I think we don't know what crystalline representations come from geometry. What about the unramified ones? Specifically let $\phi:\mathrm{Gal}(\overline{\mathbb{Q}_p}/\mathbb{Q}_p)\to GL_n(\mathbb{Q}_p)$ be a continuous unramified representation. When is $\phi$ a subquotient of $H^{i}_{{\acute{\mathrm{e}}\text {t}}}(X_{\overline{\mathbb{Q}_p}}, \mathbb{Q}_p(j))$ where $X$ is a smooth proper $\mathbb{Q}_p$-scheme?

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    $\begingroup$ My understanding is it should be only the ones with finite image, but I don't have a good references. $\endgroup$ – Will Sawin Aug 7 '20 at 13:41
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    $\begingroup$ Can't we get an infinite image unramified character as a sub of the Galois rep on $H^1$ of an elliptic curve with good ordinary reduction? Perhaps we could similarly get from abelian varieties any semi-simple unframified representation in which Frobenius with eigenvalues that are Weil numbers (at least if we allow extending the coefficients). $\endgroup$ – SashaP Aug 9 '20 at 20:40

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