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Let $k$ be a number field and denote by $J_k$ the idele group of $k$. Recall that the finite-index open subgroups of $J_k$ which contain $k^*$ are very important in class field theory. My question concerns non-open finite index subgroups.

What are examples of non-open finite-index subgroups of $J_k$? Do these have any interesting arithmetic interpretation when they contain $k^*$?

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    $\begingroup$ 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). $\endgroup$
    – grghxy
    Commented Jul 25, 2015 at 21:57
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    $\begingroup$ @grghxy You should post that as an answer! $\endgroup$
    – Myshkin
    Commented Sep 27, 2015 at 11:59
  • $\begingroup$ @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. $\endgroup$
    – grghxy
    Commented Sep 27, 2015 at 16:38

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