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

TL;DR

This paper investigates the count of rational points on two-parameter Calabi-Yau hypersurfaces over finite fields, revealing a connection between these counts and fundamental periods, with implications under dualities like F-theory.

Contribution

It establishes a link between rational point counts and fundamental periods for Calabi-Yau hypersurfaces, and explores duality invariances in these counts.

Findings

Fundamental period equals the number of rational points in zeroth order p-adic expansion.

Under dualities, 3D and 4D Calabi-Yau manifolds share the same rational point counts.

Numerical examples support the theoretical results.

Abstract

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.