Let $G=Gal(\bar K/K)$ be the absolute Galois group of a number field $K$. Let $v$ be a finite place of $K$ and $w$ a place of $\bar K$ extending $v$. Take an $\ell$-adic representation $$ \rho: G \longrightarrow \mathrm{GL}_n(\mathbb{Q}_{\ell}).$$

Assume $\rho$ is unramified at $v$ and denote with $F_{w,\rho}$ the image of the Frobenius element associated at $w$.

How the conjugacy class of $F_{w,\rho}$ does depend only by $v$?

  • $\begingroup$ That is because if you pick a different $w'$ lying above $v$, the Frobenius of $w$ over $v$ and the Frobenius of $w'$ over $v$ are conjugate in $G$. $\endgroup$ – Ferra Oct 11 '15 at 23:00
  • $\begingroup$ In finite extension I am agree with you. However in this extension I know that you have a little problem because the inertia subgroup it is not trivial. $\endgroup$ – user262440 Oct 12 '15 at 5:29
  • 1
    $\begingroup$ it doesn't matter: the decomposition groups of w and w' are conjugate! $\endgroup$ – Ferra Oct 12 '15 at 7:51

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