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p-Adic GKZ hypergeometric complex

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Abstract

To a torus action on a complex vector space, Gelfand, Kapranov and Zelevinsky introduce a system of differential equations, which are now called the GKZ hypergeometric system. Its solutions are GKZ hypergeometric functions. We study the p-adic counterpart of the GKZ hypergeometric system. The p-adic GKZ hypergeometric complex is a twisted relative de Rham complex of overconvergent differential forms with logarithmic poles. It is an over-holonomic object in the derived category of arithmetic \({{\mathcal {D}}}\)-modules with Frobenius structures. Traces of Frobenius on fibers at Techmüller points of the GKZ hypergeometric complex define the hypergeometric function over the finite field introduced by Gelfand and Graev. Over the non-degenerate locus, the GKZ hypergeometric complex defines an overconvergent F-isocrystal. It is the crystalline companion of the \(\ell \)-adic GKZ hypergeometric sheaf that we constructed before. Our method is a combination of Dwork’s theory and the theory of arithmetic \({{\mathcal {D}}}\)-modules of Berthelot.

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Authors and Affiliations

  1. Yau Mathematical Sciences Center, Tsinghua University, Bei**g, 100084, People’s Republic of China

    Lei Fu

  2. Yanqi Lake Bei**g Institute of Mathematical Sciences and Applications, Bei**g, People’s Republic of China

    Peigen Li

  3. Department of Mathematics, Tsinghua University, Bei**g, 100084, People’s Republic of China

    Peigen Li

  4. Department of Mathematics, University of California, Irvine, CA, 92697, USA

    Daqing Wan

  5. Yau Center, Southeast University, Nan**g, 210096, People’s Republic of China

    Hao Zhang

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  1. Lei Fu
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We would like to thank the referee for careful reading of the paper and for many suggestions improving the paper. The research of Lei Fu is supported by NSFC12171261 and 2021YFA 1000700.

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Fu, L., Li, P., Wan, D. et al. p-Adic GKZ hypergeometric complex. Math. Ann. 387, 1629–1689 (2023). https://doi.org/10.1007/s00208-022-02491-9

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  • DOI: https://doi.org/10.1007/s00208-022-02491-9

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