TITLE: Is every $GL_2(\mathbb{Z}/n\mathbb{Z})$-extension contained in some elliptic curve's torsion field?
QUESTION [16 upvotes]: I suppose this question could be phrased in terms of Galois representations, but I'm asking it this way.
Let $n>1$ be an integer.    If $K$ is a number field with $\operatorname{Gal}(K/\mathbb{Q}) \cong GL_2(\mathbb{Z}/n\mathbb{Z})$ (edit) and containing the $n$-th roots of unity (/edit), must there exist an elliptic curve $E$ defined over $\mathbb{Q}$ such that $K \subseteq \mathbb{Q}(E[n])$?  It's not necessary to produce the curve, although it would be cool.
If not, is there a good way to test if this holds for a given $K$?
(If you want to restrict to the case where $n$ is a prime power, that's fine.)

REPLY [4 votes]: Two comments. First, there is a trick used by Serre, Ribet and others to prove, using an old result of Goursat, that under certain assumptions such as absolute irreducibility, two mod p Galois representations cut the same Galois extension if and only if they are twists of each other. I guess using this combined with the reference in a previous answer should be enough to answer your question.
Secondly, if you take n to be a prime power with exponent at least 2, it is not even clear that a mod n representation can be lifted at all to any p-adic representation, even if you admit coefficients in extensions of the p adic field. Further assumptions such as finite ramification set make the question even harder. My guess is that in general you should not expect existence of any geometric p adic lift, in particular this rules out elliptic curves.