Timeline for Why are $S$-arithmetic groups interesting?
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| Sep 10, 2012 at 4:58 | comment | added | grp | 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. | |
| Sep 9, 2012 at 11:32 | history | answered | Jim Humphreys | CC BY-SA 3.0 |