Modularity of residual Galois extensions and the Eisenstein ideal

For a totally real field $F$, a finite extension $\mathbf{F}$ of ${\mathbf{F}}_p$ and a Galois character $\chi :G_F\to {\mathbf{F}}^{\times }$ unramified away from a finite set of places $\Sigma \supset {\mathfrak{p} \mid p}$considertheBloch-KatoSelmergroup$H:=H^1_{\Sigma}(F, \chi^{-1})$. In an earlier paper of the authors it was proved that the number $d$ of isomorphism classes of (non-semisimple, reducible) residual representations $\bar{\rho }$ giving rise to lines in $H$ which are modular by some ${\rho }_f$ (also unramified outside $\mathrm{\Sigma }$) satisfies $d \geq n:= \dim_{$\mathbf{F}$} H$. This was proved under the assumption that the order of a congruence module is greater than or equal to that of a divisible Selmer group. We show here that if in addition the relevant local Eisenstein ideal $J$ is non-principal, then $d>n$. When $F=\mathbf{Q}$ we prove the desired bounds on the congruence module and the Selmer group. We also formulate a congruence condition implying the non-principality of $J$ that can be checked in practice, allowing us to furnish an example where $d>n$.