Let $X$ be a smooth projective variety over a p-adic local field $K$, and let $\bar{K}$ be the algebraic closure of $K$. Fix an isomorphism $\sigma:\bar{K}\to\mathbb{C}$. Do $\sigma$ induces an (iso)morphiam between the Hodge class $Hdg(X_{\mathbb{C}})\otimes \mathbb{Z}_\ell$ and the Tate class $H^{2k}_{et}(X,\mathbb{Z_\ell}(k))^{G_K}$ ( or with coefficients in $\mathbb{Q}_\ell$ and $\mathbb{Z}/\ell^n\mathbb{Z}$)?
If not, please explain it to me, thanks.