Modularity of residual Galois extensions and the Eisenstein ideal

Tobias Berger, Krzysztof Klosin

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…