Timeline for Finite-index subgroups of the ideles
Current License: CC BY-SA 3.0
4 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 27, 2015 at 16:38 | comment | added | grghxy | @Myshkin: The last part of my comment is well-documented (e.g., in Artin-Tate) and overall this is very standard stuff (at least seems to be to me). If someone else wants to put some part of it as an actual "answer" then that is fine with me; I simply prefer not to do so. | |
| Sep 27, 2015 at 11:59 | comment | added | Myshkin | @grghxy You should post that as an answer! | |
| Jul 25, 2015 at 21:57 | comment | added | grghxy | It has no arithmetic interest; the abelianized Galois group has many non-open finite-index subgroups, with no relevance to anything. For example, $G_{\mathbf{Q}}^{\rm{ab}}=\prod_p\mathbf{Z}_p^{\times}$ has the direct product of countably infinitely many copies of $\mathbf{F}_2$ as a quotient, and that is of uncountable dimension as an $\mathbf{F}_2$-vector space yet only countably many open subgroups. Hence, lots of its hyperplanes give rise to non-open subgroups of index 2. More interesting is that all open subgroups of $J_k/k^{\times}$ are of finite index (false for global function fields). | |
| Jul 25, 2015 at 21:34 | history | asked | Daniel Loughran | CC BY-SA 3.0 |