Comparison of component groups of $\ell$-adic and mod $\ell$ monodromy groups
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866918426804486144 |
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| author | Dai, Boyi Hui, Chun Yin |
| author_facet | Dai, Boyi Hui, Chun Yin |
| contents | Let $\{ρ_{\ell}:\mathrm{Gal}_K\to\mathrm{GL}_n(\mathbb{Q}_{\ell})\}_{\ell}$ be a semisimple compatible system of $\ell$-adic representations of a number field $K$ that is arising from geometry. Let $\textbf{G}_{\ell}\subset\mathrm{GL}_{n,\mathbb{Q}_{\ell}}$ and $\widehat{\underline{G_{\ell}}}\subset\mathrm{GL}_{n,\mathbb{F}_\ell}$ be respectively the algebraic monodromy group and full algebraic envelope of $ρ_{\ell}$. We prove that there is a natural isomorphism between the component groups $π_0(\textbf{G}_{\ell}) \simeq π_0(\widehat{\underline{G_\ell}})$ for all sufficiently large $\ell$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_20907 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Comparison of component groups of $\ell$-adic and mod $\ell$ monodromy groups Dai, Boyi Hui, Chun Yin Number Theory Let $\{ρ_{\ell}:\mathrm{Gal}_K\to\mathrm{GL}_n(\mathbb{Q}_{\ell})\}_{\ell}$ be a semisimple compatible system of $\ell$-adic representations of a number field $K$ that is arising from geometry. Let $\textbf{G}_{\ell}\subset\mathrm{GL}_{n,\mathbb{Q}_{\ell}}$ and $\widehat{\underline{G_{\ell}}}\subset\mathrm{GL}_{n,\mathbb{F}_\ell}$ be respectively the algebraic monodromy group and full algebraic envelope of $ρ_{\ell}$. We prove that there is a natural isomorphism between the component groups $π_0(\textbf{G}_{\ell}) \simeq π_0(\widehat{\underline{G_\ell}})$ for all sufficiently large $\ell$. |
| title | Comparison of component groups of $\ell$-adic and mod $\ell$ monodromy groups |
| topic | Number Theory |
| url | https://arxiv.org/abs/2407.20907 |