv))e1 < δ p

for a unit u(π1 (kv)) ∈ Up . This implies that, v v −1 −1 < δ. − k (u(π (kv))e ) − u(π (kv))(k e ) = 1 1 1 1 kvk⋄ kvk⋄ p

p

p

p

Since ν is continuous and satisfies (Iν ), by choosing δ > 0 small enough we can assure that if t > t0 (δ), and f (at kv) 6= 0, then the following holds: −1 v (9) ν(k e1 ) − ν( kvk⋄ ) < ǫ. p For sufficiently large t, if f (at kv) 6= 0, then the vector kvk⋄p pt , π2 (kv), . . . , πn−1 (kv), vn will be in a bounded distance from e1 . As the Kp -invariant measure on the orbit Kp .v is the push-forward of the normalized Haar measure on K, one can approximate the integral in (6) by the integration on the Kp -orbit Kp .(kvk⋄p e1 ) ⊂ Qnp . For t > max{t′ , t′′ }, Z (10) f (at kv)ν(k−1 e1 )dm(k) Kp Z v )dλ f kvk⋄p pt , π2 (kv), . . . , πn−1 (kv), vn ν( − ≤ ǫ, n−1 ⋄ kvk Kp .(kvk⋄p e1 ) p

14


where λn−1 is the normalized Haar measure on K.(kvk⋄p e1 ).

Remark that (Iν ) is an essential condition but (If ) is not a real restriction since we can easily modify the give function to satisfy (If ). Proposition 4.2. Suppose h is a real-valued continuous function of compact support defined on (Qnp − {0}) × Qp . Then Z Z Z X 1 h(p−z ke1 , ζ)dζdm(k). h(pt v, q(v))dv = vol(K · e1 ) p(n−2)z lim (n−2)t t→∞ p n K Qp Qp z∈Z where vol = voln−1 denote the induced (n − 1)-dimensional volume from Qnp .

Proof. Let us make the substitution w = pt v. This gives dw = 1/pnt dv, hence we have Z Z Z 1 1 t −2t nt 2t h(w, p−2t q(w))dw. h(p v, q(v))dv = h(w, p q(w))p dw = p n p(n−2)t Qnp p(n−2)t Qnp Qp Replacing p−2t by p−t , the left integral in (11) can be changed to Z t h(w, p−t q(w))dw. L1 (h) := lim p t→∞

With the decomposition Qnp = (11)

S

z∈Z p

Qn p

−z Un , p

write X h= hz , z∈Z

where hz (v, x) = h(v, x)1p−z Unp (v). Since h has compact support, all but finitely many of hz are zero. We first claim that for h with supp h ⊆ Unp × Qp , Z Z Z 1 h(ke1 , ζ)dζdm(k). h(pt v, q(v))dv = vol(K.e1 ) lim (n−2)t t→∞ p K Qp Qn p Then, for h with supp h ⊆ p−z0 Unp where z0 ∈ Z, define h′ (v, x) = h(p−z0 v, p−2z0 x), which has a compact support in Unp × Qp . Using change of variables, we have Z Z h(v, p−t q(v))dv h(v, p−t q(v))dv = lim pt L1 (h) = lim pt t→∞

Qn p

= pnz0 lim pt t→∞

= pnz0 lim pt t→∞

t→∞

Z

p−z0 Un p

h(p−z0 v, p−t−2z0 q(v))dv

Un p

Z

Un p

= vol(K.e1 )pnz0

h′ (v, q(v))dv = vol(K.e1 )

Z Z K

Z Z

Qp

h′ (ke1 , ζ)dζdm(k)

h(p−z0 ke1 , p−2z0 ζ)dζdm(k) K Qp Z Z h(p−z0 ke1 , ζ ′ )dζ ′ dm(k), = vol(K.e1 )p(n−2)z0 K

Qp

where the last line relies on the change of variables ζ ′ = p−2z0 ζ. Now, using the decomposition (11), Proposition 4.2 follows for an arbitrary h.


Now let us prove the claim. Let h satisfy supp h ⊆ Unp × Qp . Furthermore, since we put q(v) in the second variable and q is the quadratic form with p-adic integral coefficients, we can add the assumption that supp h ⊆ Unp × Zp . Let us denote by Ak the subspace of the space of continuous functions on Unp × Zp which is spanned by the functions of the form h(v, x) = f (v)g(x), with f (v) = 1A (v) and g(x) = 1B (x), ¯ ⊆ where A ⊆ Unp and B ⊆ Zp are respectively the preimages of subsets A¯ ⊆ (Z/pk Z)n and B S∞ k Z/p Z by the reduction map redk . We will also set A = k=1 Ak . Using Stone-Weierstrass theorem for p-adic fields ([5]), A is dense in the space of all continuous functions on Unp × Zp . Let us denote the reduction maps Zp → Zp /pk Zp (and also Znp → (Zp /pk Zp )n ) by redk . Let h be a real-valued continuous function of compact support contained in Unp × Zp . We will do some reduction steps: we may assume that h(v, x) = f (v)g(x), and f (v) = 1A (v) and g(x) = 1B (x), with n A = red−1 k (A) ⊆ Up ,

(12)

B = red−1 k (B) ⊆ Zp .

In other words, v ∈ A, w ∈ Znp with kwk < p−k =⇒ v + w ∈ A, x ∈ B, y ∈ Zp with |y| < p−k =⇒ x + y ∈ B. It thus suffices to show that L1 (h) = lim pt µn ({v ∈ A, q(v) ∈ pt B) t→∞

= vol(K.e1 )mK ({k ∈ K : ke1 ∈ A})µ1 (B). Let us define Ct (A, B) = {v ∈ Znp : v ∈ A, p−t q(v) ∈ B}, Ct′ (A, B) = {u + pt w : u ∈ A, q(u) = 0, w ∈ Znp , 2β(u, w) ∈ B}. Here β is the associated bilinear form to the quadratic form q. Recall the polarization identity: q(v + w) = q(v) + 2β(v, w) + q(w). We claim that Ct (A, B) = Ct′ (A, B) for any t > k. First, we show that Ct′ (A, B) ⊆ Ct (A, B). Assume that v = u + pt w with u ∈ A, w ∈ Znp satisfying q(u) = 0, 2β(u, w) ∈ B. This implies that q(u + pt w) = 2pt β(u, w) + p2t q(w)2 = pt (2β(u, w) + pt q(w)2 ) Since u ∈ A and kpt vk ≤ p−t , for t > k, we have v ∈ A. Moreover, p−t q(v) = 2β(u, w) + pt q(w)2 ∈ B. Conversely, if v ∈ Ct (A, B), set v′ = e1 + 12 q(v)en . Note that q(v′ ) = q(v) and kvk = kv′ k = 1. Hence by Proposition 2.2, there exists k ∈ K such that v = kv′ . Set u = ke1 and w = 12 p−t q(v)ken , so that v decomposes as v = k(e1 + 21 q(v)en ) = u + pt w. Furthermore, since v ∈ A and ku − vk = k 12 q(v)ken k ≤ p−t < p−k , we have u ∈ A. Clearly, we have q(v) = q(ke1 ) = 0 and 1 2β(u, w) = 2β(ke1 , p−t q(v)ken ) = p−t q(v)β(ke1 , ken ) = p−t q(v) ∈ B. 2

