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Hasse–Witt matrices, unit roots and period integrals

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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.

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Acknowledgements

The authors are grateful to Mao Sheng, Zijian Yao, Dingxin Zhang and Jie Zhou for their interests and helpful discussions. B. Lian and S.-T. Yau are partially supported by the Simons Collaboration on Homological Mirror Symmetry 2015–2019.

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Authors and Affiliations

  1. Department of Mathematics, Brandeis University, 415 South Street, Waltham, MA, 02453, USA

    An Huang & Bong Lian

  2. YMSC, Tsinghua University, JingZhai Tsinghua University, Haidian, 100084, Beijing, China

    Shing-Tung Yau & Chenglong Yu

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  1. An Huang
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Huang, A., Lian, B., Yau, ST. et al. Hasse–Witt matrices, unit roots and period integrals. Math. Ann. 387, 145–173 (2023). https://doi.org/10.1007/s00208-022-02464-y

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