I'd like to ask for references on the status of modularity results for elliptic curves with CM which are not necessarily defined over $\mathbb Q$. In the case of an elliptic curve with CM defined over $\mathbb Q$ I'm aware of a nice article by Shimura, where this is explained.

Due to the fact of being an outsider in this business I would highly appreciate any hints/help towards the literature (or sharing of "common knowledge" in that field).

Maybe I should say that for me modularity means to ask for a "nice parametrization" of the elliptic curve $E$ in hand in terms of an appropriate moduli space (of (elliptic) curves). More precisely, I would also be very interested in "nice (algebraic) parametrizations" which don't match up necessarily the Hasse-Weil zeta function of $E$ with an appropriate modular form, i.e. are there "reasonable" weak forms of modularity known for elliptic curves with CM?

Thanks a lot in advance (and all my apologies in case this question is way too naive)!

Generalized Q-curves and factors of $J_1(N)$dx.doi.org/10.1007/BF02940901 These CM elliptic curves are sometimes called of Shimura type. I'm not sure but I think these are exactly the CM elliptic curves whose restriction of scalars appear inside $J_1(N)$ over $\mathbf{Q}$. Moreover, every CM elliptic curve is isomorphic over $\mathbf{Q}$ to a curve of Shimura type. $\endgroup$ – François Brunault Jun 19 '13 at 8:48