极大环面的局部发散轨道与形式在整点的值
Locally divergent orbits of maximal tori and values of forms at integral points
摘要：
令$\G$是定义在数域$K$上的半单代数群，$\te$是最大$K$-$\G$的分裂环面，$\mathcal{S}$是$K$的包含阿基米德赋值的有限集，$\OO$是$\mathcal{S}$的环-$K$的整数，$K_\mathcal{S}$是完备化的直积$K_v, v \in \mathcal{S}$。表示$G = \G(K_\mathcal{S})$、$T = \te(K_\mathcal{S})$和$Γ= \G(\OO)$。设$Tπ(g)$是$T$对$G/Γ$作左平移作用的局部发散轨道。我们证明：（$1$）如果$\# S = 2$，则闭包$\overline{Tπ(g)}$是所有按照$\G \times \G$的抛物子群分层的多个$T$ -轨道的并集，因此，$\overline{Tπ(g)}$只有在${Tπ(g)}$是封闭的时才是齐次的，（$2$）如果$\# \mathcal{S} > 2$和$K$不是$\mathrm{CM}$ -场，则$\overline{Tπ(g)}$在两个半单秩相等的约化群的闭轨道之间被压缩，这意味着$\overline{Tπ(g)}$是齐次的，$\G = \mathbf{SL}_{n}$。作为一个应用，如果$f = (f_v)_{v \in \mathcal{S}} \in K_{\mathcal{S}}[x_1, \cdots, x_{n}]$，其中$f_v$在$K$齐次形式上是非成对比例可分解的，则$f(\OO^{n})$在$K_{\mathcal{S}}$中是稠密的。
英文摘要：
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 $Γ= \G(\OO)$. Let $Tπ(g)$ be a locally divergent orbit for the action of $T$ on $G/Γ$ by left translations. We prove: ($1$) if $\# S = 2$ then the closure $\overline{Tπ(g)}$ is a union of finitely many $T$-orbits all stratified in terms of parabolic subgroups of $\G \times \G$ and, therefore, $\overline{Tπ(g)}$ is homogeneous only if ${Tπ(g)}$ is closed, ($2$) if $\# \mathcal{S} > 2$ and $K$ is not a $\mathrm{CM}$-field then $\overline{Tπ(g)}$ is squeezed between closed orbits of two reductive groups of equal semisimple ranks implying that $\overline{Tπ(g)}$ 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 $K_{\mathcal{S}}$.
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