16


This equality implies that |redℓ (Ct′ (A, B))| . ℓ→∞ pnℓ

µn (Ct (A, B)) = µn (Ct′ (A, B)) = lim

Set Aℓ = redℓ (A) ⊆ Z/pℓ Z and Bℓ = redℓ (B) ⊆ Z/pℓ Z. We will denote the induced quadratic form over Zp /pℓ Zp by q as well. First observe that for ℓ/2 > t > k, we have redℓ (Ct′ (A, B)) = {u + pt w : u ¯ ∈ Aℓ , q(¯ u) = 0, 2β(¯ u, w) ¯ ∈ Bℓ }. For any choice of u ∈ Aℓ ∩ {u : q(u) = 0} there exist exactly |Bℓ | choices for w such that 2β(¯ u, w) ¯ ∈ Bℓ . Hence, in order to find µn (Ct (A, B)), it suffices to find the cardinality of the fibers of the map Φ : (Z/pℓ Z)n × (Z/pℓ Z)n → (Z/pℓ Z)n defined by Φ(u, w) = u + pt w. We will show that all the fibers have the same size. If Φ(u1 , w 1 ) = Φ(u2 , w 2 ), then u1 ≡ u2 (mod pt ). Conversely, suppose that u1 ∈ Aℓ , q(u1 ) and v = u1 + pt w1 for some w1 which satisfies 2β(u¯1 , w¯1 ) ∈ Bℓ . Let u2 ∈ Aℓ be such that q(u2 ) = 0, u1 ≡ u2 (mod pt ). Write u2 − u1 = pt z. Set w2 = w1 − z, hence v = u2 + pt w2 . First note that q(u1 ) = q(u2 ) = 0 combined with q(u2 ) = q(u1 + pt z) = q(u1 ) + 2pt β(u1 , z) + p2t q(z) implies that β(u1 , z) is divisible by pt . This implies that 2β(u2 , w2 ) = 2β(u1 − pt z, w1 + z) = 2β(u1 , w1 ) − 2pt β(z, w1 + z) + 2β(u1 , z) ∈ Bℓ This shows that the fibers of Φ are in one-to-one correspondence with the equivalence classes of u ∈ Aℓ module pt . This implies that for all sets A, B satisfying (12), for ℓ ≥ t > k, we have |redℓ (Ct (A, B))| = Nt |Aℓ ∩ {u : q(u) = 0}| × |Bℓ |. Note that Nt does not depend on ℓ. In the special case A = Unp and B = Zp , Ct (A, B) is precisely the p−t -neighborhood of the set {u ∈ Unp : q(u) = 0}. It follows that lim pt µn (Ct (Unp , Zp )) = µn−1 ({u ∈ Unp : q(u) = 0}) = vol(K.e1 ).

t→∞

From here, for general A and B we obtain lim pt µn (Ct (A, B)) = lim pt µn (Ct (Unp , Zp ))

t→∞

t→∞

µn (Ct (A, B)) µn (Ct (Unp , Zp ))

|Aℓ ∩ {v ∈ (Z/pℓ Z)n : q(v) = 0} · |Bℓ | ℓ→∞ |{v ∈ (Z/pℓ Z)n : q(v) = 0} · |Z/pℓ Z| = vol(K.e1 )mK {k ∈ K : ke1 ∈ A}µ1 (B).

= vol(K.e1 ) lim

5. Counting Results In this section, we prove Theorems 1.4 and 1.5. The proofs depend on an ergodic result which will be proven in the next section. In the course of these proofs we will deal Q with compactly supported continuous function h : (QnS − {0}) × QS → R of the form h(v, ζ) = p∈S hp (vp , ζp ). For h in this space, define Z v −(n−2) , q(v) dv. L(h) = lim kTk h t→∞ T⋄ Qn S


Proof of Theorem 1.4. Without loss of generality, we may assume that there exists a compact set C ⊂ G so that every quadratic form in D is of the form qg for some g ∈ C, where q is a fixed standard form, since there are finitely many equivalence classes of quadratic forms over QS . Since C is compact, there exists β such that for every g = (gp )p∈S ∈ C and every v ∈ QnS , β −1 kvp kpQ≤ kgp vp kp ≤ βkvp kp for every p ∈ S. Let ǫ > 0 be given and g ∈ C be arbitrary. Let f = p∈S fp be a continuous non-negative function of compact support on QnS such that Jf∞ ≥ (1 + ǫ)1/|S| on [β −1 , 2β] × I∞ and Jfp ≥ (1 + ǫ)1/|S| on pZ ∩ {x ∈ Qp : β −1 ≤ |x|p ≤ 2β} × Ip for p ∈ Sf . It is clear that if v ∈ QnS satisfies the conditions T∞ ≤ kv∞ k ≤ 2T∞ , kvp kp = Tp ,

(13)

q(gv) ∈ I

then Jf (kgvk⋄ /|T|⋄ , q(gv)) ≥ 1 + ǫ. By Lemma 4.1 and Lemma 3.6 in [6], for sufficiently large t, Z c(K)kTkn−2 f (at kgv)dm(k) ≥ 1 K

if v satisfies (13). Summing over v ∈ ZnS , it follows that the number of vectors v ∈ ZnS satisfying the conditions (13) is bounded from above by Z Z X n−2 n−2 c(K)kTk f (at kgv)dm(k) = c(K)kTk f˜(at kg)dm(k). K

v∈ZS

The statement follows from a geometric sum argument.

K

Proposition 5.1. Let fp : Qnp → R be a positive continuous function with a compact support in Q Qnp,+ , and f = p∈S fp . Assume further that fp satisfies the property (If ) when p ∈ Sf . Let Q νp : Unp → R be positive continuous functions satisfying the condition (Iν ) and ν = p∈S νp . For a non-exceptional quadratic form q, ǫ > 0 and every g ∈ G, there exists a positive S-time t0 so that for t ≻ t0 , Z ⋄ X kgvk gv −(n−2) −1 ˜ ≤ ǫ, Jf (14) , q(gv) ν f (a kg)ν(k e )dm(k) − c(K) t 1 kTk T⋄ kgvk⋄ K n v∈ZS Q where c(K) = p∈S c(Kp ). Q Proof. Since Jf = p∈S Jfp has compact support, by Theorem 1.4 for given g ∈ G, the number of v ∈ ZnS such that Jf (g, v, t)ν(g, v) 6= 0 is bounded by ckTkn−2 for some c > 0. Take t0 such that Lemma 4.1 and Lemma 3.6 in [6] hold for gv and ǫ/c. Then for t ≻ t0 , Z ⋄ X gv kgvk −1 −(n−2) kTk , q(gv) ν − c(K) f˜(at kg)ν(k e1 )dm(k) Jf ⋄ ⋄ T kgvk K v∈Zn S Z X gv kgvk⋄ n−2 −1 −(n−2) , q(gv) ν − c(K)kTk f (a kgv)ν(k e )dm(k) ≤ kTk J t 1 f T⋄ kgvk⋄ K n v∈ZS Y p−(n−2)tp · ǫ/c · ckTkn−2 = C(f )ǫ. ≤ C(f )e−(n−2)t0 p∈Sf

Here C(f ) is a constantQdepending Q on f which comes from the fact that if |ai − bi | < ǫ and |ai | < A, |bi | < B, then | ai − bi | < C(A, B)ǫ. Note that both terms in the second line of the equations are bounded by a constant depending on f .

18


Lemma 5.2. Let f and ν be as in Theorem 5.1. Let h∞ (v∞ , ζ∞ ) =QJf∞ (kv∞ k∞ , ζ∞ )ν∞ (v∞ /kv∞ k∞ ), hp (vp , ζp ) = Jfp (kvp k⋄p , ζp )νp (vp /kvp k⋄p ) for p ∈ Sf . Set h(v, ζ) = p∈S hp (vp , ζp ). Then we have Z YZ ˜ νp (kp−1 e1 )dm(kp ) f (g) dg L(h) = c(K) G/Γ

p∈S

Proof. Since for any unit u, Z Z f (p−r , x2 , . . . , xn−1 , xn )dx2 . . . dxn = Qn−1 p

we have that Z

Qn p

f dx1 . . . dxn =

Z

Qp

Z

Qn−1 p

Qn−1 p

f (p−r u, x2 , . . . , xn−1 , xn )dx2 . . . dxn ,

f (x1 , x2 , . . . xn−1 , xn )dx2 . . . dxn · dx1

= pr (1 − p−1 )

XZ r∈Z

p−r x

Kp

Qn−1 p

f (p−r , x2 , . . . xn−1 , xn )dx2 . . . dxn .

r The change of variables n + q(0, x2 , . . . , xn−1 , 0) = ζ yields p dxn = dζ, and hence Z Z Z X f (p−r , x2 , · · · , xn−1 , xn )dx2 . . . dxn−1 (1 − p−1 )dζ f dx1 . . . dxn = Qn p

r∈Z

=

XZ r∈Z

Qn−2 p

Qp

Qp

Jf (p−r , ζ)pr(n−2) (1 − 1/p)dζ.

