Timeline for Why are $S$-arithmetic groups interesting?
Current License: CC BY-SA 3.0
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| Sep 6, 2014 at 22:17 | answer | added | paul garrett | timeline score: 12 | |
| Sep 6, 2014 at 19:26 | history | edited | KConrad | CC BY-SA 3.0 |
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| Sep 10, 2012 at 0:19 | comment | added | grp | @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 $F$. | |
| Sep 9, 2012 at 23:47 | comment | added | Ian Agol | Finitely generated subgroups of $GL_n(\overline{Q})$ actually lie inside an $S$-arithmetic subgroup. | |
| Sep 9, 2012 at 21:58 | comment | added | grp | 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 $K$. | |
| Sep 9, 2012 at 17:48 | history | edited | user14211 | CC BY-SA 3.0 |
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| Sep 9, 2012 at 11:32 | answer | added | Jim Humphreys | timeline score: 20 | |
| Sep 9, 2012 at 5:13 | history | asked | user14211 | CC BY-SA 3.0 |