2006/06/28 by Shabnam Kadir, Kadir, Shabnam, Noriko Yui +1
Mathematics · #11F80 #11G40 #14G10 #14J32 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometry and complex manifolds #Number Theory (math.NT) #math.AG #math.NT #msc:11F80 #msc:11G40 #msc:14G10 #msc:14J32
paper · pdf · doi:10.48550/arxiv.math/0606707
48 pages
arxiv created 2006/06/28 · openalex publication_date 2006/06/28 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider certain families of Calabi-Yau orbifolds and their mirror partners constructed from Fermat hypersurfaces in weighted projective 4-spaces. Our focus is the topological mirror symmetry. There are at least three known ingredients to describe the topological mirror symmetry, namely, integral vertices in reflexive polytopes, monomials in graded polynomial rings (with some group actions), and periods (and Picard-Fuchs differential equations). In this paper we will introduce Fermat motives associated to these Calabi-Yau orbifolds and then use them to give motivic interpretation of the topological mirror symmetry phenomenon between mirror pairs of Calabi-Yau orbifolds. We establish, at the Fermat (the Landau-Ginzburg) point in the moduli space, the one-to-one correspondence between the monomial classes and Fermat motives. This is done by computing the number of \bf Fq-rational points on our Calabi-Yau orbifolds over \bf Fq in two different ways: Weil's algebraic number theoretic method involving Jacobi (Gauss) sums, and Dwork's p-adic analytic method involving Dwork characters and Gauss sums. We will discuss specific examples in detail.