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Let $K$ be a finite extension of $\mathbb{Q}_p$ and let $X$ be a smooth variety over $K$. Dr. Yamashita announced that he had proved the Galois representation of $p$-adic étale cohomology group $H^*_{\acute{e} t}(X_{\overline{K}},\mathbb{Q}_p)$ is de Rham (Cf. https://www.sciencedirect.com/science/article/pii/S1631073X11002998).

But his paper remains unpublished yet. Without using his result, is it possible to prove that $H^*_{\acute{e} t}(X_{\overline{K}},\mathbb{Q}_p)$ is de Rham?

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  • $\begingroup$ Are you asking whether or not some other publication or announcement on this topic has appeared? $\endgroup$
    – Jason Starr
    Commented Apr 5, 2023 at 16:48
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    $\begingroup$ Yes, see for instance arxiv.org/abs/1801.01779 $\endgroup$
    – Satan's Minion
    Commented Apr 5, 2023 at 17:04
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    $\begingroup$ @Satan'sMinion The link you give is for rigid-analytic spaces, not algebraic varieties. Are the $p$-adic etale cohomology of $X$ and its rigid analytification the same? In any case the result for open algebraic varieties long pre-dates that reference, see this 2011 paper of Beilinson: arxiv.org/pdf/1102.1294.pdf, $\endgroup$
    – David Loeffler
    Commented Apr 5, 2023 at 21:47
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    $\begingroup$ @DavidLoeffler Just to answer the question in your comment: the etale cohomology of $X$ and its rigid-analytification do indeed agree in this case. See $\S7$ of de Jong and van der Put’s 1995 Documenta paper. $\endgroup$
    – Oli Gregory
    Commented Apr 6, 2023 at 4:07
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    $\begingroup$ @David Loeffler: Yes, there are general comparison theorems of Huber which give this easily. And I said "for instance" because I am well aware there are other references for this... $\endgroup$
    – Satan's Minion
    Commented Apr 6, 2023 at 8:06

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