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A-hypergeometric series and a p-adic refinement of the Hasse-Witt matrix

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Abstract

We identify the p-adic unit roots of the zeta function of a projective hypersurface over a finite field of characteristic p as the eigenvalues of a product of special values of a certain matrix of p-adic series. That matrix is a product \(F(\varLambda ^p)^{-1}F(\varLambda )\), where the entries in the matrix \(F(\varLambda )\) are A-hypergeometric series with integral coefficients and \(F(\varLambda )\) is independent of p.

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References

  1. Adolphson, A., Sperber, S.: \(A\)-hypergeometric series and the Hasse-Witt matrix of a hypersurface. Finite Fields Appl. 41, 55–63 (2016)

    Article  MathSciNet  Google Scholar 

  2. Adolphson, A., Sperber, S.: A generalization of the Hasse-Witt matrix of a hypersurface. Finite Fields Appl. 47, 203–221 (2017)

    Article  MathSciNet  Google Scholar 

  3. Adolphson, A., Sperber, S.: Distinguished-root formulas for generalized Calabi-Yau hypersurfaces. Algebra Number Theory 11(6), 1317–1356 (2017)

    Article  MathSciNet  Google Scholar 

  4. Beukers, F., Vlasenko, M.: Dwork crystals I. Int. Math. Res. Not., to appear

  5. Beukers, F., Vlasenko, M.: Dwork crystals II. Int. Math. Res. Not., to appear

  6. Dwork, B.: On the zeta function of a hypersurface. Inst. Hautes Études Sci. Publ. Math. No. 12, 5–68 (1962)

  7. Dwork, B.: \(p\)-adic cycles. Inst. Hautes Études Sci. Publ. Math. No. 37, 27–115 (1969)

  8. Katz, N.: Algebraic solutions of differential equations (\(p\)-curvature and the Hodge filtration). Invent. Math. 18, 1–118 (1972)

    Article  MathSciNet  Google Scholar 

  9. Koblitz, N.: \(p\)-adic variation of the zeta-function over families of varieties defined over finite fields. Compositio Math. 31(2), 119–218 (1975)

    MathSciNet  MATH  Google Scholar 

  10. Miller, L.: Über gewöhnliche Hyperflächen. I. J. Reine Angew. Math. 282, 96–113 (1976)

    MATH  Google Scholar 

  11. Miller, L.: Über gewöhnliche Hyperflächen. II. J. Reine Angew. Math. 283(284), 402–420 (1976)

    MATH  Google Scholar 

  12. Serre, J.-P.: Endomorphismes complètement continus des espaces de Banach \(p\)-adiques. Inst. Hautes Études Sci. Publ. Math. 12, 69–85 (1962)

    Article  Google Scholar 

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Correspondence to Alan Adolphson.

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Communicated by Jens Funke.

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Adolphson, A., Sperber, S. A-hypergeometric series and a p-adic refinement of the Hasse-Witt matrix. Abh. Math. Semin. Univ. Hambg. 91, 225–256 (2021). https://doi.org/10.1007/s12188-021-00243-1

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  • DOI: https://doi.org/10.1007/s12188-021-00243-1

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