Let ${\mathbf {G}}$ 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 ${\mathcal O}$ the ring of $\mathcal {S}$ -integers of K. Let $G = {\mathbf {G}}(K_{\mathcal {S}})$ , $\Gamma = {\mathbf {G}}({\mathcal O})$ and $\pi :G \rightarrow G/\Gamma $ the quotient map. We describe the closures of the locally divergent orbits ${T\pi (g)}$ 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 ${\mathbf {G}} \times {\mathbf {G}}$ 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 ${\mathbf {G}} = \mathbf {SL}_{n}$ and, generally, non-homogeneous but squeezed between closed orbits of two reductive subgroups of equal semisimple K-ranks for ${\mathbf {G}} \neq \mathbf {SL}_{n}$ . As an application, we prove that $\overline {f({\mathcal O}^{n})} = K_{\mathcal {S}}$ for the class of non-rational locally K-decomposable homogeneous forms $f \in K_{\mathcal {S}}[x_1, \ldots , x_{n}]$ .

中文翻译：

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最大环面局部发散轨道的闭包和齐次形式的值

让${\mathbf {G}}$是数域上的半单代数群ķ,$\数学{S}$一组有限的地方ķ,$K_{\数学{S}}$完井的直接产物$K_{v}, v \in \mathcal {S}$， 和${\数学 O}$的戒指$\数学{S}$- 的整数ķ. 让$G = {\mathbf {G}}(K_{\mathcal {S}})$,$\Gamma = {\mathbf {G}}({\mathcal O})$和$\pi :G \rightarrow G/\Gamma $商图。我们描述了局部发散轨道的闭合${T\pi (g)}$在哪里吨是一个最大值$K_{\数学{S}}$-分裂环面G. 如果$\# S = 2$然后关闭$ \overline{T\pi (g)}$是一个有限并集吨-根据抛物线子群分层的轨道${\mathbf {G}} \times {\mathbf {G}}$因此，$\overline{T\pi (g)}$是同质的（即$\overline{T\pi (g)}= H\pi (g)$对于一个子组H的G) 当且仅当${T\pi (g)}$已经关闭。另一方面，如果$\# \mathcal {S}> 2$和ķ不是一个$\mathrm {CM}$- 场然后$\overline {T\pi (g)}$是同质的${\mathbf {G}} = \mathbf {SL}_{n}$并且，通常是非齐次的，但压缩在两个相等半单的约简子群的闭合轨道之间ķ- 排名${\mathbf {G}} \neq \mathbf {SL}_{n}$. 作为应用程序，我们证明$\overline {f({\mathcal O}^{n})} = K_{\mathcal {S}}$对于局部非理性类ķ-可分解的同质形式$f \in K_{\mathcal {S}}[x_1, \ldots , x_{n}]$.