Bornes de torsion et un théorème effectif du pgcd

2025/11/01 by Kweon, Hyuk Jun, Venkatesh, Madhavan · 1 citation
#Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2511.00431

Abstract

We prove an effective, probabilistic version of Deligne's `théorème du pgcd' for a smooth, projective, geometrically integral (nice) variety X₀⊂ ℙ^N over \mathbbF_q of dimension n and degree D, obtained via good reduction from a nice variety X₀ over a number field K at a prime \mathfrakp⊂ O_K. The main ingredients include bounding torsion in the Betti cohomology of X₀, a mod -- ℓ big monodromy result and equidistribution of Frobenius in the representation associated to the sheaf of vanishing cycles modulo ℓ.

Citations

• Counting points on surfaces in polynomial time
• Complexity of counting points on curves and the factor P₁(T) of the zeta function of surfaces
• Bounds on the Torsion Subgroups of Néron-Severi Groups
• Improved explicit estimates on the number of solutions of equations over a finite field
• Torsion cohomology classes and algebraic cycles on complex projective manifolds

Cited by

• Counting points on surfaces in polynomial time

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