Divisibility of Frobenius eigenvalues on $\ell$-adic cohomology

TL;DR

This paper proves new divisibility properties of Frobenius eigenvalues on $\, ext{l}$-adic cohomology of varieties over finite fields, sharpening existing theorems and extending results to complex varieties and the affine case.

Contribution

It establishes stronger divisibility bounds for Frobenius eigenvalues and Hodge levels beyond the middle dimension, improving upon prior theorems.

Findings

Eigenvalues of Frobenius have higher than known divisibility by q beyond the middle dimension.

New lower bounds for Hodge levels are established for complex varieties.

Results extend to the affine case, refining previous theorems.

Abstract

v2: For a projective variety defined over a finite field with $q$ elements, it is shown that as algebraic integers, the eigenvalues of the geometric Frobenius acting on $\ell$-adic cohomology have higher than known $q$-divisibility beyond the middle dimension. This sharpens both Deligne's integrality theorem and the cohomological divisibility theorem proven by the first author and N. Katz. Similar lower bounds are proved for the Hodge level for a complex variety beyond the middle dimension, improving earlier results in this direction. We discuss the affine case. The previous version contained a gap at this place. We are thankful to Dingxin Zhang for noticing it.