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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.

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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

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    $\begingroup$ Excellent---thanks. I could see the argument that Serre implied that ab vars of GL_2 type were modular, but I was missing Ribet's 6.1. $\endgroup$
    – Kevin Buzzard
    Commented Nov 3, 2009 at 15:37
  • $\begingroup$ I had exactly that conversation with KB a couple of weeks ago, except that I didn't know the punchline. $\endgroup$
    – TSG
    Commented Nov 3, 2009 at 21:31

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