Lifting Galois representations via Kummer flags
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866917299336773632 |
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| author | Conti, Andrea Demarche, Cyril Florence, Mathieu |
| author_facet | Conti, Andrea Demarche, Cyril Florence, Mathieu |
| contents | Let $Γ$ be either i) the absolute Galois group of a local field $F$, or ii) the topological fundamental group of a closed connected orientable surface of genus $g$. In case i), assume that $μ_{p^2} \subset F$. We give an elementary and unified proof that every representation $ρ_1: Γ\to \mathbf{GL}_d(\mathbb{F}_p)$ lifts to a representation $ρ_2: Γ\to \mathbf{GL}_d(\mathbb{Z}/p^2)$. [In case i), it is understood these are continuous.] The actual statement is much stronger: for all $r \geq 1$, under "suitable" assumptions, triangular representations $ρ_r: Γ\to \mathbf{B}_d(\mathbb{Z}/p^r)$ lift to $ρ_{r+1}: Γ\to \mathbf{B}_d(\mathbb{Z}/p^{r+1})$, in the strongest possible step-by-step sense. Here "suitable" is made precise by the concept of $\textit{Kummer flag}$. An essential aspect of this work is to identify the common properties of groups i) and ii) that suffice to ensure the existence of such lifts. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_08888 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Lifting Galois representations via Kummer flags Conti, Andrea Demarche, Cyril Florence, Mathieu Number Theory Algebraic Topology 11F80 Let $Γ$ be either i) the absolute Galois group of a local field $F$, or ii) the topological fundamental group of a closed connected orientable surface of genus $g$. In case i), assume that $μ_{p^2} \subset F$. We give an elementary and unified proof that every representation $ρ_1: Γ\to \mathbf{GL}_d(\mathbb{F}_p)$ lifts to a representation $ρ_2: Γ\to \mathbf{GL}_d(\mathbb{Z}/p^2)$. [In case i), it is understood these are continuous.] The actual statement is much stronger: for all $r \geq 1$, under "suitable" assumptions, triangular representations $ρ_r: Γ\to \mathbf{B}_d(\mathbb{Z}/p^r)$ lift to $ρ_{r+1}: Γ\to \mathbf{B}_d(\mathbb{Z}/p^{r+1})$, in the strongest possible step-by-step sense. Here "suitable" is made precise by the concept of $\textit{Kummer flag}$. An essential aspect of this work is to identify the common properties of groups i) and ii) that suffice to ensure the existence of such lifts. |
| title | Lifting Galois representations via Kummer flags |
| topic | Number Theory Algebraic Topology 11F80 |
| url | https://arxiv.org/abs/2403.08888 |