# Is the group of integer points on a finite-type group scheme over Z finitely presented?

Let $G$ be a group scheme of finite type over $\mathbf{Z}$. Must $G(\mathbf{Z})$ be finitely presented?

(The question is inspired by a not yet successful attempt to answer a question of Brian Conrad.)

A few special cases:

1. If $G$ is the Néron model of an abelian variety over $\mathbf{Q}$, then a positive answer amounts to the Mordell-Weil theorem (combine with restriction of scalars to get the full Mordell-Weil theorem).

2. If $G$ is the restriction of scalars of $\mathbf{G}_m$ from the ring of integers of a number field down to $\mathbf{Z}$, then a positive answer follows from Dirichlet's unit theorem.

It follows from Theorem 6.12 of Borel and Harsh-Chandra, "Arithmetic subgroups of algebraic groups", that $G(\mathbb{Z})$ is finitely generated if $G$ is affine. Perhaps one can combine this with Chevalley's theorem to deduce finite generation in the general (not necessarily affine) case.

EDIT (Proof completed (assuming $G$ is separated) and simplfied using comments of BCnrd below)

As discussed in the comments below, we may assume $G$ is flat and we may also assume it is connected. By Chevalley's theorem, there is an affine subgroup scheme $H_{\mathbb{Q}}$ of the generic fibre $G_\mathbb{Q}$ of $G$ such that the quotient $G_{\mathbb{Q}}/H_\mathbb{Q}$ is an abelian variety. Let $H$ be the Zariski closure of $H_{\mathbb{Q}}$ in $G$ with the reduced induced structure. Then $H$ is a closed subgroup scheme of $G$. By a theorem of Raynaud (see comment of BCnrd below for the reference) $H$ is also affine.

We have an incusion of groups

$G(\mathbb{Z})/H(\mathbb{Z}) \subset G(\mathbb{Q})/H(\mathbb{Q}) \subset (G_{\mathbb{Q}}/H_{\mathbb{Q}})(\mathbb{Q})$.

Since $G_{\mathbb{Q}}/H_{\mathbb{Q}}$ is an abelian variety, by the Mordell-Weil theorem $(G_{\mathbb{Q}}/H_{\mathbb{Q}})(\mathbb{Q})$ is a finitely generated abelian group, hence so is $G(\mathbb{Z})/H(\mathbb{Z})$. Since $H(\mathbb{Z})$ is finitely generated by the Borel-Harish-Chandra theorem, it follows that $G(\mathbb{Z})$ is finitely generated.

• I think it's a bit more subtle. For cocompact lattices, I think you still need to use Weil's paper, and, as I recall, one of the reasons Kazhdan defined Property (T) was to prove that nonuniform lattices were finitely generated (a group with Property (T) is compactly generated, so discrete implies finitely generated). Apr 28, 2010 at 5:31
• Well, I am not an expert and am only quoting the theorem; here the question is one about arithmetic groups so one does not have to deal with general (non-arithmetic) lattices. (If one embeds G in $GL_{n,Z}$ as closed subgroupscheme then if I am not mistaken, $G(\mathbb{Z})$ is the same as the $G_Z$ in their theorem.)
– naf
Apr 28, 2010 at 6:00
• Well, first of all, $G(\mathbb{Z} \otimes \mathbb{R})$ might not be reductive, as in the examples given in the question. Perhaps I'm missing something, but I don't see why the argument works even in the semisimple setting. It is a lattice, which is not accidentally analogous to Dirichlet's unit theorem, but I still don't see how you get to finite generation for a generic discrete subgroup of finite covolume in $G(\mathbb{R})$. Apr 28, 2010 at 6:10
• @Torsten Ekedahl. There is no reductivity assumption in the theorem I quoted (Theorem 6.5 deals with the reductive case.) I admit to not having read the proof so perhaps I am still missing something.
– naf
Apr 28, 2010 at 6:30
• Let's assume $G$ separated. A $G/H$ is not needed to prove finite generation, since $G(\mathbf{Z})/H(\mathbf{Z})$ is subgroup of $G(\mathbf{Q})/H(\mathbf{Q})$, a subgroup of (finitely generated) Mordell-Weil group of ab var. $G_{\mathbf{Q}}/H_{\mathbf{Q}}$. A more serious point is to prove $H$ is affine! Since finite type with affine generic fiber, it is affine over $\mathbf{Z}[1/N]$ for some $N$. To prove affine is problem over $\mathbf{Z}_{(p)}$ for each $p|N$. Now use SGA3 result of Raynaud (needs sep'tdness) that appears with proof as Prop. 3.1 of Prasad-Yu paper "Quasi-reductive groups". Apr 28, 2010 at 14:55