Asymptotic distribution of values of isotropic quadratic forms at S-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]. © 2017 AIMSCIENCES.