On Constructing Extensions of Residually Isomorphic Characters

This is an exposition of our joint work with Kakde, Silliman, and Wang, in which we prove a version of Ribet's Lemma for $\mathrm{GL}_2$ in the residually indistinguishable case. We suppose we are given a Galois representation taking values in the total ring of fractions of a complete reduced Noetherian local ring $\mathbf{T}$, such that the characteristic polynomial of the representation is reducible modulo some ideal $I \subset \mathbf{T}$. We assume that the two characters that arise are congruent modulo the maximal ideal of $\mathbf{T}$. We construct an associated Galois cohomology class valued in a $\mathbf{T}$-module that is "large" in the sense that its Fitting ideal is contained in $I$. We make some simplifying assumptions that streamline the exposition -- we assume the two characters are actually equal, and we ignore the local conditions needed in arithmetic applications.