The Number of Rational Points of Two Parameter Calabi-Yau manifolds as Toric Hypersurfaces

The number of rational points in toric data are given for two-parameter Calabi-Yau $n$-folds as toric hypersurfaces over finite fields ${\mathbb{F}}_p$ . We find that the fundamental period is equal to the number of rational points of the Calabi-Yau $n$-folds in zeroth order $p$-adic expansion. By analyzing the solution set of the GKZ-system given by the enhanced polyhedron, we deduce that under type II/F-theory duality the 3D and 4D Calabi-Yau manifolds have the same number of rational points in zeroth order. Taking the quintic and its duality as an example, the number of rational points in some specific complex moduli are given by numerical calculation to support our results.