Zeta Functions of Projective Toric Hypersurfaces over Finite Fields

TL;DR

This paper derives a formula for the zeta function of projective toric hypersurfaces over finite fields and uses it to compute rational points on Calabi-Yau manifolds relevant to Mirror Symmetry.

Contribution

It provides a new explicit formula for zeta functions of toric hypersurfaces and applies it to count rational points on Calabi-Yau families.

Findings

Formula for zeta function of projective toric hypersurfaces

Estimate of the Newton polygon for these zeta functions

Exact count of rational points on certain Calabi-Yau manifolds

Abstract

I give a formula for the zeta function of a projective toric hypersurface over a finite field and estimate its Newton polygon. As an application this formula allows us to compute the exact number of rational points on the families of Calabi-Yau manifolds in Mirror Symmetry.