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

TL;DR

This paper establishes an effective probabilistic version of Deligne's gcd theorem for certain algebraic varieties over finite fields, using advanced cohomological and monodromy techniques.

Contribution

It introduces an effective, probabilistic approach to Deligne's gcd theorem for varieties over finite fields, leveraging bounds on torsion, big monodromy, and Frobenius equidistribution.

Findings

Bounded torsion in Betti cohomology.

Proved big monodromy result.

Established Frobenius equidistribution.

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 $\ell$.