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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.

flag	asked Nov 3 2009 at 12:22 Kevin Buzzard 22.8k●2●69●136

Lavender Honey · Accepted Answer · 2009-11-03 14:07:10Z UTC

17

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

answered Nov 3 2009 at 14:07

Lavender Honey
6,024●30●69

	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. – Kevin Buzzard Nov 3 2009 at 15:37
	How this conversation would have gone outside Math Overflow: K: Are all Q-curves modular? FC: Of course! That's a consequence of Serre's conjecture. K: Oooh, gotta be careful. FC: But Serre has a nice argument proving this for every abelian variety A of GL_2-type! K: Hang on, wait a minute, how do you know that Q-curves give abelian varieties A/Q of GL_2-type? FC: Ooh, err... um... I think Ken wrote a paper about this once. – Lavender Honey Nov 3 2009 at 16:31
	I had exactly that conversation with KB a couple of weeks ago, except that I didn't know the punchline. – TG Nov 3 2009 at 21:31

Are Q-curves now known to be modular?

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