Suppose $K$ is a unramified finite extension of $\mathbb Q_p$, and $X$ is a projective smooth curve defined over $K$. By $p$-adic Hodge theory we know $D_{cris}(H_{et}^i(X,\mathbb Q_p))=H_{dR}^i(X)$. Thus the Frobenius naturally acts on $H_{dR}^i(X)$. I wonder is this action semisimple? (that is, Frobenius is diagonalizable in $\bar K$).
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3$\begingroup$ You might find the Coleman--Edixhoven paper "On the semisimplicity of the Up operator" illuminating (the title is about the Hecke operator U(p) on modular forms, but the proofs proceed by relating the problem to semisimplicity of Frobenius on DdR). $\endgroup$– David LoefflerSep 21 at 5:28
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