The result will now follow from Proposition 3.10), Proposition 4.2, and Lemma 3.9 in [6]. The computations proceeds verbatim those in Lemma 3.9 in [6]. Q Proof of Lemma 1.2. Since Ω = p∈S Ωp ⊂ QnS , it suffices to show that for any choice of intervals Ip ⊆ Qp , p ∈ S, there exists a constant λ(qp , Ωp ) such that vol{v ∈ Tp Ωp : qp (v) ∈ I} is asymptotically λ(qp , Ωp )Tpn−2 . The case p = ∞ is in Lemma 3.8 (ii) of [6]. Hence we may assume that p ∈ Sf , and write Ip = a + pb Zp , for a ∈ Qp and b ∈ Z. Recall that Ωp = {v ∈ Qnp : kvkp ≤ ρp (v/kvk⋄p )}, where ρ(uv) = ρ(v) for any v ∈ Unp and u ∈ Up . Substituting g −1 Ωp by Ωp , we may assume that qp is standard. Moreover, the function ρ′ determining g−1 Ωp satisfies ρ′ (v) = ρ′ (uv) for u ∈ Up . It would be slightly more convenient ˆ p = {v ∈ Qn : kvkp = ρ(v/kvk⋄ )}. to work with the set Ω p p We now apply Proposition 4.2 to find sequences (hm ), (h′m ) of compactly supported continuous positive functions such that hm ≤ 1Ωˆ p ×Ip ≤ h′m and |hm −h′m | converges uniformly to zero as m → ∞. Since hm , h′m are compactly supported, by definition, limm→∞ L1 (hm ) = limm→∞ L1 (h′m ). Hence Z ˆ p : qp (v) ∈ Ip } = lim p−(n−2)t 1Ωˆ p ×Ip (pt v, qp (v))dv lim p−(n−2)t vol({v ∈ p−t Ω t→∞

t→∞

= vol(K.e1 )p−b P

R (n−2)z

X z∈Z

p(n−2)z

Z

K

Qn p

1Ωˆ p (p−z ke1 )dm(k) =: λqp ,Ωˆ p p−b .

1Ωˆ p (p−z ke1 )dm(k) is a summation over a finite set of z’s. The result S ˆ is a disjoint union. will now follows from the fact that Ω = i≥0 pi Ω Note that

z∈Z p

K


Proof of Theorem 1.5. We will start by introducing two function spaces: let L denote the space of real-valued functions F (v, ζ) on QnS − {0} × QS with support in an implicitly fixed compact set C, satisfying F (uv, ζ) = F (v, ζ) for all u ∈ Up , p ∈ Sf . We equip L with the topology of uniform convergence on C. By L0 , we denote the subspace of L consisting of the functions of Q Q the form F (v, ζ) = Jf (kvk⋄ , ζ)ν(v/kvk⋄ ), where f = p∈S fp , ν = p∈S νp , fp is continuous with a compact support on Qnp,+ satisfying (If ) and νp is a non-negative continuous function on Unp satisfying (Iν ). We claim that L0 is dense in L. In fact, it is easy to see from a p-adic Stone-Weierstrass Theorem (see [5]) that the approximation holds if the invariance properties are dropped from L and L0 . With the invariance properties, one only needs to integrate the Q approximating function ν over p∈S Up with respect to the Haar measure to obtain the desired approximating function. This implies that for each x ∈ D, there are hx , h′x in L0 such that for all v ∈ QnS and ζ ∈ QS , hx (xv, ζ) ≤ 1Ωˆ ≤ h′x (xv, ζ) and |L(hx ) − L(h′x )| < ǫ. where 1Ωˆ is the characteristic function of ˆ = v ∈ Qn : ρ∞ (v∞ /kv∞ k∞ )/2 < kv∞ k∞ 0, there exist finitely many points x1 , . . . , xℓ with Hxi closed and for an arbitrary compact subset F ⊆ D − ∪ℓi=1 Hxi , there exists T0 ≻ 0 such that for every x = gΓ ∈ F and every T ≻ T0 , we obtain (1 − θ)VI,q,Ωˆ (T) ≤ NI,q,Ωˆ (T) ≤ (1 + θ)VI,q,Ωˆ (T). The claim will now follows from Lemma 1.2 and a standard geometric series argument.

20


6. Equidistribution of translated orbits Let G, H, and K be as before. In this section, we prove equidistribution of the translated orbit at Kg in G/Γ, where g ∈ G is fixed (or varies over a compact set), and t tends to infinity. The main result will be proven in two steps. In the first step–dealt with in this section–we assume that the test function in Theorem 1.8 is bounded. The main argument, which is purely dynamical, is based on approximating the integral above, with a similar one involving orbits shifted by unipotent elements. The advantage is that the latter can be handled using an Sarithmetic version of Dani-Margulis theorem, which is generally false for unbounded functions. We will eventually need to prove Theorem 1.8 for test functions with a moderate blowup at the cusp. This step requires an analysis of certain integrals involving the α-function, which is carried out in a later section. Theorem 6.1. Let n ≥ 3 be a positive integer, and let G, H, K and at be as above. Let φ : G/ΓQ → R be a bounded continuous function, and qp be a positive continuous function on Unp , and q = p∈S qp . Let x0 ∈ G/Γ be such that the orbit Hx0 is not closed. Then Z Z Z φ(y)dµ(y) q(k) dm(k). φ(at kx0 )q(k)dm(k) = lim t→∞

K

G/Γ

K

Here dµ denotes the G-invariant probability measure on G/Γ. The proof of this theorem closely follows [3]. The integral of φ along the at -translates of the K-orbits is approximated by a similar integral with at replaced by the unipotent flow to which a generalized theorem of Dani and Margulis will be applied. One difference between the situation here and the one in [6] is that as {at } is a Z-flow, the orbit needs to be “fattened up” in order to obtain rough approximations of the orbit of a unipotent flow, which is similar to intervals in Qp . Fix a prime p ∈ Sf . For simplicity, let us suppress the dependence of matrices introduced below on p. For α ∈ Up = {α : |α| = 1} and t ∈ Z, we set dt =

p−t 0 1 αp−t 1 αpt , , u = , k = t,α t,α 0 1 −α−1 pt 1 − p2t 0 pt 1 0 0 −α bt,α = , wα = α−1 (p3t − pt ) 1 α−1 0

Note that dt = bt,α ut,α kt,α wα , {kt,α : t ∈ N, α ∈ Up } and {wα : α ∈ Up } embeds into Kp (with the embedding of SL2 (Qp ) into SO(q)). This is parallel to a similar formula in the real case. The only different is that w is constant in the real case, whereas here it depends on α. Proof of Theorem 6.1. Without loss of generality, we may assume that φ and q are uniformly continuous. We first prove an analogous statement for (uz )z∈QS instead of (at )t∈TS . We will then deduce the statement for the latter from a similar statement for the former by an approximation process. First, we will show that under the assumptions of Theorem 1.8, we have Claim 1: There exist x1 , . . . , xℓ ∈ G/Γ such that the orbits Hxi are closed and each carries S a finite H-invariant measure, such that for each compact set F ⊆ D − ℓi=1 Hxi , for any T with m(T) sufficiently large, and all x ∈ F, we have ) ( Z Z 1 φ dµ > ǫ < ǫ. φ(uz kx) dz − m k∈K: λS (I(T)) I(T) G/Γ


Indeed, let P1 , . . . , Pk be the subgroups provided by Theorem 1.10 for the date (φ, KD, ǫ), and let Ci ⊆ X(Pi , U) be the corresponding compact sets. Define Yi = {g ∈ G : Kg ⊆ X(Pi , U)}. Let M be the subgroup of G generated by k−1 Uk, where k ∈ K. The normalizer L = NG (M ) contains K, hence is of the form L = EK where E is a normal subgroup of H. Since U ⊆ L and U is not included in any proper normal subgroup of H, we obtain M = H. Note that for any y ∈ Yi , we have k−1 Uk ⊆ yPi y−1 , for all k ∈ K. So H ⊆ yPi y−1 . It follows that there exists a subgroup of finite index P′i in Pi such that H = yP′i y−1 for all y ∈ Yi . This implies that y1−1 y2 ∈ NG (H) for all y1 , y2 ∈ Yi . Hence Yi can be covered with a finite number of right cosets of H. Therefore, [ [ Yi Γ/Γ ⊆ Hxj . 1≤i≤k

1≤j≤l

Note that Pi ∩ Γ is a lattice in Pi . Since X(Pi , U) is an analytic subset of G, we have m({k ∈ K : kg ∈ X(Pi , U)γ}) = 0 for each g ∈ G − Yi γ. Therefore     [ m k ∈ K : kg ∈ X(Pi , U)Γ = 0,   1≤i≤k

holds for any g ∈ G −

S

1≤i≤k

Yi Γ. Since Ci ⊆ X(Pi , U) and F ⊆ D − (

m k ∈ K : kx ∈

k [

i=1

Ci Γ/Γ

)

Sℓ

j=1 Hxj ,

we have

= 0,

for every x ∈ F. From here it easily follows (using an analogue of Lemma 4.2. in [6]) that there S exists an open subset W ⊆ G/Γ containing ki=1 Ci Γ/Γ such that m({k ∈ K : kx ∈ W }) < ǫ

for any x ∈ F. From the choice of C1 , . . . , Ck , we have that for every y ∈ KD − W and all T with m(T) ≫ 0, we have Z Z 1 φdµ ≤ ǫ. φ(uz x)dλS (z) − (15) λS (I(T)) I(T) G/Γ

