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Let $X$ be a smooth curve over $\operatorname{Spec}\mathbb{Q}_p$ and $P\in X(\mathbb{Q}_p)$. Let $K_P\simeq \mathbb{Q}_p((T))$ denote the completion of the function field of $X$ at $P$. What is known about the class field theory of $K_P$. Tate duality for the field $\mathbb{Q}_p$ states that there is a perfect pairing $H^i(\operatorname{G}_{\mathbb{Q}_p},M)\simeq H^{2-i}((\operatorname{G}_{\mathbb{Q}_p},M^*(1))^\vee$. Is there a version of local Tate duality for the field $K_P$?

I believe that the Galois group $\operatorname{G}_{K_p}=\operatorname{Gal}(\bar{K}_P/K_P)$ is close to a semidirect product of $\hat{\mathbb{Z}}$ with $\operatorname{Gal}(\bar{\mathbb{Q}}_p((T))/\mathbb{Q}_p((T)))\simeq \operatorname{Gal}(\bar{\mathbb{Q}}_p/\mathbb{Q}_p)$ where $\hat{\mathbb{Z}}$ is generated by a "loop" $\gamma$ about the point $P$. The semidirect product taken w.r.t $\tilde{g}\gamma\tilde{g}^{-1}=\gamma^{\chi(g)}$ where $\tilde{g}$ is a lift of $g\in \operatorname{G}_{\mathbb{Q}_p}$ to $\operatorname{G}_{K_P}$ and $\chi$ is the cyclotomic character. Hence $\operatorname{G}_{K_P}$ has a nice enough description for such questions to be in reach in this particular case.

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  • $\begingroup$ The typographical difference between what you see with \text{Spec}\mathbb Q_p and what you see with \operatorname{Spec}\mathbb Q_p was really conspicuous here before I edited. The difference in general is not merely that some space appears to the right of "Spec", but also that the spacing depends on the context. And it effects the positions of superscripts and subscripts in some cases. $\endgroup$
    – Michael Hardy
    Commented Aug 21, 2019 at 20:05
  • $\begingroup$ The completion $K_P$ is isomorphic to $\mathbb{Q}_p((T))$ for any curve $X$: smoothness at $P$ implies the existence of a neighborhood $P\in U\subset X$ with an etale morphism $U\to \mathbb{A}^1_{\mathbb{Q}_p}$ that induces an isomorphism between the completed local ring $\hat{\mathcal{O}_{U,p}}=\hat{\mathcal{O}_{X,p}}$ and the completed local ring of a point in $\mathbb{A}^1$. $\endgroup$
    – SashaP
    Commented Aug 21, 2019 at 20:26
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    $\begingroup$ There is a theory of "higher local fields"; you can try searching for this to find references. $\endgroup$
    – naf
    Commented Aug 22, 2019 at 7:22
  • $\begingroup$ @ulrich Thank you, I was able to find the local class field theory statement I was looking for due to Parshin and Kato. There is a reciprocity map from $K_2(K_P)\rightarrow \text{Gal}(K^{\text{ab}}/K)$. There are global analogs due to Saito et al. I haven't come across generalizations of local duality theorems so far. $\endgroup$
    – user130124
    Commented Aug 22, 2019 at 14:46

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