Strong arithmetic mirror symmetry and toric isogenies

TL;DR

This paper investigates strong arithmetic mirror symmetry in Calabi-Yau varieties, especially toric hypersurfaces, showing that certain pairs exhibit point count congruences over finite fields, with evidence extending to higher dimensions.

Contribution

It characterizes when strong arithmetic mirror symmetry occurs in toric hypersurface pencils and provides experimental evidence of its generalization beyond elliptic curves.

Findings

Strong arithmetic mirror symmetry occurs in specific elliptic curve pencils.

Experimental evidence suggests the phenomenon extends to higher-dimensional Calabi-Yau varieties.

Pencils of K3 surfaces with the same Picard-Fuchs equation have related point counts.

Abstract

We say a mirror pair of Calabi-Yau varieties exhibits strong arithmetic mirror symmetry if the number of points on each variety over a finite field is equivalent, modulo the order of that field. We search for strong mirror symmetry in pencils of toric hypersurfaces generated using polar dual pairs of reflexive polytopes. We characterize the pencils of elliptic curves where strong arithmetic mirror symmetry arises, and provide experimental evidence that the phenomenon generalizes to higher dimensions. We also provide experimental evidence that pencils of K3 surfaces with the same Picard-Fuchs equation have related point counts.