On lifting and modularity of reducible residual Galois representations over imaginary quadratic fields

In this paper we study deformations of mod $p$ Galois representations $\tau$ (over an imaginary quadratic field $F$) of dimension $2$ whose semi-simplification is the direct sum of two characters $\tau_1$ and $\tau_2$. As opposed to our previous work we do not impose any restrictions on the dimension of the crystalline Selmer group $H^1_{\Sigma}(F, {\rm Hom}(\tau_2, \tau_1)) \subset {\rm Ext}^1(\tau_2, \tau_1)$. We establish that there exists a basis $\mathcal{B}$ of $H^1_{\Sigma}(F, {\rm Hom}(\tau_2, \tau_1))$ arising from automorphic representations over $F$ (Theorem 8.1). Assuming among other things that the elements of $\mathcal{B}$ admit only finitely many crystalline characteristic 0 deformations we prove a modularity lifting theorem asserting that if $\tau$ itself is modular then so is its every crystalline characteristic zero deformation (Theorems 8.2 and 8.5).