The modularity conjecture for elliptic curves over totally real fields

Let $F$ be a totally real field, meaning that $F$ is a finite extension of ${\mathbb Q}$ such that every embedding $F\hookrightarrow{\mathbb C}$ has image in ${\mathbb R}$. An elliptic curve $E$ over $F$ is modular if its Hasse--Weil $L$-function coincides with the $L$-function of a Hilbert modular newform of parallel weight $2$ over $F$.

Modularity conjecture. Every elliptic curve over a totally real field $F$ is modular.

This conjecture generalizes the Shimura--Taniyama conjecture. It is known to be true for certain totally real fields of higher degree, but remains open in general when $[F:{\mathbb Q}]\geq 4$.