On Ribet’s lemma for GL2 modulo prime powers

Let ρ:G→GL₂(K) be a continuous representation of a compact group G over a complete discretely valued field K with ring of integers O and uniformiser π. We prove that trρ is reducible modulo πⁿ if and only if ρ is reducible modulo πⁿ. More precisely, there exist characters χ₁,χ₂:G→(O/πnO)^× such that det(t-ρ(g))≡(t-χ₁(g))(t-χ₂(g))(modπⁿ) for all g∈G, if and only if there exists a G-stable lattice Λ⊆K² such that Λ/πⁿΛ contains a G-invariant, free, rank one O/πⁿO-submodule. Our result applies in the case that ρ is not residually multiplicity-free, in which case it answers a question of Bellaïche and Chenevier (J Algebra 410:501–525, 2014, pp. 524). As an application, we prove an optimal version of Ribet’s lemma, which gives a condition for the existence of a G-stable lattice Λ that realises a non-split extension of χ₂ by χ₁.