Modularity of residual Galois extensions and the Eisenstein ideal

For a totally real field F, a finite extension F of Fp and a Galois character χ:GF→F× unramified away from a finite set of places Σ⊃{p∣p} consider the Bloch-Kato Selmer group H:=H1Σ(F,χ−1). In an earlier paper of the authors it was proved that the number d of isomorphism classes of (non-semisimple, reducible) residual representations ρ¯¯¯ giving rise to lines in H which are modular by some ρf (also unramified outside Σ) satisfies d≥n:=dimFH. 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=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.