Calabi-Yau Manifolds Over Finite Fields, II

TL;DR

This paper investigates the zeta-functions of a family of quintic threefolds over finite fields and their mirrors, revealing structures related to genus 4 curves and connections to mirror symmetry and arithmetic properties.

Contribution

It uncovers the structure of zeta-functions for quintic threefolds and their mirrors, highlighting the influence of mirror symmetry and arithmetic phenomena in finite field contexts.

Findings

Zeta-functions involve genus 4 Riemann curves.

Mirror symmetry affects the form of zeta-functions.

Zeta-functions exhibit an arithmetic analogue of the large complex structure limit.

Abstract

We study zeta-functions for a one parameter family of quintic threefolds defined over finite fields and for their mirror manifolds and comment on their structure. The zeta-function for the quintic family involves factors that correspond to a certain pair of genus 4 Riemann curves. The appearance of these factors is intriguing since we have been unable to `see' these curves in the geometry of the quintic. Having these zeta-functions to hand we are led to comment on their form in the light of mirror symmetry. That some residue of mirror symmetry survives into the zeta-functions is suggested by an application of the Weil conjectures to Calabi-Yau threefolds: the zeta-functions are rational functions and the degrees of the numerators and denominators are exchanged between the zeta-functions for the manifold and its mirror. It is clear nevertheless that the zeta-function, as classically defined, makes an essential distinction between Kahler parameters and the coefficients of the defining polynomial. It is an interesting question whether there is a `quantum modification' of the zeta-function that restores the symmetry between the Kahler and complex structure parameters. We note that the zeta-function seems to manifest an arithmetic analogue of the large complex structure limit which involves 5-adic expansion.