The modularity conjecture for elliptic curves and QM surfaces over imaginary quadratic fields

Let $K$ be an imaginary quadratic field. A Bianchi newform over $K$ is a weight-$2$ cuspidal eigenform of level $[?]$ with rational Hecke eigenvalues; here $[?]$ denotes the ideal level, and a QM surface is an abelian surface with quaternion multiplication. Modularity conjecture.

1. If $f$ is a Bianchi newform over $K$ of weight $2$ and level $\Gamma_0(\mathfrak{n})$ with rational Hecke eigenvalues, then either there is an elliptic curve $E/K$ without CM by $K$ of conductor $\mathfrak{n}$ such that

$$L(E/K,s)=L(f,s),$$

or there is a QM surface $A/K$ of conductor $\mathfrak{n}^2$ such that

$$L(A/K,s)=L(f,s)^2.$$

2. Conversely, if $E/K$ is an elliptic curve without CM by $K$ of conductor $\mathfrak{n}$, then there is an $f$ as above such that

$$L(E,s)=L(f,s).$$

Moreover, if $A/K$ is a QM surface of conductor $\mathfrak{n}^2$, then there is an $f$ as above such that

$$L(A,s)=L(f,s)^2.$$

This conjecture extends the correspondence between elliptic curves over $\mathbb{Q}$ and classical weight-$2$ newforms to imaginary quadratic fields, where quaternion-multiplication surfaces must also be included. The paper gives explicit examples of genuine QM abelian surfaces that are modular, but the general correspondence asserted here remains unresolved.