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.
Similar content being viewed by others
Availability of data and material
Not applicable.
References
Adolphson, A., Sperber, S.: \(A\)-hypergeometric series and the Hasse-Witt matrix of a hypersurface. Finite Fields Appl. 41, 55–63 (2016)
Adolphson, A., Sperber, S.: A generalization of the Hasse-Witt matrix of a hypersurface. Finite Fields Appl. 47, 203–221 (2017)
Adolphson, A., Sperber, S.: Distinguished-root formulas for generalized Calabi-Yau hypersurfaces. Algebra Number Theory 11(6), 1317–1356 (2017)
Beukers, F., Vlasenko, M.: Dwork crystals I. Int. Math. Res. Not., to appear
Beukers, F., Vlasenko, M.: Dwork crystals II. Int. Math. Res. Not., to appear
Dwork, B.: On the zeta function of a hypersurface. Inst. Hautes Études Sci. Publ. Math. No. 12, 5–68 (1962)
Dwork, B.: \(p\)-adic cycles. Inst. Hautes Études Sci. Publ. Math. No. 37, 27–115 (1969)
Katz, N.: Algebraic solutions of differential equations (\(p\)-curvature and the Hodge filtration). Invent. Math. 18, 1–118 (1972)
Koblitz, N.: \(p\)-adic variation of the zeta-function over families of varieties defined over finite fields. Compositio Math. 31(2), 119–218 (1975)
Miller, L.: Über gewöhnliche Hyperflächen. I. J. Reine Angew. Math. 282, 96–113 (1976)
Miller, L.: Über gewöhnliche Hyperflächen. II. J. Reine Angew. Math. 283(284), 402–420 (1976)
Serre, J.-P.: Endomorphismes complètement continus des espaces de Banach \(p\)-adiques. Inst. Hautes Études Sci. Publ. Math. 12, 69–85 (1962)
Funding
Not applicable.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflicts of interest or competing interests.
Code availability
Not applicable.
Additional information
Communicated by Jens Funke.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
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
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12188-021-00243-1