TITLE: Dirichlet's unit theorem for reductive schemes
QUESTION [15 upvotes]: 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?

REPLY [4 votes]: nfdc23 answered the question:
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.