Are Q-curves now known to be modular? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T21:21:04Z http://mathoverflow.net/feeds/question/3927 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/3927/are-q-curves-now-known-to-be-modular Are Q-curves now known to be modular? Kevin Buzzard 2009-11-03T12:22:40Z 2009-11-03T14:07:10Z <p>I really should know the answer to this, but I don't, so I'll ask here.</p> <p>A <em>Q-curve</em> is an elliptic curve E over Q-bar which is isogenous to all its Galois conjugates. A Q-curve is <em>modular</em> 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).</p> <p>Has current machinery proved the well-known conjecture that all Q-curves are modular yet?</p> <p>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.</p> http://mathoverflow.net/questions/3927/are-q-curves-now-known-to-be-modular/3936#3936 Answer by Lavender Honey for Are Q-curves now known to be modular? Lavender Honey 2009-11-03T14:07:10Z 2009-11-03T14:07:10Z <p>Yes, this is a consequence of Serre's conjecture. The canonical reference is probably Corollary 6.2 of Ribet's paper on Q-curves:</p> <p><a href="http://math.berkeley.edu/~ribet/Articles/korea.pdf" rel="nofollow">http://math.berkeley.edu/~ribet/Articles/korea.pdf</a></p>