Timeline for The existence of an elliptic curve with a specific Galois representation induced by a character

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Oct 28, 2012 at 21:21	comment	added	David Loeffler		PS: Don't be tempted to read "$A[\ell] \cong \rho$" as "the image of $\Gal(\overline{F} / F)$ acting on $\rho$ and of $\Gal(\overline{F} / F')$ acting on $A[\ell]$ coincide as subgroups of $GL_2(\mathbb{F}_\ell)$". Although plausible, this reading isn't what's meant here at all.
Oct 28, 2012 at 21:18	comment	added	David Loeffler		If that doesn't address your question, I'm not entirely sure what your question is. Certainly not every mod $\ell$ representation of $Gal(\overline{F} / F)$ can be realized in the $\ell$-torsion of an elliptic curve over $F$; but the restriction of any such representation to $Gal(\overline{F} / F')$ can be realized by an elliptic curve over some big enough $F'$, simply because that restriction can be made trivial.
Oct 28, 2012 at 19:18	comment	added	Jonah Sinick		Good point. This doesn't address my question of whether one can get any representation induced by a character can be gotten from an elliptic curve, but it helps highlight how potential modularity theorems can be proved.
Oct 28, 2012 at 10:38	history	edited	David Loeffler	CC BY-SA 3.0	changed rho to ell (thanks Chandan)
Oct 28, 2012 at 10:34	comment	added	Chandan Singh Dalawat		David, the $A[\rho]$ in the third line should be $A[l]$.
Oct 28, 2012 at 9:46	history	answered	David Loeffler	CC BY-SA 3.0