proved (see [@DGP]), and under some mild hypothesis it can be shown that any such curve is potentially modular (see [@Calegari-Geraghty Theorem 1.1]). Curves with complex multiplication correspond to Hecke characters and belong to non-cuspidal modular forms. These will be considered in Section \[section CM\].

Following the classical result of Ribet ([@Ribetlowering]), we make the following conjecture.

Let $F$ be a weight $2$ Bianchi newform of level $\Gamma_0(\id{n}\id{p})$, with $\id{p}$ a prime number. Suppose that $\ell\ge 5$ is a prime for which the representation $\rho_{F,\ell}$ satisfies:

1.  The residual representation is absolutely irreducible.

2.  The residual representation is finite at $\id{p}$.

Then there is an automorphic form $G \in H^1(Y_0(\id{n}),\F_{\ell})$ whose Galois representation is isomorphic to $\overline{\rho_{F,\ell}}$. \[conj:loweringlevel\]

Some results in the direction of the conjecture are proven in [@Calegari]. Note that the conjecture refers to an element in the mod $\ell$ cohomology. In contrast to modular forms for $\operatorname{SL}_2(\ZZ)$, for Bianchi modular forms the homology (and the cohomology) of $Y_0(\id{n})$ contains a big torsion part. By results of Scholze ([@Scholze]) the torsion forms do have Galois representations attached, but these are residual representations that in general do not admit a lift of level $\id{n}$ to characteristic zero, so the 