论文标题
Chevalley组方案的算术子组I：$ \ {p \} $ -Arithmetic子组建立的Bruhat-titt的商
Arithmetic subgroups of Chevalley group schemes over function fields I: quotients of the Bruhat-Tits building by $\{P\}$-arithmetic subgroups
论文作者
论文摘要
令$ \ mathbf {g} $为一个还原的Chevalley组方案（定义在$ \ mathbb {z} $上）。令$ \ mathcal {c} $成为一个平稳，投影，几何整数曲线，在字段$ \ mathbb {f} $上。令$ p $为$ \ MATHCAL {C} $上的封闭点。令$ a $为$ \ lbrace p \ rbrace $的常规功能环。 $ a $的分数字段$ k $具有离散估值$ν=ν_{p}：k^{\ times} \ rightarrow \ mathbb {z} $关联到$ p $。在这项工作中，我们研究了$ a $ a $ a $ a $ a $ a $ a $ \ mathbf {g} $的组$ \ textbf {g}（a）$在构建$ \ mathbf {g} $上的$ - \ textbf {g}（a）\ backslash \ mathcal {x} $。我们得到这个轨道空间是与某些部门腔室的封闭连接的CW复合物的``胶合''。后者由$ \ Mathcal {C} \ SmallSetMinus \ {p \} $的PICARD组和$ \ Mathbf {g} $的等级进行参数。此外，我们观察到，任何理性部门的尖端是一个特殊的顶点都包含一个嵌入此轨道空间的子部门面。
Let $\mathbf{G}$ be a reductive Chevalley group scheme (defined over $\mathbb{Z}$). Let $\mathcal{C}$ be a smooth, projective, geometrically integral curve over a field $\mathbb{F}$. Let $P$ be a closed point on $\mathcal{C}$. Let $A$ be the ring of functions that are regular outside $\lbrace P \rbrace$. The fraction field $k$ of $A$ has a discrete valuation $ν=ν_{P}: k^{\times} \rightarrow \mathbb{Z}$ associated to $P$. In this work, we study the action of the group $ \textbf{G}(A)$ of $A$-points of $\mathbf{G}$ on the Bruhat-Tits building $\mathcal{X}=\mathcal{X}(\textbf{G},k,ν_{P})$ in order to describe the structure of the orbit space $ \textbf{G}(A)\backslash \mathcal{X}$. We obtain that this orbit space is the ``gluing'' of a closed connected CW-complex with some sector chambers. The latter are parametrized by a set depending on the Picard group of $\mathcal{C} \smallsetminus \{P\}$ and on the rank of $\mathbf{G}$. Moreover, we observe that any rational sector face whose tip is a special vertex contains a subsector face that embeds into this orbit space.
