Let $p$ be a prime. From what I understand, the p-adic cyclotomic character $\varepsilon:Gal(\overline{\mathbb{Q}}/\mathbb{Q}) \to \mathbb{Z}_p^\times$ is given by choosing a compatible sequence of $p$-power roots of unity $(\zeta_{p^n})_n$ with $\zeta_{p^n}^p=\zeta_{p^{n-1}}$ in $\overline{\mathbb{Q}}$ then for $\sigma$, $$\sigma(\zeta_{p^n}) = \zeta_{p^n}^{\varepsilon(\sigma) \bmod p^n}$$
If we take $F=Frob_p$ - an arithmetic Frobenius element of at $p$, i.e an element of the decomposition group $Gal(\overline{\mathbb{Q}_p}/\mathbb{Q}_p)$ at p such that $F(\zeta_{p^n}) = \zeta_{p^n}^p \bmod p$, so what is $\varepsilon(F) \bmod p$?
More generally, can anyone explain in detail the underlined sentence in the following paragraph (taken from Breuil's ICM 2010 paper, at the end of the intro where he sets the basic notations/conventions)
There are many possible Frobenius elements (and decomposition groups). There is a canonical one on $\Bbb{Q}(\zeta_\infty)$ which is $\phi(\zeta_{p^n-1}) =\zeta_{p^n-1}^p,\phi( \zeta_{p^n})= \zeta_{p^n}$ which gives $\varepsilon(\phi)=1$ (then extend $\phi$ to $\overline{\Bbb{Q}}$ as you wish). The others are of the form $\sigma \phi$ for some $\sigma\in Gal(\overline{\Bbb{Q}}/\Bbb{Q}(\zeta_{p^n-1}))$ and $\varepsilon(\sigma\phi)=\varepsilon(\sigma)$ doesn't have to be $1$.
About the sentence underlined in red:
The previous $\phi$ (the one on $\Bbb{Q}(\zeta_\infty)$) extends naturally to $\Bbb{Q}_p(\zeta_\infty)$.
$Gal(\Bbb{Q}_p^{ab}/\Bbb{Q}_p)$ is non-trivial because of the non-cyclotomic wildly ramified abelian extensions (linked to abelian formal group laws linked to elliptic curves), in contrary to $Gal(\Bbb{Q}_p(\zeta_\infty)/\Bbb{Q}_p)$ which is $=Gal(\Bbb{Q}_p(\zeta_\infty)/\Bbb{Q}_p(\zeta_{p^\infty-1}))\times Gal(\Bbb{Q}_p(\zeta_\infty)/\Bbb{Q}_p(\zeta_{p^\infty})) \cong \Bbb{Z}_p^\times\times \phi^{\hat{\Bbb{Z}}} $ then $\Bbb{Z}_p^\times\times \phi^{\Bbb{Z}}$ is dense and this latter group is isomorphic to $\Bbb{Q}_p^\times$, they are saying a canonical choice is to send $(a, \phi^n)$ to to $a p^{-n}$. Your character is sending $Gal(\Bbb{Q}_p^{ab}/\Bbb{Q}_p)\to Gal(\Bbb{Q}_p(\zeta_\infty)/\Bbb{Q}_p)\to \Bbb{Z}_p^\times\times \phi^{\hat{\Bbb{Z}}} \to \Bbb{Z}_p^\times $.