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In Kevin Buzzard's survey article on potential modularity Buzzard writes:

Let us say that we have an elliptic curve $E$ over a totally real field $F$, and we want to prove that $E$ is potentially modular (that is, that $E$ becomes modular over a finite extension field $F^{′}$ of $F$, also assumed totally real). Here is a strategy. Say $p$ is a large prime such that $E[p]$ is irreducible. Let us write down a random odd $2$-dimensional mod $ℓ$ Galois representation $\rho_{ℓ} : > Gal(\overline{F}/F) → GL(2,\mathbf{F}_ℓ )$ which is induced from a character; because this representation is induced it is known to be modular. Now let us consider the moduli space parametrising elliptic curves $A$ equipped with

  1. An isomorphism $A[p] \cong E[p] $
  2. An isomorphism $A[ℓ]\cong ρ_ℓ$

This moduli problem will be represented by some modular curve, whose connected components will be twists of $X(pℓ)$ and hence, if $p$ and $ℓ$ are large, will typically have large genus. However, such a curve may well still have lots of rational points, as long as I am allowed to look for such things over an arbitrary finite extension $F^{′}$ of $F$ !

It's not immediately obvious to me that there's an elliptic curve $A$ over some $F^{′}$ satisfying the second condition alone (never mind satisfying both conditions simultaneously). Is there a simple explanation for why there should be such an $A$? Did Professor Buzzard mean "consider the set of A such that $A[ℓ]\cong ρ_ℓ$ for some representation induced by a character" (as opposed to a particular one)?

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1 Answer 1

In this context, if $\rho$ is a mod $\ell$ representation of $Gal(\overline{F} / F)$, and $A$ is an elliptic curve over an extension $F' / F$, then the statement "$A[\ell] \cong \rho$" needs a little bit of interpretation, because the two sides are representations of different things: $A[\ell]$ is a mod $\ell$ representation of the subgroup $Gal(\overline{F} / F') \subset Gal(\overline{F} / F)$. So the statement is to be read as "$A[\ell]$ is isomorphic as a $Gal(\overline{F} / F')$-representation to the restriction of $\rho$". Now, the bigger $F'$ is, the weaker this condition becomes: in particular, if we take any elliptic curve $A$ over $F$ and define $F'$ to be the extension of $F$ generated by the $\ell$-torsion points of $A$ and the splitting field of $\rho$, then the statement is automatic (both sides are the trivial representation).

(This is kind of a stupid example, but maybe you can believe now that there exist non-stupid examples as well!)

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David, the $A[\rho]$ in the third line should be $A[l]$. –  Chandan Singh Dalawat Oct 28 '12 at 10:34
    
So it should -- thanks! –  David Loeffler Oct 28 '12 at 10:38
    
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. –  Jonah Sinick Oct 28 '12 at 19:18
    
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. –  David Loeffler Oct 28 '12 at 21:18
    
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. –  David Loeffler Oct 28 '12 at 21:21

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