# What unramified Galois representations come from geometry?

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?

• My understanding is it should be only the ones with finite image, but I don't have a good references. – Will Sawin Aug 7 '20 at 13:41
• 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). – SashaP Aug 9 '20 at 20:40