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)?