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- W3210183045 abstract "Let $rhocolon Gto mathrm{GL}_2(K)$ be a continuous representation of a compact group $G$ over a complete discretely valued field $K$ with residue characteristic not equal to $2$, 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 $chi_1,chi_2 colon Gto(mathcal O/pi^nmathcal O)^{times}$ with $operatorname{tr}rho equiv chi_1+chi_2pmod{pi^n}$ if and only if there exists a $G$-stable lattice $Lambdasubset K^2$ such that $Lambda/pi^nLambda$ contains a $G$-invariant, free, rank $1$ $mathcal O/pi^nmathcal O$-submodule. This answers a question of Bellaiche--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 $chi_1$." @default.
- W3210183045 created "2021-11-08" @default.
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- W3210183045 date "2021-11-02" @default.
- W3210183045 modified "2023-09-27" @default.
- W3210183045 title "On Ribet's Lemma for $mathrm{GL}_2$ modulo prime powers" @default.
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