**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?