Closures of locally divergent orbits of maximal tori and values of homogeneous forms

Abstract

Let $\G$ be a semisimple algebraic group over a number field $K$, $\mathcal{S}$ a finite set of places of $K$, $K_\mathcal{S}$ the direct product of the completions $K_v, v \in \mathcal{S}$, and $\OO$ the ring of $\mathcal{S}$-integers of $K$. Let $G = \G(K_\mathcal{S})$, $\Gamma = \G(\OO)$ and $\pi:G \rightarrow G/\Gamma$ the quotient map. We describe the closures of the locally divergent orbits ${T\pi(g)}$ %in $G/\Gamma$ where $T$ is a maximal $K_\mathcal{S}$-split torus in $G$. If $\# S = 2$ then the closure $\overline{T\pi(g)}$ is a finite union of $T$-orbits stratified in terms of parabolic subgroups of $\G \times \G$ and, consequently, $\overline{T\pi(g)}$ is homogeneous (i.e., $\overline{T\pi(g)}= H\pi(g)$ for a subgroup $H$ of $G$) if and only if ${T\pi(g)}$ is closed. On the other hand, if $\# \mathcal{S} > 2$ and $K$ is not a $\mathrm{CM}$-field then $\overline{T\pi(g)}$ is homogeneous for $\G = \mathbf{SL}_{n}$ and, generally, non-homogeneous but squeezed between closed orbits of two reductive subgroups of equal semisimple $K$-ranks for $\G \neq \mathbf{SL}_{n}$. As an application, we prove that $\overline{f(\OO^{n})} = K_{\mathcal{S}}$ for the class of non-rational locally $K$-decomposable homogeneous forms $f \in K_{\mathcal{S}}[x_1, \cdots, x_{n}]$.