Hasse–Witt matrices and mirror toric pencils

Mirror symmetry suggests unexpected relationships between arithmetic properties of distinct families of algebraic varieties. For example, Wan and others have shown that for some mirror pairs, the number of rational points over a finite field matches modulo the order of the field. In this paper, we obtain a similar result for certain mirror pairs of toric hypersurfaces. We use recent results describing the relationship between the Picard–Fuchs equations of these varieties and their Hasse–Witt matrices, which encapsulate information about the number of points, to compute the number of points modulo the order of the field explicitly. We classify pencils of K3 hypersurfaces in Gorenstein Fano toric varieties where the point count coincides and analyze examples related to classical hypergeometric functions.

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Published 30 October 2023