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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2511.00431 |
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| _version_ | 1866914129773592576 |
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| author | Kweon, Hyuk Jun Venkatesh, Madhavan |
| author_facet | Kweon, Hyuk Jun Venkatesh, Madhavan |
| contents | We prove an effective, probabilistic version of Deligne's `théorème du pgcd' for a smooth, projective, geometrically integral (\textit{nice}) variety $X_{0}\subset \mathbb{P}^{N}$ over $\mathbb{F}_{q}$ of dimension $n$ and degree $D$, obtained via good reduction from a nice variety $\mathcal{X}_{0}$ over a number field $K$ at a prime $\mathfrak{p}\subset \mathcal{O}_{K}$. The main ingredients include bounding torsion in the Betti cohomology of $\mathcal{X}_{0}$, a mod -- $\ell$ big monodromy result and equidistribution of Frobenius in the representation associated to the sheaf of vanishing cycles modulo $\ell$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_00431 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Bornes de torsion et un théorème effectif du pgcd Kweon, Hyuk Jun Venkatesh, Madhavan Algebraic Geometry Number Theory We prove an effective, probabilistic version of Deligne's `théorème du pgcd' for a smooth, projective, geometrically integral (\textit{nice}) variety $X_{0}\subset \mathbb{P}^{N}$ over $\mathbb{F}_{q}$ of dimension $n$ and degree $D$, obtained via good reduction from a nice variety $\mathcal{X}_{0}$ over a number field $K$ at a prime $\mathfrak{p}\subset \mathcal{O}_{K}$. The main ingredients include bounding torsion in the Betti cohomology of $\mathcal{X}_{0}$, a mod -- $\ell$ big monodromy result and equidistribution of Frobenius in the representation associated to the sheaf of vanishing cycles modulo $\ell$. |
| title | Bornes de torsion et un théorème effectif du pgcd |
| topic | Algebraic Geometry Number Theory |
| url | https://arxiv.org/abs/2511.00431 |