# Dirichlet's unit theorem for reductive schemes

Let $O_{K,S}$ be the ring of $S$-integers in a number field $K$. Dirichlet's unit theorem implies that the group of units in $O_{K,S}$ is a finitely generated group. In other words, the group $\mathbb G_{m}(O_{K,S})$ is finitely generated. In particular, if $T$ is a torus over $K$, then $T(O_{K,S})$ is finitely generated.

My question is whether this statement generalizes to other group schemes. [Edit: Daniel Litt's comment makes it clear that unipotent groups don't satisfy this finiteness property.]

Question. Let $G$ be a reductive group scheme over $O_{K,S}$. Is $G(O_{K,S})$ finitely generated?

The analogous question for proper group schemes has a positive answer by the Mordell-Weil theorem.

What if we restrict to semi-simple group schemes?

• This is not true already for $\mathbb{G}_a$ (consider $\mathbb{Z}[1/2]$), but may be true in the reductive case...? Oct 6, 2016 at 15:52
• I think mathoverflow.net/questions/22798/… might be relevant. Indeed, $G(\mathbb Z)$ is finitely presented. Oct 6, 2016 at 16:28
• For connected reductive $G$ over a number field $K$ and a finite set $S$ of places of $K$ containing the archimedean places, any affine flat finite type $O_{K,S}$-group scheme $\mathcal{G}$ with $K$-fiber $G$ is the schematic-closure in ${\rm{GL}}_{n,O_{K,S}}$ of some $G \hookrightarrow {\rm{GL}}_{n,K}$ since $O_{K,S}$ is Dedekind, so $\mathcal{G}(O_{K,S})$ is an $S$-arithmetic subgroup of $G(K)$. All $S$-arithmetic groups are finitely presented, by Theorem 5.11 in the book of Platanov-Rapinchuk. Over global function fields some hypotheses are needed, see H. Behr's 1998 paper in Crelle 495. Oct 7, 2016 at 6:33
• @nfdc23 I think you should post your comment as an answer. Commenting on your comment, over global function field we understand now better than in 1998, see arxiv.org/abs/1102.0428. Oct 7, 2016 at 12:04

For connected reductive $G$ over a number field $K$ and a finite set $S$ of places of $K$ containing the archimedean places, any affine flat finite type ,$O_{K,S}$-group scheme $\mathcal G$ with $K$-fiber $G$ is the schematic-closure in $\mathrm{GL}_{n,O_{K,S}}$ of some $G \to \mathrm{GL}_{n,K}$, since $O_{K,S}$ is a Dedekind domain. Therefore, $\mathcal G(O_{K,S})$ is an $S$-arithmetic subgroup of $G(K)$. All $S$-arithmetic groups are finitely presented, by Theorem 5.11 in the book of Platanov-Rapinchuk. Over global function fields some hypotheses are needed, see H. Behr's 1998 paper in Crelle 495.