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?

allthe obvious assumptions on $\rho$ may not be enough. – eric Jan 15 at 20:39