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arXiv · 2511.00431

Bornes de torsion et un th\'eor\`eme effectif du pgcd

Hyuk Jun Kweon·Madhavan Venkatesh
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Abstract

We prove an effective, probabilistic version of Deligne's `th\'eor\`eme 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 $\ell$.

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Hyuk Jun Kweon, Madhavan Venkatesh. 2025-11-01. Bornes de torsion et un th\'eor\`eme effectif du pgcd. https://arxiv.org/abs/2511.00431

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