# modularity of elliptic curves with cm

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)!

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Not all CM elliptic curves are $\mathbb{Q}$-curves, so there is no hope in general of finding them inside the Jacobian of a modular curve. However, CM elliptic curves are "modular" in the sense that we can associate to them a Hecke character, which gives all the nice properties of its $L$-function. –  Kevin Ventullo Jun 18 '13 at 23:35
There is a nice article where Shimura's result is generalized for a wider class of CM elliptic curves, see S. Wortmann, *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. –  François Brunault Jun 19 '13 at 8:48
In the last sentence, I meant "isomorphic over $\overline{\mathbf{Q}}$"... –  François Brunault Jun 19 '13 at 8:48
What is true is that every elliptic curve over $\overline{\mathbf{Q}}$ with CM by $K$ is isomorphic over $\overline{\mathbf{Q}}$ to a $K$-curve (a curve which is isogenous over $\overline{\mathbf{Q}}$ to all its $\operatorname{Gal}(\overline{\mathbf{Q}}/K)$-conjugates), see Wortmann's article and the references. –  François Brunault Jun 19 '13 at 8:53
Note that if $\operatorname{Res}_{F/\mathbf{Q}} E$ appear inside $J_1(N)$ over $\mathbf{Q}$ then $E$ itself appears inside $J_1(N)$ over $F$. So in this case you can find a modular parametrization $X_1(N) \to E$ which is defined over $F$. –  François Brunault Jun 19 '13 at 10:57