By Eichler-Shimura over totally real fields I mean the following conjecture.

Conjecture. Let $K$ be a totally real field. Let $f$ be a Hilbert eigenform with rational eigenvalues, of parallel weight $2$, that is new at level $\mathfrak{N}$. Then there is an elliptic curve $E_f$ over $K$ of conductor $\mathfrak{N}$ such that $$\mathrm{L}(E_f,s)=\mathrm{L}(f,s)$$.

I understand from this paper of Zhang that the conjecture holds if $[K:\mathbb{Q}]$ is odd, or if $\upsilon_{\mathfrak{P}}(\mathfrak{N})=1$ for some prime $\mathfrak{P}$.

Zhang's paper appeared in 2001. What is the state of the art regarding this conjecture? What progress have been made since?


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