Lifting residual Galois representations with the same semi-simplification
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918059323686912 |
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| author | Nikoloski, Stefan |
| author_facet | Nikoloski, Stefan |
| contents | If a $p$-adic Galois representation $ρ_{f,ν}:Γ_{\mathbb Q} \to \GL_2(E_{f,ν})$ attached to some eigenform $f$ is residually reducible it will have 2 non-isomorphic reductions, which have the same semi-simplification. In this paper, we answer a version of the inverse question, first brought up by Toby Gee and Alice Pozzi. Starting with two modulo $p$ non-semi-simple representations $\overline{ρ_1},\overline{ρ_2}:Γ_{\mathbb Q} \to \GL_2(k)$, which have the same semi-simplification we show that under some mild conditions that they are reductions of representations attached to newforms of the same weight $r \ge 2$, the same level $N \ge 1$, and the same Neben character $\varepsilon$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_12644 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Lifting residual Galois representations with the same semi-simplification Nikoloski, Stefan Number Theory 11F80, 11R39 If a $p$-adic Galois representation $ρ_{f,ν}:Γ_{\mathbb Q} \to \GL_2(E_{f,ν})$ attached to some eigenform $f$ is residually reducible it will have 2 non-isomorphic reductions, which have the same semi-simplification. In this paper, we answer a version of the inverse question, first brought up by Toby Gee and Alice Pozzi. Starting with two modulo $p$ non-semi-simple representations $\overline{ρ_1},\overline{ρ_2}:Γ_{\mathbb Q} \to \GL_2(k)$, which have the same semi-simplification we show that under some mild conditions that they are reductions of representations attached to newforms of the same weight $r \ge 2$, the same level $N \ge 1$, and the same Neben character $\varepsilon$. |
| title | Lifting residual Galois representations with the same semi-simplification |
| topic | Number Theory 11F80, 11R39 |
| url | https://arxiv.org/abs/2506.12644 |