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