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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.

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    $\begingroup$ are you taking the tensor product over $\mathbb{Z}$? If so, note that Hodge conjecture can fail integrally (I think there are examples due to Kollar, maybe other people too). Not sure how relevant is this, just a thought. $\endgroup$ – user140765 May 31 '19 at 19:22
  • $\begingroup$ possibly see this question (mathoverflow.net/q/40486/140765), I am not knowledgeable enough to determine whether it is relevant $\endgroup$ – user140765 May 31 '19 at 19:24
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    $\begingroup$ Over local fields Tate conjecture is very false: for any smooth proper scheme $\mathcal{X}/\mathcal{O}_K$ the Galois invariants on the etale cohomology of the generic fiber $\mathcal{X}_{\bar{K}}$ coincide with that of the special fiber $\mathcal{X}_{\bar{k}}$, while the dimension of the subspaces of algebraic classes might be different: e.g. for an elliptic curve $E$ without CM and with good reduction over $\mathcal{O}_K$ the generic fiber of $X=E\times E$ has $1$-dimensional Neron-Severi group, while it is at least two-dimensional for the special fiber. $\endgroup$ – SashaP Jun 1 '19 at 0:38
  • $\begingroup$ Correction: NS group of the generic fiber is 3 dimensional and for the special fiber it is at least 4 dimensional(these dimensions are composed from the rank of the endomorphism ring and the 2 dimensional piece given by vertical and horizontal divisors) $\endgroup$ – SashaP Jun 1 '19 at 1:54

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