Arithmetic subgroups of Chevalley group schemes over function fields I: quotients of the Bruhat-Tits building by $\{P\}$-arithmetic subgroups

Abstract

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 $\{P\}$. The fraction field $k$ of $A$ has a discrete valuation $\nu ={\nu }_P:k^{\times }\to \mathbb{Z}$ associated to $P$. In this work, we study the action of the group $\text{G}(A)$ of $A$-points of $\mathbf{G}$ on the Bruhat-Tits building $\mathcal{X}=\mathcal{X}(\text{G},k,{\nu }_P)$ in order to describe the structure of the orbit space $\text{G}(A)\backslash \mathcal{X}$. 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 $\mathcal{C}\setminus \{P\}$ 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.

This preprint contains the sections from 1 to 9 of arXiv:2207.06546v3. Section 10 will be extended to a finite set of places in a work in progress