2016/10/04 by Christopher Magyar, Ursula Whitcher, Magyar, Christopher +1 · 1 citation
Mathematics · #11G42 #14H52 #14J28 #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT) #math.AG #math.NT #msc:11G42 #msc:14H52 #msc:14J28
paper · pdf · doi:10.48550/arxiv.1610.01011
13 pages, 3 figures. Comments welcome
arxiv created 2016/10/04 · arxiv updated 2016/10/05
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.