Comparing Galois representations in the residually reducible case
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2025
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| _version_ | 1866917015125491712 |
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| author | Freitas, Nuno Sánchez-Rodríguez, Ignasi |
| author_facet | Freitas, Nuno Sánchez-Rodríguez, Ignasi |
| contents | Let $n \geq 2$ and $p$ be a prime. Let $K$ be a number field and consider two Galois representations $ρ_1, ρ_2 : \operatorname{Gal}(\overline{K} / K) \to \operatorname{GL}_n(\mathbb{Z}_p)$ having residual image a $p$-group. We explain and implement an algorithm that makes effective a result of Loïc Grenié to decide wether the semisimplifications of $ρ_1$ and $ρ_2$ are isomorphic.
As an application, we show that an irreducible representation $ρ: G_{\mathbb{Q}(\sqrt{-3})} \to \operatorname{GL}_2(\mathbb{Z}_3)$ unramified outside 3 is determined by the characteristic polynomials of Frobenius elements at five primes of small norm. As an additional check, we apply it to a 2-adic example studied by Grenié, recovering Grenié's result in a fully automated way. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_12956 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Comparing Galois representations in the residually reducible case Freitas, Nuno Sánchez-Rodríguez, Ignasi Number Theory 11F80, 11Y40 (Primary) 11G05 (Secondary) Let $n \geq 2$ and $p$ be a prime. Let $K$ be a number field and consider two Galois representations $ρ_1, ρ_2 : \operatorname{Gal}(\overline{K} / K) \to \operatorname{GL}_n(\mathbb{Z}_p)$ having residual image a $p$-group. We explain and implement an algorithm that makes effective a result of Loïc Grenié to decide wether the semisimplifications of $ρ_1$ and $ρ_2$ are isomorphic. As an application, we show that an irreducible representation $ρ: G_{\mathbb{Q}(\sqrt{-3})} \to \operatorname{GL}_2(\mathbb{Z}_3)$ unramified outside 3 is determined by the characteristic polynomials of Frobenius elements at five primes of small norm. As an additional check, we apply it to a 2-adic example studied by Grenié, recovering Grenié's result in a fully automated way. |
| title | Comparing Galois representations in the residually reducible case |
| topic | Number Theory 11F80, 11Y40 (Primary) 11G05 (Secondary) |
| url | https://arxiv.org/abs/2510.12956 |