  We say a mirror pair of Calabi-Yau varieties exhibits strong arithmetic
mirror symmetry if the number of points on each variety over a finite field is
equivalent, modulo the order of that field. We search for strong mirror
symmetry in pencils of toric hypersurfaces generated using polar dual pairs of
reflexive polytopes. We characterize the pencils of elliptic curves where
strong arithmetic mirror symmetry arises, and provide experimental evidence
that the phenomenon generalizes to higher dimensions. We also provide
experimental evidence that pencils of K3 surfaces with the same Picard-Fuchs
equation have related point counts.
