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Given a finite extension $K$ of $\mathbb{Q}_p$, is there some conjectural statement characterizing which finite-dimensional $p$-adic representations of the absolute Galois group of $K$ are (Tate twists of) subquotients of the cohomology groups of $K$-varieties?

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    $\begingroup$ Varieties give rise to de Rham and thus potentially semi-stable representations, but not every such representation comes from geometry: for example, the eigenvalues of Frobenius should be Weil numbers. I don't know any conjectural description. $\endgroup$ May 19, 2019 at 7:06
  • $\begingroup$ The answer to the question in the body of the text is "at present, no" (even for crystalline representations). $\endgroup$
    – Olivier
    May 24, 2019 at 13:24
  • $\begingroup$ @Olivier is there a reference discussing this? What are the obstructions beyond those provided by Weil conjectures? $\endgroup$
    – user140765
    May 24, 2019 at 15:56

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