TITLE: The existence of an elliptic curve with a specific Galois representation induced by a character
QUESTION [6 upvotes]: 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 

An isomorphism $A[p] \cong E[p] $
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)?

REPLY [8 votes]: 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!)