Calabi-Yau manifolds over finite fields, II

Abstract

We study ζ-functions for a one parameter family of quintic threefolds defined over finite fields and for their mirror manifolds and comment on their structure. The ζ-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 ζ-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 ζ-functions is suggested by an application of the Weil conjectures to Calabi-Yau threefolds: the ζ-functions are rational functions and the degrees of the numerators and denominators are exchanged between the ζ-functions for the manifold and its mirror. It is clear nevertheless that the ζ-function, as classically defined, makes an essential distinction between Kähler parameters and the coefficients of the defining polynomial. It is an interesting question whether there is a 'quantum modification' of the ζ-function that restores the symmetry between the Kähler and complex structure parameters. We note that the ζ-function seems to manifest an arithmetic analogue of the large complex structure limit which involves 5-adic expansion.

Mini-Workshop: Geometry and Duality in String Theory

“…In [CdOV1] it was shown that the number of rational points of the quintic Calabi-Yau manifolds over F p can be given in terms of the periods; our calculations verify this relation for the octic. The periods satisfy a system of differential equations known as the Picard-Fuchs equations, with respect to the parameters.…”

“…In order to study these questions, we shall be considering families with up to two parameters, and use methods very similar to [CdOV1,CdOV2]. In particular, we study a one parameter family of K3 surfaces and a two parameter family of CalabiYau threefolds, octic hypersurfaces in weighted projective space P 4 (1,1,2,2,2) [8], the mirror symmetry of which was studied in detail in [CdOFKM].…”

“…We shall find the same phenomenon for the octic where there is a sextic, R (0,0,0,0,0) , that appears in both the original family and the mirror. For the octic, on the mirror side, there was also a contribution related to a zero-dimensional Calabi-Yau manifold(studied in [CdOV1]) which was sensitive to a particular type of (non-conifold) singularity. This contrasts with the the fact that zeta function of the original family had a contribution related to a particular monomial (not on the polyhedron) that is sensitive to the presence of conifold points.…”

The Langlands program and string modular K3 surfaces

“…It is therefore of interest to consider the behavior of families of varieties and analyze their arithmetic structure as their moduli change. This line of thought has already been followed in recent work [34,35,36]. These interesting papers address, among other issues, the question of what happens to the reduced variety at the conifold locus.…”

Computing Zeta Functions via p-Adic Cohomology

“…For instance, Candelas et al [6] have studied the zeta functions of some Calabi-Yau threefolds occurring as toric complete intersections, motivated by considerations of mirror symmetry.…”