Why are $S$-arithmetic groups interesting? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T09:07:11Z http://mathoverflow.net/feeds/question/106709 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/106709/why-are-s-arithmetic-groups-interesting Why are $S$-arithmetic groups interesting? unknown (yahoo) 2012-09-09T05:13:05Z 2012-09-09T17:48:53Z <p>Let $K$ be a number field and $S$ a finite set of valuations of $K$, including $\infty$.</p> <p>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 $s$.</p> <p>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 $s \notin S$.</p> <p>Example: Let $K = \mathbb{Q}$ and $S = {\infty, p_1, \ldots, p_n }$. Then we have</p> <p>$$K_S = \mathbb{R} \times \mathbb{Q}_{p_1} \times \cdots \times \mathbb{Q}_{p_n}$$ $$\mathcal{O}_S = \mathbb{Z}[p_1^{-1},\ldots,p_n^{-1}]$$</p> <p>Note that these rings come with topologies induced from the topologies on the completions $K_s$.</p> <p>Furthermore, we can define algebraic groups over $K_S$ such as, for example</p> <p>$$\mathbf{GL}_m(K_S) = \prod_{s \in S} \mathbf{GL}_m(K_s)$$</p> <p>Here are my questions:</p> <p>Why is it interesting to study groups in the $S$-arithmetic setting such as $\mathbf{GL}_m(\mathcal{O}_S)$ or $\mathbf{GL}_m(K_S)$?</p> <p>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.</p> <p>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.</p> <p>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.</p> http://mathoverflow.net/questions/106709/why-are-s-arithmetic-groups-interesting/106722#106722 Answer by Jim Humphreys for Why are $S$-arithmetic groups interesting? Jim Humphreys 2012-09-09T11:32:23Z 2012-09-09T11:32:23Z <p>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 <em>Cohomologie des groupes discrets</em> 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. </p> <p>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 <code>$p$</code> is a fixed prime and <code>$S$</code> consists of the infinite prime together with <code>$p$</code>, the <code>$S$</code>-arithmetic group <code>$\mathrm{SL}_2(\mathbb{Z}[1/p])$</code> fails to be discrete in <code>$\mathrm{SL}_2(\mathbb{R})$</code> as well as in <code>$\mathrm{SL}_2(\mathbb{Q}_p)$</code>. But it is discrete in the direct product of these two locally compact groups.</p> <p>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. </p>