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There are lots of references on global unramified geometric class field theory (following Deligne's $\ell$-adic sheaves approach). There are also some notes talking about how to extend Deligne's approach to the ramified case e.g. Bhatt's Oberwolfach 2016 report titled "geometric class field theory". In his article, Bhatt sketched the main descend step in the ramified cases which is the main technical step in GCFT. By certain consideration about the fundamental groups, he reduced the global statement to a statement in local geometric class field theory. However,the article is rather sketchy. Could someone point me to some references about local geometric class field theory if there is any? Thanks!

Edit: In his article, Bhatt mentioned that the reduction from global ramified theory to local theory is due to D. Gaitsgory, S. Raskin and J. Campbell, without further reference available. After doing some literature research, I only found one paper available online that is relevant, namely "Unramified geometric class field theory and Cartier duality" by Campbell, in which he mentioned he will extend his method to ramified and local cases. However, personally, it doesn't seem to me that the approach adopted in Campbell's current paper is like the Deligne's argument nor the argument sketched in Bhatt's article. Could someone help explain what's going on here?

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  • $\begingroup$ Any help is appreciated! $\endgroup$
    – wkf
    Dec 20, 2018 at 12:26
  • $\begingroup$ Have you seen this? It has a treatment of both (ramified) local and global GCFT $\endgroup$
    – Exit path
    Nov 2, 2020 at 4:34

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