I really should know the answer to this, but I don't, so I'll ask here.


A *Q-curve* is an elliptic curve E over Q-bar which is isogenous to all its Galois conjugates. A Q-curve is *modular* if it's isogenous (over Q-bar) to some factor of the Jacobian of X\_1(N) for some N>=1 (here X\_1(N) is the compact modular curve over Q-bar).


Has current machinery proved the well-known conjecture that all Q-curves are modular yet?


Remark: I know there are many partial results. What I'm trying to establish is whether things like Khare-Wintenberger plus best-known modularity lifting theorems are strong enough to give the full conjecture yet, or whether we're still waiting.



Are Q-curves now known to be modular?
Yes, this is a consequence of Serre's conjecture. The canonical reference is probably Corollary 6.2 of Ribet's paper on Q-curves:


<http://math.berkeley.edu/~ribet/Articles/korea.pdf>