Let q be a bounded continuous function on K. It follows that (16) Z Z Z Z 1 φdµ q(k) dm(k) ≤ sup |q(k)|(1 + 2 sup |φ(y)|)ǫ. φ(uz x)q(k)dλS (z) − k∈K λS (I(T)) K I(T) K y∈G/Γ G/Γ

22


We will now turn to the approximation part. Fix p ∈ Sf . In what follows dα is the Haar R measure on Zp normalized such that Up dα = 1. First note that Z Z Z φ(bt,α ut,α kt,α wα kx)q(k)dm(k)dα φ(dt kx)q(k) dm(k) = Up K p Kp Z Z φ(bt,α ut,α kt,α wα kx)q(k)dm(kt,α wα k)dα = (17) Up K p Z Z φ(bt,α ut,α kx)q(wα−1 kt,α k)dm(k)dα. = Up

Kp

We can now write Z Z Z φ(ut,α kx)q(wα−1 k) dm(k)dα φ(dt kx)q(k) dm(k) − Kp Up K p Z Z (18) φ(bt,α ut,α kx)q(wα−1 kt,α k) − φ(ut,α kx)q(wα−1 k) dm(k)dα . = Up K p

Note that as t → ∞, bt,α and kt,α converge to 1 uniformly in α. Hence, for large values of t, we have Z Z Z ǫ φ(ut,α kx)q(wα−1 k) dm(k)dα < . φ(dt kx)q(k) dm(k) − 2 Kp Up K p

Since q is continuous, we can partition Up into intervals I1 , . . . , Ir such that for each 1 ≤ i ≤ r, there exists wi ∈ Kp such that |q(wα−1 k) − q(wi k)| < ǫ/4 for every α ∈ Ii . This implies that Z Z −1 φ(u kx)q(w k) − φ(u kx)q(w k) dm(k)dα < λp (Ii )ǫ/4. t,α t,α i α Ii Kp

It is clear that Ii,t = {p−t α : α ∈ Ii } has a measure comparable to pt . So, for sufficiently large t, we have Z Z Z Z φ(ut kx)q(wi k)dθ(t). φ(ut,α kx)q(wi k) dm(k)dα = Ii

Ii,t

Kp

Kp

We can now apply (15) to obtain the result.

7. Establishing upper bounds for the certain integrals In this section, we will prove Theorem 1.7. In the course of the proof we will need to consider a number of integrals that are intimately connected to the integral featuring in Theorem 1.7. In the rest of this section, we fix p ∈ Sf and denote by at the linear transformation apt defined in equation (2): at e1 = pt e1 , at en = p−t en , and at ej = ej , 2 ≤ j ≤ n − 1. Lemma 7.1. Let V be a finite dimensional vector space over Qp and K be a compact subgroup of GL(V ). Let v ∈ V and W be a proper subspace of V such that for any finite index subgroup K ′ of K and any non-zero subspace W ′ ⊂ W , K ′ W ′ 6⊆ W ′ . Then the subset (19) has Haar measure zero.

tran(v, W ) = {k ∈ K : kv ∈ W } ⊆ K


Proof. Denote the Haar measure on K by m. Since K is compact, m(K) < ∞. Suppose that m(tran(v, W )) > 0. Take W ′ ⊆ W of the smallest dimension such that tran(v, W ′ ) has a positive measure. For each k ∈ K, set kW ′ = {kw : w ∈ W ′ }. For each k′ ∈ K, we have tran(v, k ′ W ′ ) = k′ tran(v, W ′ ) and hence m(tran(v, k′ W ′ )) = m(tran(v, W ′ )) for all k′ ∈ K. If k′ W ′ 6= k′′ W ′ , since W ′ ∩ k′ −1 k′′ W ′ has dimension strictly lower than that of W ′ , by minimality, the intersection tran(v, k ′ W ′ ) ∩ tran(v, k ′′ W ′ ) =k′ tran(v, W ′ ) ∩ k′ tran(v, k ′ ⊆k′ tran(v, W ′ ∩ k′

−1 ′′

k W ′)

−1 ′′

k W ′)

has measure zero. Thus {kW ′ : k ∈ K} is a finite set and K ′ := {k : kW ′ = W ′ } is of finite index in K. K ′ W ′ = W ′ ⊂ W ′ contradiction to the condition that K ′ W ′ * W ′ . Lemma 7.2. Let ρ be an analytic representation of SLn (Qp ) on a finite dimensional normed space (V, k k) over Qp such that all elements of ρ (SLn (Zp )) preserve the norm on V . For g ∈ SLn (Qp ) and v ∈ V , we will denote ρ(g)(v) by gv. Let K be a closed analytic subgroup of SLn (Zp ). Assume that V has a decomposition V = W − ⊕ W 0 ⊕ W +,

(20) where

W 0 = {v ∈ V : at v = v, t ∈ Z} W ± = v ∈ V : at v = p∓t v, t ∈ Z .

Let Q be a closed subset of {v ∈ V : kvkp = 1}. Suppose that the following conditions are satisfied: (1) The subspace W − + W 0 satisfies the assumption of Lemma 7.1; (2) There is a positive integer ℓ such that for any nonzero v ∈ W − , (21) codim x ∈ Lie(K) : xv ∈ W − ≥ ℓ. Then for any s, 0 < s < ℓ,

lim sup

t→+∞ v∈Q

Z

K

dm(k) = 0, kat kvks

where m is the normalized Haar measure on K. Proof. Let p : V → W + ⊕ W 0 and p+ : V → W + denote the natural projections associated to the decomposition (20). For any v ∈ V and r ≥ 0, define D(v, r) = {k ∈ K : kp(kv)k ≤ r} D + (v, r) = k ∈ K : kp+ (kv)k ≤ r .

We will show that the measure of D(v, r) is bounded by Cr ℓ , where C is uniform over all v ∈ Q. Recall that the Lie algebra Lie(K) is defined as the tangent space to the p-adic analytic manifold K at the identity. Consider the map fv : K → W + ⊕ W 0 defined by fv (k) = p(kv). Since K acts by linear transformations, the derivative of fv at the identity is given by (22)

de fv (x) = p(xv),

x ∈ Lie(K) = Te K ⊆ sln (Qp ).

24


For every v ∈ V , set Lv = {x ∈ Lie(K) : xv ∈ W − } = ker de fv . For k ∈ K, also define the map mk : K → K by mk (k′ ) = k′ k. Note that mk (e) = k and de mk : Te K → Tk K is an isomorphism. Differentiating fkv = fv ◦mk yields de fkv = dk (fv )◦de mk , showing that rank de fkv = rank dk fv . From the assumption, if fv (k) = 0, then kv ∈ W − , hence by the assumption of the theorem, codim ker de fkv ≥ ℓ, which implies that codim ker dk fv ≥ ℓ. Let us denote by P(V ) the projective space on V , and by [v] the image of the non-zero vector v in P(V ). We know that if f : U → V is a p-adic analytic function, then the map x 7→ rank dx f is lower semi-continuous, that is, lim inf y→x rank dy f ≥ rank dx f . This implies that for every v ∈ W − , there exists an open subset U containing [v] ∈ P(V ) such that for [w] ∈ U , we have codim ker dk fw ≥ ℓ. By compactness, there exists ǫ > 0 such if kp(kv)k ≤ ǫkvk, then the map fv has a rank at least ℓ. Using the implicit function theorem for local fields (e.g. [21]), for any k ∈ K for which kp(kv)k ≤ ǫkvk, we can choose local coordinate systems for Uk ⊂ K at k and for W + ⊕ W 0 in which fv is given by fv (u1 , . . . , um ) = (u1 , . . . , uℓ , ψℓ+1 , . . . , ψm′ ), where m = dim K, m′ = dim W 0 + dim W + and ψj (u1 , . . . , um ) is an analytic function for j = ℓ + 1, . . . , m′ depending smoothly on v. If r ≤ ǫ, for such a neighborhood Uk , k ∈ D(v, r), there is a nonnegative α = α(k) satisfying that; r ℓ+α m (D(v, ǫ) ∩ Uk ) m (D(v, r) ∩ Uk ) ∼ ǫ so that m (D(v, r)) 1 ≤ ℓ m (D(v, r)) . ℓ r ǫ If r ≥ ǫ, since m(D(v, r)) ≤ 1, 1 m (D(v, r)) ≤ ℓ. ℓ r ǫ By compactness of the ball {v ∈ V : kvk = 1}, we have (23)

C :=

m (D(v, r)) < ∞. rℓ kvk=1,r>0 sup

We will now have to consider the set D + (v, 0) = {k ∈ K : p+ (kv) = 0} = {k ∈ K : kv ∈ W 0 ⊕ W − }. By Lemma 7.1, we have m(D + (v, 0)) = 0. We claim that this and the compactness of Q imply that (24)

lim sup m(D + (v, r)) = 0.

r→0 v∈Q

Otherwise, pick a sequence rn → 0, and vn ∈ Q such that m(D + (vn , rn )) > ǫ for some ǫ. Upon passing to a subsequence, we can assume that vn converges to v. ǫ ≤ m({k : kp+ (kvn )k ≤ rn }) ⊆ m({k : kp+ (kv)k ≤ rn + kvn − vk}). Since rn + kvn − vk → 0, we obtain m(D + (v, 0)) ≥ ǫ, which is a contradiction.


