Are affine groups over rings of integers finitely generated? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T07:06:00Z http://mathoverflow.net/feeds/question/54028 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/54028/are-affine-groups-over-rings-of-integers-finitely-generated Are affine groups over rings of integers finitely generated? Dror Speiser 2011-02-01T22:15:53Z 2011-02-02T00:20:21Z <p>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.</p> <p>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\$?).</p> <p>1) What is a known positive generalisation?</p> <p>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" (<a href="http://arxiv.org/abs/math/0311310" rel="nofollow">arxiv</a>). The paper contains a proof that the unit group of a quadratic number field is finitely generated - using heights.</p> <p>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.</p> <p>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?</p> <p>I think (2) is more far fetched than (1), so feel free to ignore it :)</p> http://mathoverflow.net/questions/54028/are-affine-groups-over-rings-of-integers-finitely-generated/54044#54044 Answer by profilesdroxford54 for Are affine groups over rings of integers finitely generated? profilesdroxford54 2011-02-02T00:20:21Z 2011-02-02T00:20:21Z <p>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). </p> <p>This includes examples such as \$SL_2(\mathbb{Z})\$, as well as \$S\$-units in number fields. </p> <p>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.</p> <p>However the Mordell-Weil theorem (finite generation of rational (or equivalently integral) points on an abelian variety, is a rather deeper result. </p>