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Hello,

my question is simple: do we have a "Dirichlet's unit theorem" for the group of units of a maximal order of a central division algebra ?

In other words: let $k$ be a number field, let $D$ be a central division $k$-algebra (i.e. a skew field with center $k$), and let $\Lambda$ be a maximal order over $\mathcal{O}_k$.

Is $\Lambda^\times$ a finitely generated group ? what is known about its group structure ?

I browsed the web and looked at Reiner's "Maximal orders" but didn't find anything.

I'm happy to assume that $D$ satisfies Eichler's condition if necessary.

In fact, my original question is even more precise: $k/\mathbb{Q}$ is quadratic imaginary, $D$ carries a unitary involution $\tau$ (which therefore restricts to complex conjugation on $k$), and I am interested in the structure of UNITARY units in a maximal order $\Lambda$.

If anyone knows any results/references, I would be happy to know them.

Thanks in advance!

Greg

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  • $\begingroup$ I think it's true that $\Lambda^\times$ is finitely generated for any maximal order $\Lambda$ of a finite dimensional division algebra over $\mathbf{Q}$. See Eichler, Über die Einheiten der Divisionsalgebren, Math. Ann. 114 (1937), n°1, 635-654. I don't know about your particular setting. $\endgroup$
    – François Brunault
    Commented Oct 16, 2012 at 11:59
  • $\begingroup$ I feel like the place to look would be Weil's Basic Number Theory. The modern way to prove both Dirichlet and the finiteness of the class number is via a Fujisaki's lemma argument: compactness of a norm one idele group. This sort of analysis can be repeated in the situation of a $k$-central division algebra, and this was the aim of Weil's basic number theory. $\endgroup$
    – stankewicz
    Commented Oct 16, 2012 at 12:15
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    $\begingroup$ Let $G$ be the finite type affine $O_k$-group scheme representing the functor $A \mapsto (\Lambda\otimes A)^{\times}$ on commutative $O_k$-algebras. The Weil restriction $\mathcal{G} = {\rm{R}}_{O_k/\mathbf{Z}}(G)$ is a finite type affine $\mathbf{Z}$-group scheme (perhaps not $\mathbf{Z}$-flat), and $\mathcal{G}(\mathbf{Z}) = \Lambda^{\times}$. Thus, $\Lambda^{\times}$ is an arithmetic subgroup of the rational points of the reductive generic fiber of $\mathcal{G}$. Arithmetic groups are finitely generated (see Borel's beautiful thin book on arithmetic groups), even finitely presented. $\endgroup$
    – user27056
    Commented Oct 16, 2012 at 12:25
  • $\begingroup$ thanks for your answers. I will try to look in the direction of arithmetic groups, then. $\endgroup$
    – GreginGre
    Commented Oct 16, 2012 at 12:47
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    $\begingroup$ There is a general survey article on unit groups of orders that might interest you: E. Kleinert, "Units of Classical Orders: A Survey", L'Enseignement Mathématique 40 (1994), 205–248. $\endgroup$
    – KConrad
    Commented Oct 21, 2012 at 17:44

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Hello Greg!

I'm not a specialist on the subject (and might have misunderstood something), but I did search on this issue while ago so here are my impressions.

I think that the answer to the first general question is negative (at least in strong sense). Dirichlet's theorem describes the unit group algebraically almost completely in terms of signature. In most cases (and in particular in the case you are interested in) the unit group of a maximal order of a division algebra is a very complicated object and as far as I know there is no general theorem that gives a good idea of the algebraic structure of this group.

An example of troubles one encounters in division algebras: "Presentations of the unit group of an order in a non-split quaternion algebra" Capi Corrales,a, Eric Jespers, Guilherme Leal, and Angel del Riod, Advances in Mathematics 186 (2004) 498–524.

Probably the best overall sources on the subject are Ernst Kleinert's book (Units in skew fields) and a survey article(Units of classical orders, Enseigment mathematique, 1994). One of his central themes is consideration of what should an analogue of Dirichlet unit theorem look like in a division algebra. So these references are probably quite a good answer to your general question.

While the algebraic side of Dirichlet's theorem seem to be quite hard to generalize, there is also the geometric side, which describes how "dense" the unit group is geometrically if we consider the ring of algebraic integers as a lattice through the usual Minkowski embedding.

In the case you are interested the unit group has a subgroup of finite index (the norm 1 group), which is a co-compact subgroup in $SL_n(C)$. The "density" of this norm 1 group is decided by algebraic invariants of the division algebra. So in this sense we can generalize the geometric side of Dirichlet's theorem. Here the key word is point counting in Lie groups. There is a recent book on the subject: The Ergodic Theory of Lattice Subgroups, A. Gorodnik and A. Nevo, Princeton University Press, 2010.

If you are interested in for example of the number of unitary units (if I understood what you are asking it is indeed a finite number) I don't think this approach helps much. Actually even the original Dirichlet's theorem does not directly tell much about the roots of unity part, except that it exist and is generated by a single element.

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