Since at |W 0 = IdW 0 and (23), we see that for a positive integer t, kat kvks ≥ kp(kv)ks hence Z ℓ−s m (D(v, r)) dm(k) −s ≤ ≤ p C r , s (r/p)s D(v,r)−D v, r kat kvk p

for any v ∈ V , kvk = 1 and any r > 0. From the fact that m (D(v, 0)) = 0, Z ∞ Z X dm(k) dm(k) (25) = s s ka kvk ka −n −n−1 t t kvk D(v,r) D(v,p r)\D(v,p r) n=0

≤

p−s

C r ℓ−s . − p−ℓ

On the other hand, since at |W + = p−t IdW + , we get that kat kvks ≥ pts kp+ (kv)ks so that for any v ∈ V , r > 0 and r1 > 0, Z Z Z dm(k) dm(k) dm(k) ≤ + s s s K\D(v,r) kat kvk K−D + (v,r1 ) kat kvk D = (v,r1 )\D(v,r) kat kvk ≤ p−ts r1−s + m D + (v, r1 ) r −s . Therefore for a given ǫ0 > 0, since ℓ−s > 0, take r > 0 and r1 > 0 such that Cr ℓ−s/ p−s − p−ℓ < ǫ0 /3 and m (D + (v, r1 )) r −s < ǫ0 /3. Then we can take t′ > 0 such that if t > t′ , then p−ts r1−s < ǫ0 /3 so that Z dm(k) < ǫ0 . s K kat kvk Vi n Qp , and denote by ρi the i-th exterior power of We now write Vi for the i-th exterior power n the natural representation SLn (Qp ) on Qp . Recall that for an i-tuple J = (j1 , . . . , ji ), we define eJ = ej1 ∧ · · · ∧ eji . It is easy to see that Vi decomposes into at -eigenspaces Vi = Wi− ⊕ Wi0 ⊕ Wi+ , and the eigenspaces are given by Wi− = span {ej1 ···ji : j1 = 1, ji < n} , Wi+ = span {ej1 ···ji : 1 < j1 , ji = n} , Wi0 = span {ej1 ···ji : j1 = 1, ji = n or j1 > 1, ji < n} Proposition 7.3. Let qp be the standard quadratic form on Qnp , n ≥ 4. Let K = SLn (Zp ) ∩ SO(qp ). Let A = Ap and let Vi , Wi− , Wi0 and Wi+ be as above. For each Vi , define (26) F (i) = x1 ∧ x2 ∧ · · · ∧ xi : x1 , x2 , · · · , xi ∈ Qnp ⊂ Vi and Qi = F (i) ∩ {v ∈ Vi : kvkp = 1}. Then for any s, 0 < s < 2, Z dm(k) (27) lim sup = 0. t→∞ v∈Qi K kat kvks

Proof. Let us first consider the special case of n = 4, in which qp is given by qp (x1 , x2 , x3 , x4 ) = x1 x4 + α2 x22 + α3 x23 , where αi ∈ {±1, ±u0 , ±p, ±u0 p}. Here u0 ∈ Up is a representative of a quadratic non-residue in the quotient Zp /pZp . Define the following subgroups of K:

26


(28)

(29)

   



1 0  z 1 U = uz =   0 0    −α2 z 2 −2α2 z

      U′ = u′z =     

1 0 z −α3 z 2

0 0 1 0

0 0 1 0 0 1 0 −2α3 z

Direct calculation yields

uz e123 = e123 + 2α2 z e134 + α2 z 2 e234 ,

  0    0   : z ∈ Qp 0     1

  0    0   : z ∈ Qp . 0     1

u′z e123 = e123 − 2α3 ze124 + α3 z 2 e234 ,

Similarly, for v = x12 e12 + x13 e13 + x23 e23 , we have uz v = x12 e12 + 2α2 zx12 e14 − α2 z 2 x12 e24 + x13 e13 + (zx13 + x23 )e23 + α2 (z 2 x13 + 2zx23 )e34 , u′z v = x12 e12 − (zx12 + x23 )e23 + α3 (z 2 x12 − 2t3 x23 )e24 + x13 e13 − 2α3 zx13 e14 − α3 z 2 x13 e34 . Note that if K ′ is a finite index subgroup of K, then it is open in K and K ′ always intersects U and U′ . The above computation shows that the subspace Wi− + Wi0 of Vi , for any i, satisfies the assumption of Lemma 7.1 with respect to K. Moreover, from the fact that subgroups U and U′ are contain we obtain the condition (21) at least for ℓ = 2 of Lemma 7.2. By Lemma 7.2, for 0 < s < 2, we can see that Z dm(k) = 0, lim sup t→∞ v∈Qi K kat kvks

for any i. Now let us show that (27) holds for general quadratic forms when n ≥ 4. For a general standard quadratic form qp on Qnp , for any j2 , j3 fixed, consider the embedding ι(j2 , j3 ) from SO(x1 x4 + αj2 x22 + αj3 x33 ) to SO(qp ) induced by the following injection from Q4p into Qnp ; e1 7→ e1 , e2 7→ ej2 , e3 7→ ej3 , e4 7→ en .

Clearly, the embedding ι(j2 , j3 ) maps subgroups U(Zp ) and U′ (Zp ) into subgroups of the maximal compact subgroup K of SO(qp ). For any v ∈ Wi− + Wi0 , we can find an appropriate embedding ι(j2 , j3 ) so that orbits ι(j2 , j3 )(U(Zp )).v and ι(j2 , j3 )(U′ (Zp )).v escape the subspace Wi− + Wi0 except the identity. Therefore by the same argument with the case n = 4, we can come to a conclusion (27). Note that if K ′ is a finite index subgroup of K, then it is open in K and K ′ always intersects U and U′ . The above computation shows that the subspace Wi− + Wi0 of Vi , for any i, satisfies the assumption of Lemma 7.1 with respect to K. Moreover, from the fact that subgroups U and U′ intersect any neighbourhood of the identity, we obtain the condition (21) at least for ℓ = 2 of Lemma 7.2. The claim will now follow from Lemma 7.2. In the next lemma, we will combine the results over different places p ∈ S. Recall that for t = (tp )p∈S , we set at = (atp )p∈S . For p ∈ S, we denote by ǫp the element of T for which tp = 1 and tq = 0 for q 6= p. Note that p ∈ Sf , aǫp = ap1 = diag p, 1, . . . , 1, p−1 as an element in SO(qp ) (or as an element embedded in SO(q)).


