Question

I'll begin by saying that I'm not sure what I want to ask specifically, but pretty sure what in general, so please don't hold my misunderstandings against me, but do comment on them.

I know that the unit group of a number field is finitely generated, and so is $SL_2(\mathbb{Z})$. I understand that so are $SL_n(\mathbb{Z})$ (or was it $GL$?).

1) What is a known positive generalisation?

I also know that the subgroup of an abelian variety of points over a number field is finitely generated. I noticed that this is relevant after reading Franz Lemmermeyer's "Higher Descent on Pell Conics III. The First 2-Descent" (arxiv). The paper contains a proof that the unit group of a quadratic number field is finitely generated - using heights.

The way I think about it is this: the norm equation isn't a projective variety, so we make up for that by considering it over the integers. So we have heights and parallelogram laws and a proof of finitely generated.

2) Is there a single proof for Mordell-Weil, Dirichlet's Unit Theorem, and any to answer to (1), at the same time, that uses some kind of underlying concept to projective-ness and integral-ness?

I think (2) is more far fetched than (1), so feel free to ignore it :)

Answer 1

It seems to me that if you are thinking of affine groups, then the appropriate result is that $S$-arithmetic subgroups of reductive linear algebraic groups over number fields are finitely generated. Over function fields there are exceptions (which I think are known explicitly).

This includes examples such as $SL_2(\mathbb{Z})$, as well as $S$-units in number fields.

For number fields the proof depends on the existence of compact (equivariant) retracts of fundamental domains for the action of the group on suitable spaces -- reduction theory, combined with the finite generation of the group of units.

However the Mordell-Weil theorem (finite generation of rational (or equivalently integral) points on an abelian variety, is a rather deeper result.