Cohomological divisibility and point count divisibility

2003/05/08 by Hélène Esnault, Esnault, Hélène, Nicholas M. Katz +1
Mathematics · #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Finite Group Theory Research #Number Theory (math.NT) #math.AG #math.NT

paper · pdf · doi:10.48550/arxiv.math/0305127

10 pages, latex

openalex publication_date 2003/05/08 · arxiv created 2003/05/10 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show that Ax-Katz divisibility on the number of rational points of a variety defined over a finite field by equations of low degrees comes from divisibility of the eigenvalues of the Frobenius action of the ℓ-adic cohomology with compact supports. This completes the motivc picture, as we knew already that in characteristic 0, the Hodge type was the expected one. [version 2 as the title contained a typo..., else no changes]

Citations

• La conjecture de Weil : I
• Decomposition of the diagonal and eigenvalues of Frobenius for Fano hypersurfaces

Related

• Divisibility of Frobenius eigenvalues on ℓ-adic cohomology
• Eigenvalues of Frobenius acting on the ℓ-adic cohomology of complete intersections of low degree
• On a theorem of Ax and Katz
• Revisiting Dwork cohomology: Visibility and divisibility of Frobenius eigenvalues in rigid cohomology
• Appendix to "Congruences for rational points on varieties over finite fields" by N. Fakhruddin and C. S. Rajan