Lemma 7.4. Let n ≥ 4, q be a non-degenerate isotropic standard quadratic form over QnS , and K be the maximal compact subgroup of SO(q) defined above. If q is non-exceptional, then for any s ∈ (0, 2) and any c > 0, there exists a positive integer m such that for every lattice ∆ in QnS , and every t ∈ R = {mǫp : p ∈ S} the following inequality holds: Z c max (αi+j (∆)αi−j (∆))s/2 . αi (at k∆)s dm(k) < αi (∆)s + ω 2 (30) 2 0 0 such that for any v ∈ F (i), where F (i) is defined in (26) with kvkp = 1, and any tp > mp . Z dm(kp ) < c/2. p s Kp katp kp vkp Note that

Z

dm(kp ) ≤1 p s ka Kp 0 kp vkp This implies that for t = mǫp , p ∈ S, we have Z YZ dm(k) dm(k) c = < . p s s ka kvk 2 t Kp katp kp vkp K p∈S

The argument in the case where q∞ is of signature (2, 2) is similar. In this case, one cannot find a multiple of ǫ∞ for which the inequality holds, but using Proposition 7.3, one can choose m large enough so that Z Z dm(k′ ) c dm(k) < . · p ∞ ′ s s 2 K∞ ka1 k vk∞ Kp kam kvkp

In either case, we deduce that for every nonzero v ∈ F (i), and t ∈ R, the following holds: Z dm(k) c 1 (31) < . s 2 kvks K kat kvk For a given S-lattice ∆ in QnS and each i, there exists a ∆-rational subspace Li of dimension i satisfying that 1 = αi (∆). (32) d(Li ) By substituting a wedge product of ZS -generators of Li ∩ ∆ for v in (31), we have that Z dm(k) 1 c (33) < . s 2 d∆ (Li )s K dat k∆ (at kLi ) Let ω > 1 be such that for all v ∈ F (i), 1 ≤ i ≤ n − 1, and t ∈ R we have

kat vk ≤ ω. kvk Let Ψi be the set of ∆-rational subspaces L of dimension i for which

(34)

ω −1 ≤

d∆ (L) < ω 2 d∆ (Li ).

28


We will prove inequality (30) by distinguishing two cases. First, assume that Ψi = {Li }. Then by (34), we have (35) dat k∆ (at kL) = kat kvk ≥ ω −1 kvk = ω −1 d(L) ≥ ωd(Li ) = ωkv ′ k ≥ kat kv ′ k = dat k∆ (at kLi ), where v and v ′ are wedge products of ZS -generators of L and Li respectively. By inequalities (33),(35) and the definition of αi , Z c αi (at k∆)s dm(k) < αi (∆)s . (36) 2 K Now we assume that Ψi 6= {Li }. Let M be an element of Ψi different from L, and let dim(M + Li ) = i + j for some j > 0. Using Lemma 3.8 and the inequality (34), we get that for any k ∈ K, αi (at k∆) < ωαi (∆) = ≤ so that (37)

Z

ω2 ω < d(Li ) (d(Li )d(M ))1/2 ω2 (d(Li ∩ M )d(Li + M ))

αi (at k∆)s dm(k) ≤ ω 2

K

max

1/2

0 1, there exists a neighborhood V (λ), λ > 1 of the identity in H such that for all f ∈ F, λ−1 f (h) < f (gh) < λf (h) for any h ∈ H and g ∈ V (λ). (b) There exists a constant C > 0 such that for all p ∈ Sf , and all h ∈ H, we have f (aǫp h) < Cf (h), where aǫp denotes the element diag(p, 1, . . . , 1, p−1 ). (c) The functions f ∈ F are left K-invariant, that is, f (kh) = f (h), for k ∈ K and h ∈ H. (d) supf ∈F f (1) < ∞. Then there exists 0 < c = c(F) < 1 such that for any t0 > 0 and b > 0 there exists B = B(t, b) < ∞ with the following property: If f ∈ F and Z f (at0 kh)dm(k) < cf (h) + b (42) K

for any h ∈ H, then

Z

f (at k)dm(k) < B K

for all t > 0 in the T-cone generated by R.

30


Proof. First note that we can replace each f ∈ F with its right K-average Z ˜ f (hk)dm(k) f (h) = K

and F˜ = {f˜ : f ∈ F} still satisfies properties (a), (b) and (c). Hence, without loss of generality, we may and will assume that for all f ∈ F and h ∈ H, we have (43)

f (KhK) = f (h).

We need to prove (44)

sup f (at ) < B. t>0

By Corollary 7.6 and Lemma 5.11 in [6], we can take a neighborhood U of the identity in H such that at0 Uat ∈ KVat0 at K for any t0 , t ≥ 0. From (a) and (43), it follows that Z Z 1 f (at0 kat )dm(k) > mK (U ∩ K)f (at0 at ). f (at0 kat )dm(k) ≥ (45) 2 U∩K K Suppose that for all t0 ∈ R and b > 0, and all h ∈ H we have Z 1 f (at0 kh)dm(k) < mK (U ∩ K)f (h) + b (46) 4 K

Then by taking h = at in (46) we see that for any t > 0, and t0 ∈ Θ, where Θ is the semigroup generated by R, we have

1 f (at0 at ) < f (at ) + b′ ≤ max(f (at ), b′ ) 2 ′ where b = max(2b/mK (U ∩ K), f (e)). A simple induction shows that for any t in the semigroup generated by R, we have (47)

f (at ) ≤ b′ .

(48)

Note that every t in the positive cone generated by R can be decomposed as t = t1 + t2 , where t1 is in the semigroup generated by R and t2 is positive and bounded in every component. Since t2 can be written as the product of a bounded number of elements of V (1/2) and ǫp , with p ∈ Sf , the claim follows. Proof of Theorem 1.7. Define functions f0 , f1 , . . . , fn on H by Since α ≤ (49)

Pn

i=0 αi ,

fi (h) = αi (h∆),

h ∈ H.

it suffices to show that for all 0 ≤ i ≤ n, Z sup fis (at k)dm(k) < ∞. t>0

K

Let us check that f0 , f1 , . . . , fn have the properties (a), (b), (c), and (d) of Proposition 7.7. First note that for v ∈ F (i), 0 ≤ i ≤ n, and h ∈ H, we have the trivial bound khvk ≤ k ∧i hkkvk. V This proves (a) and (b). Since the action of K preserves the norm on i (QnS ), each fi is left K


invariant. It is clear that fi (1) are uniformly bounded as ∆ runs over a compact set of lattices. From Lemma 7.4 with h∆, for any i, 0 < i < n and h ∈ H, we see that Z c fis (at kh)dm(k) < fi2 + ω 2 (50) max (fi+j fi−j )s/2 . 2 0 0. proof of Theorem 1.8. We may assume that φ is nonnegative. For each r ∈ R>0 , we choose a continuous function gr on G/Γ such that 0 ≤ gr (x) ≤ 1, gr (x) = 1 if x ∈ A(r) and gr (x) = 0 outside A(r + 1). Then φ(at kx) = (gr φ)(at kx) + ((1 − gr )φ)(at kx). Following the proof of Theorem 3.5 in [6], let β = 2 − s. Since ((1 − gr )φ)(y) = 0 if y ∈ A(r), ((1 − gr )φ)(y) ≤ C(1 − gr )(y)α(y)2−β ≤ Cα(y)2−β/2 (1 − gr )(y)α(y)−β/2 ≤ Cr −β/2 α(y)2−β/2 . By Theorem 1.7 and the fact that kνk∞ < ∞ , there exists C ′ > 0 such that Z Z −β/2 ((1 − gr )φ)(at kx)ν(k)dm(k) ≤ Cr α(at kx)2−β/2 ν(k)dm(k) (53) K K ′ −β/2 ≤Cr On the other hand, since gr φ is compactly supported, we see that for sufficiently large t ∈ R>0 × Ns in the cone generated by R Z Z Z ν(k)dm(k) < ǫ/2, (gr φ)(y)dg(y) (54) (gr φ)(at kx)ν(k)dm(k) − K K G/Γ

32


where dg is the normalized Haar measure on G/Γ. Since gr φ → φ as r → ∞, (53) and (54) show the theorem. 8. Counterexamples for ternary forms In this section, we will prove Theorem 1.11 which partially shows that Theorem 1.5 is optimal in the sense that for some exceptional forms the asymptotics does not hold. We will first introduce the a family of quadratic forms which will be useful for the construction. Fix αp ∈ Qp , and define qαp (x) = x21 + x22 − α2p x23 . Similarly, for an S-vector α = (αp )p∈S ∈ QS , we set qα = (qαp )p∈S . In the case α ∈ Q, with a slight abuse of notation, we will also use qα as a shorthand for the quadratic form on QnS associated to the constant S-vector with αp = α for all p ∈ S. We will first prove an S-adic version of a well-known fact from number theory adapted to fulfill our purpose. Lemma 8.1. Given α ∈ Q, there exists a constant cα > 0 such that for a sufficiently large S-time T = (Tp )p∈S , there exists at least cα kTk log kTk vectors x = (x1 , x2 , x3 ) ∈ Z3S which satisfy (1) qα (x) = 0. (2) kxi kp = Tp for every 1 ≤ i ≤ 3 and p ∈ Sf . (3) kxk∞ ≤ T∞ . Before we proceed to the proof, we emphasize that condition (2) is a stronger form of kxk = Tp which is needed for the following arguments. Proof. It will be slightly more convenient to first work with the quadratic form q′ (x)Q= x1 x2 − x23 , which is Q-equivalent to qα for every α. For p ∈ Sf , write Tp = pnp , and set a = p∈Sf p−np = T∞ kTk−1 . We will consider triples x1 = aku2 ,

x2 = akv 2 ,

x3 = akuv

where u, k, v ∈ Z with gcd(u, v) = 1 and gcd(u, p) = gcd(v, p) = gcd(k, p) = 1 for all p ∈ Sf . It is clear that x satisfies conditions (1) and (2). Moreover, the condition (3) will also be satisfied if k is chosen such that p |u|, |v| ≤ T∞ /3ak.

