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Part I.

Say $K$ is a number field, $v$ is a finite place of $K$, $K_v$ the $v$-adic completion of $K$.

We have the local Artin map for every finite $v$:

$$\rho_v : K_v^{\times}\to\text{Gal}(K_v^{\rm ab}/K_v)$$

sending $\varpi_v$, a generator of the copy of $\mathbf{Z}$ in $K_v^{\times}$, to the Frobenius element in $\text{Gal}(K_v^{\rm unr}/K_v)$, and sending $\mathcal{O}_v^{\times}$ isomorphically onto $\text{Gal}(K_v^{\rm LT}/K_v)$, where $K_v^{\rm LT}$ is the Lubin-Tate extension of $K_v$, with $K_v^{\rm LT}\cdot K_v^{\rm unr} = K_v^{\rm ab}$.

The completion map $K\to K_v$ induces an injective map $$\alpha_v : \text{Gal}(K_v^{\rm ab}/K_v)\to\text{Gal}(K^{\rm ab}/K)$$

Let's call $c_v$ the map $K^{\times}\to K_v^{\times}$ induced by completion.

We consider the composition:

$$\mu_v := \alpha_v\circ\rho_v\circ c_v : K^{\times}\to\text{Gal}(K^{\rm ab}/K).$$

$\mu_v$ can be made independent of all choices (since $\rho_v$ can).

Can one give a fairly explicit description of $\mu_v$?

(What I have in mind when I say "explicit description" is, for instance, this: I'd guess $\mu_v^{-1}(\text{Frob}_v) = \{x\in K^{\times}\mid v(x)>0\}$, for $\text{Frob}_v$ the Frobenius element at $v$ in $\text{Gal}(K^{\rm ab}/K)$, and this ought to essentially describe $\mu_v$ "exhaustively enough" by Chebotarev)

Part II.

Is there a "higher dimensional analogue" of Part I?

For instance, for a smooth projective connected variety over a number field $X$, we call $Z_0(X)$ the free abelian group of $0$-cycles on $X$, and define:

$$\rho: Z_0(X)\to\pi_1^{\rm {e}t}(X,\bar{x})^{\rm ab}$$

by sending $x\in Z_0(X)$ to the Frobenius element $\text{Frob}_x$ at $x$, and extend by $\mathbf{Z}$-linearity. Can one describe the fibers of $\rho$ fairly explicitly? and would the fibers over Frobenius elements suffice by a higher dimensional version of Chebotarev? (ie. is there such version of Chebotarev? Any references?)

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  • $\begingroup$ There exists a unique formal group law $F$ on $\mathcal{O}_v[\![t]\!]$ with the property that $f(t) = t^q + t\varpi_v$, with $q = \#(\mathcal{O}_v/\mathfrak{m}_v)$, is an endomorphism of $F$. $F$ depends on $f$ and $\varpi_v$, a priori, but in fact only on $\varpi_v$. It turns out that for such an $F$, the resulting formal group is actually a formal $\mathcal{O}_v$-module, whose $\varpi_v$-division points generate the so called Lubin-Tate extension of $K_v$. It turns out that the group of formal module automorphisms of $F$ is exactly $\mathcal{O}_v^{\times}$, and $\endgroup$
    – user95222
    Commented Jan 7, 2018 at 3:59
  • $\begingroup$ it also turns out that such automorphism group is isomorphic to the Galois group of the Lubin-Tate extension $K_v^{\rm LT}$ of $K_v$. Such isomorphism is the $\rho_v$ in your question. If one takes the composite field $K_v^{\rm unr}\cdot K_v^{\rm LT}$, one can extend $\rho_v$ to $K_v^{\times} = \varpi_v^{\mathbf{Z}}\times\mathcal{O}_v^{\times}$ by declaring that $\rho_v(\varpi_v) = \text{Frob}_v\in\text{Gal}(K_v^{\rm unr}/K_v)$. $\endgroup$
    – user95222
    Commented Jan 7, 2018 at 4:02
  • $\begingroup$ The local-at-$v$ Kronecker Weber Thm then tells you $K_v^{\rm unr}\cdot K_v^{\rm LT}= K_v^{\rm ab}$, so you have defined $\rho_v : K_v^{\times}\to\text{Gal}(K_v^{\rm ab}/K_v)$. It turns out this $\rho_v$ agrees with the Artin map defined abstractly in Artin-Tate. This is local class field theory via Lubin-Tate theory. See Cassels-Frolich, for instance. $\endgroup$
    – user95222
    Commented Jan 7, 2018 at 4:04
  • $\begingroup$ (final comment: $K_v^{\rm LT}$ still depends on $\varpi_v$, but the composite $K_v^{\rm unr}\cdot K_v^{\rm LT}$ doesn't) $\endgroup$
    – user95222
    Commented Jan 7, 2018 at 4:06
  • $\begingroup$ (the comments above are meant to answer a question previously visible on top of them) $\endgroup$
    – user95222
    Commented Jan 7, 2018 at 6:05

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