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 $H:=H^1_{\Sigma }(F, \chi ^{-1})$. The authors previously proved that the number $d$ of isomorphism classes of (nonsemisimple, 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 nonprincipal, 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 nonprincipality of $J$ that can be checked in practice, allowing us to furnish examples where $d>n$.