It is well-known that the density of the pairs (u, v) with gcd(u, v) = 1 is c = 6/π 2 > 0. We can thus produce at least as many as X X 1 T∞ = c kTk ≫ kTk log kTk c 3ak k 1≤k≤T∞ /3a gcd(k,p)=1,p∈S

1≤k≤T∞ /3a gcd(k,p)=1, p∈S

solutions. Note that the forms qα can be obtained from q′ applying a rational linear transformation A, which changes the real and p-adic norms by a bounded factor. This completes the proof. We will need the following lemma, which is an S-arithmetic version of Lemma 3.15 in [6]. We Q will assume that Ω = p∈S Unp is the product of unit spheres, and drop it from the notation for the counting function.


Q Lemma 8.2. Let I = p∈S Ip be an S-interval. Given ǫ > 0 and an S-time T0 , the set of vectors β = (βp )p∈S ∈ QS for which there exists T ≻ T0 such that NI,qβ (T) ≥ kTk(log kTk)1−ǫ

is dense in QS . Proof. For the notational simplicity, we will set I∞ = [1/4, 1/2] and Ip = Zp for p ∈ Sf . It will be clear that the proof works for any other choice of intervals. For α ∈ Q and T large, define L(α, T) = {x ∈ Z3S : qα (x) = 0, kxk∞ ∈ [T∞ /2, T∞ ],

kxkp = Tp , ∀p ∈ Sf }.

By Lemma 8.1, for large enough T, we have cα card L(α, T) ≥ kTk log kTk ≥ kTk(log kTk)1−ǫ . 4 Note that if x = (x1 , x2 , x3 ) ∈ L(α, T) then 2 2 T∞ T∞ 2 ≤ x ≤ . 3 4(1 + α2 ) 1 + α2

By condition (2) in Lemma 8.1, we also have |x3 |p = Tp = pmp for every p ∈ S. Set 2 −2 β∞ = α2 − (1 + α2 )T∞ ,

βp2 = α2 + up pmp

where |up |p = 1. Note that for sufficiently large Tp , α2 + up pmp is a square in Qp and β is well-defined. A simple computation shows that for any x ∈ L(α, T) we have (55)

−2 2 qβ∞ (x) = qα (x) + (1 + α2 )T∞ x3 ∈ [1/4, 1],

qβp (x) = qα (x) + u2 x23 ∈ Zp .

This implies that the NI,qβ (T) ≥ card L(α, T) ≫ kTk log kT k1−ǫ . It is easy to see that the set of β obtained in this way is dense in QS . We can now ready to give a proof of the main theorem. Assume that for a given T0 , we denote by W (T0 ) the set of γ ∈ QS such that there exists β ∈ QnS satisfying |β − γ| < kTk−3 and NI,qβ (T) ≥ kTk log kTk1−ǫ . From the definition and Lemma 8.2, it is clear that W (T0 ) ⊆ QnS is an open and dense. Let Tj be a sequence of S-times going to infinity. Then W = ∩∞ j=1 W (Tj ) is a set of second category, and hence contains an irrational form. For any γ ∈ W , there exists an infinite sequence β i and Ti satisfying |γp − βpi |p < kTi k−3 for all p ∈ S and NI,qβ i (Ti ) ≥ kTi k log kTi k1−ǫ . Set Rj = kTi k. If kxk∞ < Rj , it is easy to see that i

kqβp (x) − qγp (x)k∞ = O(Ri−1 ) < 1/8 for i ≫ 1 and p ∈ S. This implies that qγ∞ (x) ∈ [1/8, 2] and qγp (x) ∈ Zp for p ∈ Sf . The claim follows from here.

34


The argument when all qp are equivalent to x1 x2 − x3 x4 is similar. An argument along the same lines as the one given above, using the parametrization of the solutions to x1 x2 − x3 x4 given by x1 = auv, x2 = azw, x3 = auz, x4 = avw establishes the result in this case. Appendix A. A version of Witt theorem Proof. For simplicity, let us denote Qp , Kp and vp by Q, K and v, etc. We need to prove that for given two vectors v1 and v2 in Qnp with the same p-norm and the same quadratic value, there is an element k in K such that k.v1 = v2 . For convenience, we may assume that v1 = e1 and let v2 = v ∈ Znp . Since the same argument will be applied to both isotropic or nonisotropic vectors, we will think of a quadratic form Q(x1 , . . . , xn ) as one of u1 x21 + · · · + ui x2i + p(ui+1 x2i+1 + · · · + un x2n ) or x1 x2 + u3 x23 + · · · + ui x2i + p(ui+1 x2i+1 + · · · + un x2n ). depending on the case we want to treat. Here ui ’s are units in Zp .

B′

, where B ′ pB ′′ and B ′′ are nondegenerate mod p. The proposition demands to find a matrix k ∈ K satisfying that Then we can write the corresponding symmetric matrix B = BQ as

(a) k.e1 = v and (b) t kBk = B. Since Zp is the inverse limit of Z/pj Z, j → ∞, we will construct a chain kj ∈ GLn (Z/pj+1 Z) such that (a’) kj .e1 = v mod pj+1 , (b’) t kj Bk j = B mod pj+1 and (c’) kj = kj+1 mod pj+1 . Then the inverse limit of (kj )∞ j=0 will be an element satisfying the conditions (a) and (b). Let us denote kj = k0 + pk1 + p2 k2 + · · ·+ pj kj . X0 Y0 0 Step 1. j = 0. Let k = k0 = depending on the size of B ′ and B ′′ . By the Z0 W0 condition (a’), the first column v0 of k0 is given by v mod p. We want to find a solution of the following equation; ′ t ′ B X0 t Z0 B X0 Y0 = mod p tY tW 0 0 Z0 W 0 0 0 t X0 B ′ X0 t X0 B ′ Y0 = mod p. tY B ′X tY B ′Y 0 0 0 0 By the assumption of our quadratic form, Q(pri (e1 )) = Q(pr i (v0 )), where pri : (x1 , . . . , xn ) ∈ (Z/pZ)n → (x1 , . . . , xi ) ∈ (Z/pZ)i . Applying Theorem ?? to (Z/pZ)i , B ′ , we can get an isometry X0 satisfying that the first column is pri (v0 ) and t X0 B ′ X0 = B ′ mod p. Since t X0 B ′ is invertible, we should take Y0 as 0.Note that In this step, we can not determine Z0 and W0 .


X0 + pX1 Y0 + pY1 Z0 + pZ1 W0 + pW1 and should satisfy the following equations; Step 2. j = 1. The matrix k1 = t

has the first column as v0 + pv1

X0 B ′ X0 + p(t X0 B ′ X1 +t X1 B ′ X0 +t Z0 B ′′ Z0 ) = B ′ mod p2 t

X0 B ′ Y0 + p(t X1 B ′ Y0 +t X0 B ′ Y1 +t Z0 B ′′ W0 ) = 0 mod p2 t

Y0 B ′ Y0 + p(t Y0 B ′ Y1 +t Y1 B ′ Y0 +t W0 B ′′ W0 ) = pB ′′ mod p2

Since Y0 = 0 and t X0 B ′ X0 = B ′ + pC111 mod p2 for some symmetric matrix C111 , we can reduce the above equations as; C111 +t X0 B ′ X1 +t X1 B ′ X0 +t Z0 B ′′ Z0 = 0 mod p t X0 B ′ Y1 +t Z0 B ′′ W0 = 0 mod p t W0 B ′′ W0 = B ′′ mod p Take any Z0 such that the first column is (vi+1 , . . . , vn ) and any W0 satisfying t W0 B ′′ W0 = mod p using Witt theorem. Then it suffices to show that there is a matrix X1 with the given first column satisfying the equation C1 +t X0 B ′ X1 +t X1 B ′ X0 +t Z0 B ′′ Z0 = 0 mod p.

