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Let $K$ be a number field with absolute Galois group $G_K$.

Let $\rho:G_K \rightarrow GL_2(\hat{\mathbb{Z}})$ be a Galois representation such that the image of $\rho$ is open in $GL_2(\hat{\mathbb{Z}})$, and the determinant of $\rho$ is cyclotomic i.e. $\det \circ \rho = \chi$ where $\chi$ is the cyclotomic character.

Then under what conditions does $\rho$ come from the Galois representation on the Tate module of an elliptic curve?

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There are no nice conditions that I know of. The problem is that even demanding all the traces of Frobenii are integers satisfying the Weil bounds is not enough, as there can exist automorphic forms for $GL(2)/K$ whose Fourier coefficients are integers but whose associated Galois rep is not attached to an ell curve. Cremona found such examples in his thesis. The motive attached to the form in his cases was part of the cohomology of an abelian surface over K with an action of an order in a non-split quat alg. TL;DR: even putting on all the obvious assumptions on $\rho$ may not be enough. –  eric Jan 15 at 20:39
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This question is related to the Fontaine-Mazur conjecture. –  Damian Rössler Jan 15 at 21:30

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