Asymptotic distribution of values of isotropic quadratic forms at $S$-integral points

Jiyoung Han, Seonhee Lim, Keivan Mallahi-Karai

Published 2016-05-09Version 1

We prove a generalization of a theorem of Eskin-Margulis-Mozes in an $S$-arithmetic setup: suppose that we are given a finite set of places $S$ over $\mathbb{Q}$ containing the archimedean place, an irrational isotropic form ${\mathbf q}$ 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 if the real part of ${\mathbf q}$ is not of signature $(2,1)$ or $(2,2)$, then the number of $S$-integral vectors ${\mathbf v} \in {\mathsf{T}} \Omega$ satisfying simultaneously ${\mathbf q}({\mathbf v}) \in {\mathbb{I}}_p$ for $p \in S$ is asymptotically $ \lambda({\mathbf q}, \Omega) |{\mathbb{I}}| \cdot \| {\mathsf{T}} \|^{n-2}$, as ${\mathsf{T}}$ goes to infinity, where $|{\mathbb{I}}|$ is the product of Haar measures of the $p$-adic intervals ${\mathbb{I}}_p$.