B ′′

Step 3. We claim that for a given invertible symmetric matrix A and any symmetric matrix C, there is a solution X of the equation t XA + AX = C. By considering the space of n by n matrices as a n2 - dimensional vector space, we can rewrite the above equation by;      [C]1 [X]1 A11 A12 · · · A1n  A21 A22 · · · A2n   [X]2   [C]2       (56)  .. ..   ..  =  ..  , .. ..  . . .  .   .  . An1 An2 · · ·

Ann

[X]n

[C]n

where a block matrix Aij is defined by    a11 a21 · · · · · · an1    .. .. .. ..    .  . . . .    ..      2a 2a · · · · · · 2a   1i 2i ni  when A = (ast ), i = j;    . .. .. .. ..   .. . . . .  Aij =   a1n a2n · · · · · · ann          All entries are zero except j-th row is  (a1i , a2i , . . . , ani ), i = 6 j.

Since t XA + AX and C are both symmetric, after removing rows repeated(for example, one of second row and (n+1)th row corresponding to c12 and c21 ) we get a linear equation from 2 (Z/pZ)n to (Z/pZ)n(n+1)/2 . Furthermore, from the fact that A is invertible, the rank of this reduced linear equation is exactly n(n + 1)/2 which tells us that there is a solution X. Now let us show that C1 +t X0 B ′ X1 +t X1 B ′ X0 +t Z0 B ′′ Z0 = 0 mod p has a solution when C1 , B ′ , X0 , B ′′ , Z0 and the first column of X1 are given. That is, the first n2 by n submatrix t (A11 , . . . , An1 ) of the equation (1) is erased together with the n variables [X]1 .

36


Before that, let us assume that the removed row among the repeated rows in the above argument is always the former one. That is, in the example, we will remove the second row and leave (n + 1)th row. Hence the entries of the n(n + 1)/2 by n submatrix of the reduced matrix are zero except the first row, and consequently the rank of the linear equation C1 +t X0 B ′ X1 +t X1 B ′ X0 +t Z0 B ′′ Z0 = 0 mod p is n(n + 1)/2 − 1. On the other hand c11 in the equation (56) can be also removed since it is determined by A11 and [X]1 . Therefore if we check that the (1,1)-entry of C1 +t X0 B ′ X1 +t X1 B ′ X0 +t Z0 B ′′ Z0 = 0 mod p holds, we can find a required matrix X1 . However this follows from the fact that Q(e1 ) = Q(v) = Q(v0 +pv1 ) mod p. Step 4. In general, suppose that there exists a solution k = k0 + pk1 + p2 k2 + · · · + pn kn + · · · satisfying (a) and (b). Then from the condition t kBk = B, ! ′ X ′ j t ∞ X Xj−i Yj−i Xi t Zi B B j p = tY tW Zj−i Wj−i pB ′′ pB ′′ i i i=0 j=0 ! j t ∞ X X Xi B ′ Xj−i t Xi B ′ Yj−i j + p = tY B ′X tY B ′Y i j−i i j−i i=0 j=0 ! j t ∞ X X Zi B ′′ Zj−i t Zi B ′′ Wj−i j+1 p . t W B ′′ Z t W B ′′ W i j−i i j−i j=0

i=0

Hence we should find Xj , Yj , Zj−1 and Wj−1 inductively. Take any Zj−1 with the first i+1 n ), where v = v + pv + · · · + pj v + · · · . Then by step3 with the fact that column t (vj−1 , . . . , vj−1 0 1 j Pj P k j+1 , we can find an appropriate X , W B(v, v) = k=0 pj j j−1 and Yj i=0 B(vi , vk−i ) mod p satisfying the following equations. t

X0 B ′ Xj +t Xj B ′ X0 = −

j−1 X

t

Xi B ′ Xj−i −

i=1

t

Wj−1 B ′′ W0 +t W0 B ′′ Wj−1 = − t

X0 B ′ Yj

= −

j−1 X

i=1 j−1 X i=0

j−1 X

t

Zi B ′′ Zj−i−1 + Cj11 mod p,

i=0

t

t

Wi B ′′ Wj−i−1 +t Yi B ′ Yj−i + Cn22 mod p,

Xi B ′ Yj−i +t Zi B ′′ Wj−i−1 + Cj12 mod p,

where Cj11 , Cj22 and Cj12 are obtained from the equations of the formal level j − 1(See the step2). Consequently, we can find k ∈ GL(n, Zp ) such that t kBk = B. Since det k = ±1, k may not be an element of K. However we can easily find k′ ∈ K from k with k′ .e1 = v using reflections in Qnp . References [1] Borel, Prasad, Values of isotropic quadratic forms at S-integral points. Compositio Math. 83(1992), no.3, 347372. [2] H. Davenport, Analytic methods for Diophantine equations and Diophantine inequalities. Second edition. With a foreword by R. C. Vaughan, D. R. Heath-Brown and D. E. Freeman. Edited and prepared for publication by T. D. Browning. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 2005. xx+140 pp.


[3] S. Dani and G. Margulis, Limit distributions of orbits of unipotent flows and values of quadratic forms, I. M. Gel’fand Seminar, 91-137, Adv. Soviet Math., 16, Part 1, Amer. Math. Soc., Providence, RI, 1993. [4] H. Davenport and D. Ridout, Indefinite quadratic forms, Proc. London Math. Soc. (3) 9 1559, 544-555. [5] J. Dieudonne, Sur les functions continues p-adiques, Bull. Sci. Math. (2) 68 (1994), 79-95. [6] A. Eskin, G. Margulis and S. Mozes, Upper bounds and asymptotics in a quantitative version of the Oppenheim conjecture, Ann. of Math. (2) 147 (1998), no. 1, 93-141. [7] A. Eskin, G. Margulis and S. Mozes, Quadratic forms of signature (2, 2) and eignevalue spacings on rectangular 2-tori. Ann. of Math. (2) 161(2005), no. 2, 679-725. [8] G. H. Hardy, Ramanujan : Twelve Lectures on subsects suggested by his life and work, Chelsea Publishing Company, New York 1959 iii+236 pp. [9] G.H. Hardy, On the Expression of a Number as the Sum of Two Squares, Quart. J. Math. 46, (1915), pp.263?283. [10] M.N. Huxley, Integer points, exponential sums and the Riemann zeta function, Number theory for the millennium, II (Urbana, IL, 2000) pp.275?290, [11] D. Kleinbock and G. Tomanov, Flows on S-arithmetic homogeneous spaces and applications to metric Diophantine approximation, Comment. Math. Helv. 82(2007), no. 3, 519-581. [12] Landau, E. Neue Untersuchungen uber die Pfeiffer’sche Methode zur Abschatzung von Gitterpunktanzahlen. Sitzungsber. d. math-naturw. Classe der Kaiserl. Akad. d. Wissenschaften, 2. Abteilung, Wien, No. 124, 469-505, 1915. [13] K. Mallahi Karai, Distribution of orbits of unipotent groups on S-arithmetic homogeneous spaces, preprint. arXiv:submit/1554931 [14] G. A. Margulis, Formes quadratriques indéfinies et flots unipotents sur les espaces homogénes. (French summary), C. R. Acad. Sci. Paris. Sér. I Math. 304 (1987), no. 10. 249-253. [15] A. Oppenheim, The minima of indefinite quaternary quadratic forms, Proc. Nat. Acad. Sci. U. S. A. 15 (9), 724-727. [16] V. Platonov, A. Rapinchuk, Algebraic Groups and Number Theory , Academic Press, INC., 1994. [17] M. Ratner, Raghunathan’s conjectures for Catesian products of real and p-adic Lie groups, Duke Math. J. 77 (1995), no. 2, 275-382. [18] D. Ridout, Indefinite quadratic forms, Mathematika, 5, 1958, 122-124. [19] W. Schmidt, Approximation to algebraic numbers, Enseignement Math. (2) 17 (1971), 187-253. [20] J.-P. Serre, A course in Arithmetic. Title of the French original edition : Cours d’Arithmétique. Graduate Texts in Mathematics Vol. 7 (New York, Springer-Verlag, 1973). [21] J.-P. Serre, Lie Algebras and Lie Groups. Lectures Notes in Mathematics, Vol. 1500. (Springer-Verlag Berlin Heidelberg New York, 1965). ´ Sci. Publ. Math., [22] J.-P. Serre, Quelques applications du théoréme de densité de Chebotarev, Inst. Hautes. Etudes No. 54 (1981), 323-401. [23] G. Tomanov, Orbits on Homogeneous Spaces of Arithmetic Origin and Approximations, Adv. Stud. Pure Math., 26, Math. Soc. Japan, Tokyo, 2000.

Asymptotic distribution of values of isotropic quadratic forms at $ S ...

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