  We prove that if two Calabi-Yau invertible pencils have the same dual
weights, then they share a common factor in their zeta functions. By using
Dwork cohomology, we demonstrate that this common factor is related to a
hypergeometric Picard--Fuchs differential equation. The factor in the zeta
function is defined over the rationals and has degree at least the order of the
Picard--Fuchs equation. As an application, we relate several pencils of K3
surfaces to the Dwork pencil, obtaining new cases of arithmetic mirror
symmetry.
