Values of decomposable forms at S-integral points and orbits of tori on homogeneous spaces

Abstract

Let $G$ be a reductive algebraic group defined over a number field $K$, and let $S$ be a finite set of valuations of $K$ containing all archimedean ones. Let $G=\prod _{v\in S}G(K_v)$, and let $\Gamma$ be an $S$-arithmetic subgroup of $G$. Let $R\subset S$ and $T_R=\prod _{v\in R}{T}_v$, where each $T_v$ is a torus of $G(K_v)$ of maximal $K_v$-rank. We prove that if $G/\Gamma$ admits a closed $T_R\pi (g)$-orbit, then either $R=S$ or $R$ is a singleton, and we describe the closed $T_R$-orbits in both cases. We apply this result to prove that if a collection of decomposable homogeneous forms $f_v\in K_v[x_1,\dots ,x_n],v\in S,$ takes discrete values at $O^n$, where $O$ is the ring of $S$-integers of $K$, then there exists a homogeneous form $g\in O[x_1,\dots ,x_n]$ such that $f_v={\alpha }_{v}g$, ${\alpha }_v\in K_v^{*}$, for all $v\in S$. Our result is also new in the simplest case of one real homogeneous form when $K=\mathbb{Q}$ and $O=\mathbb{Z}$