On Ribet's Lemma for $\mathrm{GL}_2$ modulo prime powers

Abstract

Let $\rho\colon G\to \mathrm{GL}_2(K)$ be a continuous representation of a compact group $G$ over a complete discretely valued field $K$, with ring of integers $\mathcal O$ and uniformiser $\pi$. We prove that $\operatorname{tr}\rho$ is reducible modulo $\pi^n$ if and only if $\rho$ is reducible modulo $\pi^n$. More precisely, there exist characters $\chi_1,\chi_2 \colon G\to(\mathcal O/\pi^n\mathcal O)^{\times}$ such that $\det(t - \rho(g))\equiv (t-\chi_1(g))(t-\chi_2(g))\pmod{\pi^n}$ for all $g\in G$, if and only if there exists a $G$-stable lattice $\Lambda\subset K^2$ such that $\Lambda/\pi^n\Lambda$ contains a $G$-invariant, free, rank one $\mathcal O/\pi^n\mathcal O$-submodule. Our result applies in the case that $\rho$ is not residually multiplicity free, in which case it answers a question of Bella\"iche--Chenevier. As an application, we prove an optimal version of Ribet's Lemma, which gives a condition for the existence of a $G$-stable lattice $\Lambda$ that realises a non-split extension of $\chi_2$ by $\chi_1$