Locally divergent orbits of maximal tori and values of forms at integral points

TL;DR

This paper investigates the behavior of locally divergent orbits of maximal tori in algebraic groups over number fields, characterizing their closures and applying results to the density of values of certain forms at integral points.

Contribution

It provides a detailed description of the closures of locally divergent orbits for maximal tori, revealing conditions under which these closures are homogeneous, and applies these findings to density problems of polynomial values.

Findings

For two valuations, orbit closures are unions of finitely many stratified orbits.

When more than two valuations and non-CM fields, orbit closures are squeezed between homogeneous orbits.

Application: polynomial values at integral points are dense in the product of completions.

Abstract

Let $\G$ be a semisimple algebraic group defined over a number field $K$, $\te$ a maximal $K$-split torus of $\G$, $\mathcal{S}$ a finite set of valuations of $K$ containing the archimedean ones, $\OO$ the ring of $\mathcal{S}$-integers of $K$ and $K_\mathcal{S}$ the direct product of the completions $K_v, v \in \mathcal{S}$. Denote $G = \G(K_\mathcal{S})$, $T = \te(K_\mathcal{S})$ and $\Gamma = \G(\OO)$. Let $T\pi(g)$ be a locally divergent orbit for the action of $T$ on $G/\Gamma$ by left translations. We prove: ($1$) if $\# S = 2$ then the closure $\overline{T\pi(g)}$ is a union of finitely many $T$-orbits all stratified in terms of parabolic subgroups of $\G \times \G$ and, therefore, $\overline{T\pi(g)}$ is homogeneous only if ${T\pi(g)}$ is closed, ($2$) if $\# \mathcal{S} > 2$ and $K$ is not a $\mathrm{CM}$-field then $\overline{T\pi(g)}$ is squeezed between closed orbits of two reductive groups of equal semisimple ranks implying that $\overline{T\pi(g)}$ is homogeneous when $\G = \mathbf{SL}_{n}$. As an application, if $f = (f_v)_{v \in \mathcal{S}} \in K_{\mathcal{S}}[x_1, \cdots, x_{n}]$, where $f_v$ are non-pairwise proportional decomposable over $K$ homogeneous forms, then $f(\OO^{n})$ is dense in $K_{\mathcal{S}}$.