Revisiting Dwork cohomology: Visibility and divisibility of Frobenius eigenvalues in rigid cohomology

TL;DR

This paper investigates Frobenius eigenvalues in rigid cohomology over finite fields, revealing new divisibility bounds and showing that zeta functions of related affine varieties can detect all eigenvalues, surpassing known $\, ext{l}$-adic results.

Contribution

It introduces novel divisibility bounds for Frobenius eigenvalues in rigid cohomology and demonstrates that zeta functions of related affine varieties can capture all eigenvalues, a result not known for $\, ext{l}$-adic cohomology.

Findings

Zeta functions of related affine varieties witness all Frobenius eigenvalues.

Established lower bounds for eigenvalue divisibility based on degrees of defining equations.

Divisibility bounds in rigid cohomology outperform those in $\, ext{l}$-adic cohomology.

Abstract

We study Frobenius eigenvalues of the compactly supported rigid cohomology of a variety defined over a finite field of $q$ elements via Dwork's method. A couple of arithmetic consequences will be drawn from this study. As the first application, we show that the zeta functions for finitely many related affine varieties are capable of witnessing all Frobenius eigenvalues of the rigid cohomology of the variety up to Tate twist. This result does not seem to be known for $\ell$-adic cohomology. As the second application, we prove several $q$-divisibility lower bounds for Frobenius eigenvalues of the rigid cohomology of the variety in terms of the multi-degrees of the defining equations. These divisibility bounds for rigid cohomology are generally better than what is suggested from the best known divisibility bounds in $\ell$-adic cohomology, both before and after the middle cohomological dimension.