The Langlands correspondence conjecture for weight-two eigenforms over number fields

Let $K$ be a number field, let $\mathfrak N$ be an ideal of $\mathcal{O}_K$, and let $\mathfrak f$ be a non-trivial, new, weight $2$ complex eigenform over $K$ of level $\mathfrak N$. If $K$ has a real place, an elliptic curve $E_{\mathfrak f}/K$ of conductor $\mathfrak N$ should satisfy

$$\#E_{\mathfrak f}(\mathcal{O}_K/\mathfrak q)=1+\operatorname{Norm}(\mathfrak q)-\mathfrak f(T_{\mathfrak q})$$

for every prime $\mathfrak q\nmid\mathfrak N$. If $K$ is totally complex, either such an elliptic curve of conductor $\mathfrak N$ exists, or there is a fake elliptic curve $A_{\mathfrak f}/K$ of conductor $\mathfrak N^2$ satisfying

$$\#A_{\mathfrak f}(\mathcal{O}_K/\mathfrak q)=\bigl(1+\operatorname{Norm}(\mathfrak q)-\mathfrak f(T_{\mathfrak q})\bigr)^2$$

for every prime $\mathfrak q\nmid\mathfrak N$.

Langlands correspondence conjecture. The alternatives and point-count identities above hold for every such eigenform.

The paper describes this as a special case of a fundamental conjecture from the Langlands Programme and uses it alongside the modularity conjecture. The supplied text does not indicate a resolution.