Hasse–Witt matrices, unit roots and period integrals

Abstract

Motivated by the work of Candelas et al. (Calabi–Yau manifolds over finite fields, I. arXiv:hep-th/0012233, 2000) on counting points for quintic family over finite fields, we study the relations among Hasse–Witt matrices, unit-root part of zeta functions and period integrals of Calabi–Yau hypersurfaces in both toric varieties and flag varieties. We prove a conjecture by Vlasenko (Higher Hasse–Witt matrices. Indag Math 29(5):1411–1424, 2018) on unit-root F-crystals for toric hypersurfaces following Katz’s local expansion method (1984, 1985) in logarithmic setting. The Frobenius matrices of unit-root F-crystals also have close relation with period integrals. The proof gives a way to pass from Katz’s congruence relations in terms of expansion coefficients (1985) to Dwork’s congruence relations (1969) about periods.