Asymptotic distribution of values of isotropic here quadratic forms at <i>S</i>-integral points

Abstract

We prove an analogue of a theorem of Eskin-Margulis-Mozes [10]: suppose we are given a finite set of places S over Q containing the archimedean place and excluding the prime 2, 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 unless n = 4 and q is split in at least one place, the number of S-integral vectors v ∈ TΩ satisfying simultaneously q(v) ∈ Ip for p ∈ S is asymptotically given by λ(q, Ω)|I| · T n−2 ,as T goes to infinity, where |I| is the product of Haar measures of the p-adic intervals Ip.The proof uses dynamics of unipotent flows on S-arithmetic homogeneous spaces; in particular, it relies on an equidistribution result for certain translates of orbits applied to test functions with a controlled growth at infinity, specified by an S-arithmetic variant of the α-function introduced in [10], and an S-arithemtic version of a theorem of Dani-Margulis [7].The proof of Theorem 1.5 rests upon a number of ingredients. We will use the dynamics on G/Γ, where G = SL n (Q S ) and Γ = SL n (Z S ), which can be identified with a set L S of unimodular S-lattices. First, we will relate the counting problem to a question about the asymptotic behavior of integrals of the form K f (a t k∆)dm(k), where K is a maximal compact subgroup of SO(q), x is an element in G/Γ related to q, m is the normalized Haar measure of K, and a t is a 1-parameter diagonal subgroup of SO(q) (see Equation (3) in Section 2.3). Here, f is the Siegel transform of a compactly supported function f defined on Q n S (see Definition 3.7). Such integrals, when f is replaced by a bounded continuous function can be dealt with using an S-arithmetic version of the results in [7], which we will state and prove in Section 6 of this paper.Since f is unbounded, in order to use the equidistribution result just described, one needs to also control the integral of f (a t k∆) when a t k∆ is far into the cusp. As in [10], this is dealt with using the function α S (see section 3.6), which is an analog of α introduced in the previous subsection. We prove Theorem 1.7. With the notation as above, assume that q is not exceptional. For 0 < s < 2 and for any S-lattice ∆ ∈ L S , sup t≻1 K α S (a t k∆) s dm(k) < ∞.Moreover, the bound is uniform as ∆ varies over a compact subset C of L S .

Quantitative oppenheim conjecture for $ S $-arithmetic quadratic forms of rank $ 3 $ and $ 4 $

“…The S-arithmetic space is one of the optimal candidates for a generalization of their work, since it has a lattice Z n S which is similar to the integral lattice Z n in R n . Borel and Prasad [5] generalized Margulis' theorem to the S-arithmetic version and the author, Lim and Mallahi-Karai [13] proved the quantitative version of Sarithmetic Oppenheim conjecture. See also [29] and [15] for flows on S-arithmetic symmetric spaces.…”

“…In [13], when an isotropic irrational quadratic form q S is of rank n ≥ 4 and does not contain a split form, then as T → ∞, N(T) approximates V(T). Here, we say that T → ∞ if T p → ∞ for all p ∈ S. Moreover, it is possible to estimate V(T) in terms of I S and T ([13, Proposition 1.2]).…”

“…Although [13] provides a precise condition of quadratic forms for which the quantitative Oppenheim conjecture holds, for the case of rank 4, the condition that q S does not contain any split form, forces us to exclude substantial numbers of components (3 × 7 s − 2 × 6 s among 3 × 7 s , where s = #S f ) in the space of nondegenerate isotropic quadratic forms in Q n S . Note that each of the components can be identified with the quotient space SO(q 0 S ) \ SL n (Q S ), for some q 0 S .…”

“…In fact, there are quadratic forms for which N(T) fails to approximate V(T), when q S is of rank 3, or q S is of rank 4 and contains a split form. For instance, there is an irrational quadratic form q S of rank 3 and a sequence (T j ) for a given ε > 0 so that [8,Theorem 2.2] and [13,Lemma 9.2]).…”

“…The following lemma originates from [25], stated for lattices in R n . The statement and the proof of the S-arithmetic version can be found in [13].…”