# Why are $S$-arithmetic groups interesting?

Let $K$ be a number field and $S$ a finite set of valuations of $K$, including $\infty$.

Define the $S$-numbers $K_S$ to be the direct product $\prod_{s \in S} K_s$ where $K_s$ denotes the completion of $K$ at the valuation $s$.

Define the $S$-integers $\mathcal{O}_S$ to be the subset of $K$ consisting of the elements $x$ such that $|x|_s \geq 0$ when $s \notin S$.

Example: Let $K = \mathbb{Q}$ and $S = \{\infty, p_1, \ldots, p_n \}$. Then we have

$$K_S = \mathbb{R} \times \mathbb{Q}_{p_1} \times \cdots \times \mathbb{Q}_{p_n}$$ $$\mathcal{O}_S = \mathbb{Z}[p_1^{-1},\ldots,p_n^{-1}]$$

Note that these rings come with topologies induced from the topologies on the completions $K_s$.

Furthermore, we can define algebraic groups over $K_S$ such as, for example

$$\mathbf{GL}_m(K_S) = \prod_{s \in S} \mathbf{GL}_m(K_s)$$

Here are my questions:

Why is it interesting to study groups in the $S$-arithmetic setting such as $\mathbf{GL}_m(\mathcal{O}_S)$ or $\mathbf{GL}_m(K_S)$?

In particular, is there some classical problem that is solved by using $S$-arithmetic groups, or one that served to launch the study of $S$-arithmetic groups? Perhaps some relevant (famous) names would be Borel, Harish-Chandra, Siegel, Weil, Tits, etc.

It is easy to believe that number theorists would be interested in studying a ring such as $\mathbb{Z}[p_1^{-1},\ldots,p_n^{-1}]$, although I don't really know why and I would like to hear more.

I am also aware that $\mathbf{GL}_m(K_S)$ is a natural locally compact group in which one can realize $\mathbf{GL}_m(\mathbb{Z}[p_1^{-1},\ldots,p_n^{-1}])$ as a discrete subgroup. Why one would care about this, I am also not sure. I imagine it has something to do with studying functions on the quotient and things such as Tamagawa numbers. Perhaps some representation theory is involved.

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For connected reductive $G$ over $K$ it can be useful to consider open subgroups of $G(A_K)$ of the form $G(K_S) \times U$ for a compact open subgroup $U$ of $G(A_K^S)$. This meets $G(K)$ in an $S$-arithmetic subgroup. Note also that a typical finitely generated subgroup of GL$_n(K)$ cannot be conjugated inside GL$_n(O_K)$ but lies in GL$_n(O_{K,S})$ for some $S$. Overall, $S$-arithmetic groups give a robust theory of "integrality away from $S$" inside $G(K)$ that isn't tied to a specific flat affine $O_{K,S}$-model of $G$ (no reductive one may exist!) and it's "functorial" in $G$ over $K$. –  grp Sep 9 '12 at 21:58
Finitely generated subgroups of $GL_n(\overline{Q})$ actually lie inside an $S$-arithmetic subgroup. –  Ian Agol Sep 9 '12 at 23:47
@Agol: Yes indeed, and more generally any finitely generated subgroup of $G(\overline{K})$ lies inside an $S$-arithmetic subgroup of $G(F)$ for some finite extension $F$ of $K$ inside $\overline{K}$ and a finite set $S$ of places of $F$. I prefer to think about passage to "$S$-statements" as something we do after first getting everything over a single number field; i.e., once we've gotten ourselves inside $G(F) = G_F(F)$ for a specific number field $F$ then we "spread out" to an $S$-integral statement over $F$, with $S$ living on $F$. –  grp Sep 10 '12 at 0:19

The study of discrete subgroups in real Lie groups, starting with the classical modular group, has been a natural meeting place for geometry, number theory, group theory. Analogous groups over nonarchimedean local fields have become prominent in such questions as the Congruence Subgroup Problem; but here the nature of discrete subgroups is much less obvious. Serre points out right away the difficulty one has when taking products of locally compact groups over a mixture of fields (as in the use of adeles in number theory). For example, when $p$ is a fixed prime and $S$ consists of the infinite prime together with $p$, the $S$-arithmetic group $\mathrm{SL}_2(\mathbb{Z}[1/p])$ fails to be discrete in $\mathrm{SL}_2(\mathbb{R})$ as well as in $\mathrm{SL}_2(\mathbb{Q}_p)$. But it is discrete in the direct product of these two locally compact groups.
It seems worth reflecting (a bit!) on the case of GL$_1$. Clearly $O_{K,S}^{\times}$ and its finite-index subgroups are the arithmetic subgroups of ${\rm{GL}}_1(K) = K^{\times}$ (relative to the $K$-group ${\rm{GL}}_1$). Chevalley's theorem that this satisfies the congruence subgroup property is crucial for defining Serre tori, which underlie the clean formulation of the important Artin-Weil theorem relating CM fields to general algebraic Hecke characters (the definition of which doesn't explicitly mention CM fields). It doesn't need a general theory of $S$-arithmeticity, but is so classical. –  grp Sep 10 '12 at 4:58