Bornes de torsion et un théorème effectif du pgcd
Hyuk Jun Kweon, Madhavan Venkatesh
TL;DR
This work proves an effective, probabilistic version of Deligne's gcd theorem for the middle cohomology polynomial P_{n-1}(X/ F_q, T) of smooth, projective varieties X over F_q arising from good reduction. The authors combine explicit torsion bounds on Betti cohomology, mod-ℓ big monodromy for vanishing cycles, and Katz-type equidistribution to control Frobenius actions, together with algebraic-group estimates for coprimality probabilities. They establish that for sufficiently large extension degrees, the global middle-cohomology polynomial equals the gcd of two random fibre polynomials with probability exceeding 2/3, and that the global polynomial can be recovered from two longer extensions via Kedlaya-style cyclic resultants. Algorithmically, this yields a polynomial-time reduction of zeta-function computation for nice varieties to computations on middle cohomology, enabling efficient arithmetic-geometric zeta evaluations and providing a unified treatment of symplectic and orthogonal monodromy cases.
Abstract
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 $\ell$.