Why are $S$-arithmetic groups interesting?

Question

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 $|x|_s \leq 1$ 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 1

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.

Answer 2

Supplementing other comments and @JimHumphreys' answer: Thinking of automorphic forms as living only on quotients of symmetric spaces or of real Lie groups leaves one with an extremely awkward neo-classical version of Hecke operators, and, more pointedly, no action in sight of the corresponding $p$-adic groups, so no way to take advantage of what is known about their representation theory. To "convert" automorphic forms as functions on something like $G(\mathbb Z)\backslash G(\mathbb R)$ to automorphic forms on $G(\mathbb Z[1/p])\backslash G(\mathbb R)\times G(\mathbb Q_p)$, not only makes the $p$-Hecke operators much more tractable, but, in fact, happily, allows the direct application of the representation theory of $G(\mathbb Q_p)$.

(One main virtue of the latter is the wonderful Borel-Casselman-Matsumoto theorem, that shows that not only spherical representations, but admissible repns with Iwahori-fixed vectors, are subrepresentations (and quotients) of unramified principal series. This also does account for the "square-free level" condition of many classical papers on modular forms, since these exactly correspond to Iwahori-fixed vectors...)

For that matter, the physical space $G(\mathbb Z[1/p])\backslash G(\mathbb R)\times G(\mathbb Q_p)$ arises very reasonably, as a sort of non-abelian solenoid, namely, the (projective) limit of $\Gamma(p^n)\backslash G(\mathbb R)$, as $\Gamma(p^n)$ runs over principal $p$-power congruence subgroups. One would find that the limitands in this limit arise inevitably in looking at $p$-power Hecke operators even at level one.