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

Abstract

Let $\rho :G\to {\mathrm{GL}}_2(K)$ 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 $\mathrm{tr}\rho$ is reducible modulo ${\pi }^n$ if and only if $\rho$ is reducible modulo ${\pi }^n$. More precisely, there exist characters ${\chi }_1,{\chi }_2:G\to (\mathcal{O}/{\pi }^n\mathcal{O}{)}^{\times }$ such that $\det (t-\rho (g))\equiv (t-{\chi }_1(g))(t-{\chi }_2(g))(\mathrm{mod}{\pi }^n)$ 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 Bellaïche--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 ${\chi }_1$

21 pages. Revised following referee comments. To appear in Research in the Mathematical Sciences

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• A Local-global principle for isogenies of composite degree