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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3$\begingroup$ 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

1$\begingroup$ 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 semisimple 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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