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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| Materias: | |
| Acceso en línea: | https://arxiv.org/abs/2411.08259 |
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| _version_ | 1866910695919976448 |
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| author | Huryn, Jake Zhang, Yifei |
| author_facet | Huryn, Jake Zhang, Yifei |
| contents | Let $X$ be a connected normal scheme of finite type over $\mathbf{Z}$, let $G$ be a connected reductive group over $\mathbf{Q}$, and let $\{ρ_\ell\colonπ_1(X[1/\ell])\to G(\mathbf{Q}_\ell)\}_\ell$ be a Frobenius-compatible collection of continuous homomorphisms indexed by the primes. Assume $\mathrm{Img}(ρ_\ell)$ is Zariski-dense in $G_{\mathbf{Q}_\ell}$ for all $\ell$ in a nonempty finite set $\mathcal{R}$. We prove that, under certain hypotheses on $\mathcal{R}$ (depending only on $G$), $\mathrm{Img}(ρ_\ell)$ is Zariski-dense in $G_{\mathbf{Q}_\ell}$ for all $\ell$ in a set of Dirichlet density $1$. As an application, we combine this result with a version of Hilbert's irreducibility theorem and recent work of Klevdal--Patrikis to obtain new information about the "canonical" local systems attached to Shimura varieties not of Abelian type. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_08259 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Transport of Zariski density in compatible collections of $G$-representations Huryn, Jake Zhang, Yifei Number Theory 11F80, 11G18 Let $X$ be a connected normal scheme of finite type over $\mathbf{Z}$, let $G$ be a connected reductive group over $\mathbf{Q}$, and let $\{ρ_\ell\colonπ_1(X[1/\ell])\to G(\mathbf{Q}_\ell)\}_\ell$ be a Frobenius-compatible collection of continuous homomorphisms indexed by the primes. Assume $\mathrm{Img}(ρ_\ell)$ is Zariski-dense in $G_{\mathbf{Q}_\ell}$ for all $\ell$ in a nonempty finite set $\mathcal{R}$. We prove that, under certain hypotheses on $\mathcal{R}$ (depending only on $G$), $\mathrm{Img}(ρ_\ell)$ is Zariski-dense in $G_{\mathbf{Q}_\ell}$ for all $\ell$ in a set of Dirichlet density $1$. As an application, we combine this result with a version of Hilbert's irreducibility theorem and recent work of Klevdal--Patrikis to obtain new information about the "canonical" local systems attached to Shimura varieties not of Abelian type. |
| title | Transport of Zariski density in compatible collections of $G$-representations |
| topic | Number Theory 11F80, 11G18 |
| url | https://arxiv.org/abs/2411.08259 |