Modularity conjecture for abelian varieties of GL(2)-type over totally real fields

Let $A$ be an abelian variety of $GL(2,E)$-type over a number field $K$, with conductor $\mathfrak{n}$, and let $\rho_{A,\lambda}$ denote its Galois representation for a prime $\lambda$ of $E$. A Hilbert modular form over $K$ has weight and level, and $\rho_{f,\lambda}$ denotes the Galois representation associated to such a form. Modularity conjecture. If $A$ is an abelian variety of $GL(2,E)$-type over a number field $K$ of conductor $\mathfrak{n}$, then there exists a Hilbert modular form $f$ over $K$ of weight $2$ and level $\mathfrak{n}$ such that

$$\rho_{f,\lambda} \cong \rho_{A,\lambda}$$

for all primes $\lambda$ of $E$. This is presented as a generalization of the Taniyama–Shimura–Weil conjecture and is motivated by modularity questions arising in Darmon's program for generalized Fermat equations. Its resolution is not supplied in the source.