Why are $S$-arithmetic groups interesting?

2012-09-09T05:13:05Z

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.

Answer by Jim Humphreys for Why are $S$-arithmetic groups interesting?

2012-09-09T11:32:23Z

Motivation in mathematics is always a tricky question, but I'd call attention to one name you've omitted from your list: Serre. It's definitely worthwhile to look at his paper Cohomologie des groupes discrets in: Prospects in mathematics (Proc. Sympos., Princeton Univ., Princeton, N.J., 1970), pp. 77–169. Ann. of Math. Studies, No. 70, Princeton Univ. Press, Princeton, N.J., 1971.

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.

By working in this generality, one is able to unify considerably the study of discrete subgroups of locally compact groups along with related geometry and discrete group cohomology. Here the Bruhat-Tits buildings come into play along with classical symmetric spaces, etc.

Why are $S$-arithmetic groups interesting? - MathOverflow

Why are $S$-arithmetic groups interesting?

Answer by Jim Humphreys for Why are $S$-arithmetic groups interesting?