On Ribet's Lemma for $\mathrm{GL}_2$ modulo prime powers
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2021
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| _version_ | 1866914680372461568 |
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| author | Ophir, Amit Weiss, Ariel |
| author_facet | Ophir, Amit Weiss, Ariel |
| contents | Let $ρ\colon G\to \mathrm{GL}_2(K)$ be a continuous representation of a compact group $G$ over a complete discretely valued field $K$, with ring of integers $\mathcal O$ and uniformiser $π$. We prove that $\operatorname{tr}ρ$ is reducible modulo $π^n$ if and only if $ρ$ is reducible modulo $π^n$. More precisely, there exist characters $χ_1,χ_2 \colon G\to(\mathcal O/π^n\mathcal O)^{\times}$ such that $\det(t - ρ(g))\equiv (t-χ_1(g))(t-χ_2(g))\pmod{π^n}$ for all $g\in G$, if and only if there exists a $G$-stable lattice $Λ\subset K^2$ such that $Λ/π^nΛ$ contains a $G$-invariant, free, rank one $\mathcal O/π^n\mathcal O$-submodule. Our result applies in the case that $ρ$ 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 $Λ$ that realises a non-split extension of $χ_2$ by $χ_1$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2111_01559 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | On Ribet's Lemma for $\mathrm{GL}_2$ modulo prime powers Ophir, Amit Weiss, Ariel Number Theory Representation Theory 20G25 (20E08, 20C11, 11S23) Let $ρ\colon G\to \mathrm{GL}_2(K)$ be a continuous representation of a compact group $G$ over a complete discretely valued field $K$, with ring of integers $\mathcal O$ and uniformiser $π$. We prove that $\operatorname{tr}ρ$ is reducible modulo $π^n$ if and only if $ρ$ is reducible modulo $π^n$. More precisely, there exist characters $χ_1,χ_2 \colon G\to(\mathcal O/π^n\mathcal O)^{\times}$ such that $\det(t - ρ(g))\equiv (t-χ_1(g))(t-χ_2(g))\pmod{π^n}$ for all $g\in G$, if and only if there exists a $G$-stable lattice $Λ\subset K^2$ such that $Λ/π^nΛ$ contains a $G$-invariant, free, rank one $\mathcal O/π^n\mathcal O$-submodule. Our result applies in the case that $ρ$ 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 $Λ$ that realises a non-split extension of $χ_2$ by $χ_1$ |
| title | On Ribet's Lemma for $\mathrm{GL}_2$ modulo prime powers |
| topic | Number Theory Representation Theory 20G25 (20E08, 20C11, 11S23) |
| url | https://arxiv.org/abs/2111.01559 |