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The classical language of class field theory can be generalized by class formation. So far I've seen, for example, the absolute Galois group acting on the multiplicative group of a local field, or the group of ideles of a global field. I also know there is something like geometric class field theory that describes etale isogenies of commutative algebraic groups and finite abelian geometrically connected coverings of an algebraic curve over a perfect field which looks very deep for me.

I am curious about if there is other example of class field theory and is there any reference? For example, the theory for covering space is almost the same as Galois theory for fields, so I guess there could be a topological class field theory? Maybe for general topological spaces this might be hard, but what about something like manifolds?

Ja_1941
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    Would you consider generalizations of class field theory to be "other examples" of it? See https://mathoverflow.net/questions/66500/why-is-class-field-theory-the-same-as-langlands-for-gl-1 and https://math.stackexchange.com/questions/3849429/concrete-example-of-non-abelian-class-field-theory-why-langlands-program-is – KCd Apr 23 '23 at 05:12
  • @KCd The posts look very helpful! I see how Langlands program generalizes class field theory by looking at higher dimensional representations, but is there something like a class formation? – Ja_1941 Apr 24 '23 at 04:25
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    The Langlands program is still largely conjectural, so one can't treat it as a topic with all the necessary machinery in place like class field theory for the abelian case. Moreover, the proofs of class field theory often start by checking a property for cyclic extensions and then working out the case of abelian extensions via reduction to the cyclic case thanks to every finite abelian group being a direct product of cyclic groups. There's nothing at all like that for nonabelian Galois extensions. – KCd Apr 24 '23 at 05:10

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