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For both local and global fields, we have a good handle on the abelianization of the absolute Galois group of $K$. Essentially this allows us to "understand" all maps from $G_K$ to abelian groups.

Reversing arrows: If we want to understand maps $A\to G_K$ for abelian groups $A$, then we would need to know the maximal abelian subgroups of $G_K$. My question is if we know a good description of these groups (in either the global or local setting).

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    $\begingroup$ I see no reason why "the" maximal abelian subgroup should exist. However, for abelian subgroups this question here is relevant: mathoverflow.net/q/262108/50351 $\endgroup$
    – Arno Fehm
    Commented Feb 19, 2023 at 18:45
  • $\begingroup$ @ArnoFehm you are of course right, added in the plural, thanks! $\endgroup$
    – curious math guy
    Commented Feb 19, 2023 at 19:03
  • $\begingroup$ Are there any such subgroups which are not (pro)cyclic? $\endgroup$
    – Wojowu
    Commented Feb 19, 2023 at 22:41
  • $\begingroup$ Please use a high-level tag like "nt.number-theory". I added this tag now. $\endgroup$
    – GH from MO
    Commented Feb 19, 2023 at 23:01
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    $\begingroup$ Apparently the answer to my question is no (at least if you consider closed subgroups, but a closure of an abelian subgroup would be abelian), see the final remark here. $\endgroup$
    – Wojowu
    Commented Feb 19, 2023 at 23:06

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