The Residually Indistinguishable Case of Ribet's Method for GL2

Ribet’s method provides a strategy for constructing a nontrivial extension of a $p$-adic Galois representation ${\rho }_1$ by another such representation ${\rho }_2$. Suppose we are working over a local ring. An important assumption that occurs throughout literature is that the representations ${\rho }_i$ are residually distinguishable i.e. are residually non-isomorphic. The main theorem of this paper is a general version of Ribet’s Lemma for ${\mathrm{G}\mathrm{L}}_2$ where we do not impose the assumption that the associated characters are residually distinguished.