Bornes de torsion et un théorème effectif du pgcd

We prove an effective, probabilistic version of Deligne’s `théorème du pgcd’ for a smooth, projective, geometrically integral ($\mathit{\text{nice}}$) 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 – $\mathrm{\ell }$ big monodromy result and equidistribution of Frobenius in the representation associated to the sheaf of vanishing cycles modulo $\mathrm{\ell }$.