Asymptotic distribution of values of isotropic quadratic forms at S-integral points
Jiyoung Han - Department of Mathematical Sciences, Seoul National University, Kwanak-ro 1, Kwanak-gu, Seoul 08826, Republic of Korea (email) Abstract: We prove an analogue of a theorem of Eskin-Margulis-Mozes [10]. Suppose we are given a finite set of places $S$ over $\mathbb{Q}$ containing the Archimedean place and excluding the prime $2$, an irrational isotropic form $\mathbb{q}$ of rank $n\geq 4$ on $\mathbb{Q}_S$, a product of $p$-adic intervals $\mathbb{I}_p$, and a product $\Omega$ of star-shaped sets. We show that unless $n=4$ and $\mathbb{q}$ is split in at least one place, the number of $S$-integral vectors $v \in \mathbb{T} \Omega$ satisfying simultaneously $\mathbb{q}(v) \in I_p$ for $p \in S$ is asymptotically given by \[ \lambda(\mathbb{q}, \Omega) |\,\mathbb{I}\,| \cdot ||\mathbb{T}||^{n-2} \] as $\mathbb{T}$ goes to infinity, where $|\,\mathbb{I}\,|$ is the product of Haar measures of the $p$-adic intervals $I_p$. 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 $ \alpha$-function introduced in [10], and an $S$-arithemtic version of a theorem of Dani-Margulis [7].
Keywords: Oppenheim conjecture, homogeneous dynamics.
Received: June 2016; Revised: August 2017; Available Online: October 2017. |