MODULARITY OF RESIDUAL GALOIS EXTENSIONS AND THE EISENSTEIN IDEAL

For a totally real field F, a finite extension F of F-p, and a Galois character chi : G(F) -> F-x unramified away from a finite set of places Sigma superset of {p vertical bar p}, consider the Bloch-Kato Selmer group H := H-Sigma(1)(F, chi(-1)). The authors previously proved that the number d of isomorphism classes of (nonsemisimple, reducible) residual representations (rho) over bar giving rise to lines in H which are modular by some rho(f) (also unramified outside Sigma) satisfies d >= n := dim(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 nonprincipal, 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 nonprincipality of J that can be checked in practice, allowing us to furnish examples where d > n.