Modularity of residual Galois extensions and the Eisenstein ideal

Abstract: 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 $\Sigma \supset {\mathfrak{p} \mid p}$ consider the Bloch-Kato Selmer group $H:=H^1{\Sigma}(F, \chi^{-1})$. In an earlier paper of the authors it was proved that the number $d$ of isomorphism classes of (non-semisimple, reducible) residual representations $\overline{\rho}$ giving rise to lines in $H$ which are modular by some $\rho_f$ (also unramified outside $\Sigma$) 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 $d>n$.