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Revisiting Dwork cohomology: visibility and divisibility of Frobenius eigenvalues in rigid cohomology

Part of: Zeta and $L$-functions: analytic theory Arithmetic algebraic geometry (Co)homology theory

Published online by Cambridge University Press:  01 September 2025

Daqing Wan  and
Dingxin Zhang Open the ORCID record for Dingxin Zhang [Opens in a new window]
Daqing Wan
Affiliation:
Center for Discrete Mathematics, Chongqing University, Chongqing 401331, China Department of Mathematics, University of California, Irvine, CA 92697-3875, USA dwan@math.uci.edu
Dingxin Zhang
Affiliation:
Center for Mathematics and Interdisciplinary Sciences, Fudan University, and Shanghai Institute for Mathematics and Interdisciplinary Sciences (SIMIS), Shanghai 200433, China zhang@simis.cn
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Abstract

In this paper, we investigate Frobenius eigenvalues of the compactly supported rigid cohomology of a variety defined over a finite field with q elements, using Dwork’s method. Our study yields several arithmetic consequences. First, we establish that the zeta functions of a set of related affine varieties can reveal all Frobenius eigenvalues of the rigid cohomology of the variety up to a Tate twist. This result does not seem to be known for the $\ell$-adic cohomology. As a second application, we provide several q-divisibility lower bounds for the Frobenius eigenvalues of the rigid cohomology of the variety, in terms of the dimension and multi-degrees of the defining equations. These divisibility bounds for rigid cohomology are generally better than what is suggested from the best known divisibility bounds in $\ell$-adic cohomology, both before and after the middle cohomological degree.

Keywords

Frobenius eigenvalueconiveaurigid cohomology

MSC classification

Primary: 14F30: $p$-adic cohomology, crystalline cohomology
Secondary: 11G25: Varieties over finite and local fields 11M38: Zeta and $L$-functions in characteristic $p$

Information

Type
Research Article
Information
Compositio Mathematica , Volume 161 , Issue 6 , June 2025 , pp. 1215 - 1249
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Copyright
© The Author(s), 2025. The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence

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References

Abe, T., Langlands correspondence for isocrystals and the existence of crystalline companions for curves , J. Amer. Math. Soc. 31 (2018), 921–1057.CrossRefGoogle Scholar
Abe, T. and Caro, D., Theory of weights in p-adic cohomology , Amer. J. Math. 140 (2018), 879–975.CrossRefGoogle Scholar
Adolphson, A. and Sperber, S., Newton polyhedra and the degree of the L-function associated to an exponential sum , Invent. Math. 88 (1987), 555–569.CrossRefGoogle Scholar
Adolphson, A. and Sperber, S., p-adic estimates for exponential sums and the theorem of Chevalley-Warning , Ann. Sci. École Norm. Sup. (4) 20 (1987), 545–556.CrossRefGoogle Scholar
Adolphson, A. and Sperber, S., Exponential sums and Newton polyhedra: Cohomology and estimates , Ann. Math. (2) 130 (1989), 367–406.CrossRefGoogle Scholar
Adolphson, A. and Sperber, S., Dwork cohomology, de Rham cohomology, and hypergeometric functions , Amer. J. Math. 122 (2000), 319–348.CrossRefGoogle Scholar
Ax, J., Zeroes of polynomials over finite fields , Amer. J. Math. 86 (1964), 255–261.CrossRefGoogle Scholar
Baldassarri, F. and Berthelot, P., On Dwork cohomology for singular hypersurfaces , in Geometric aspects of Dwork theory, vol. I, II (Walter de Gruyter, Berlin, 2004), 177–244.CrossRefGoogle Scholar
Baldassarri, F. and D’Agnolo, A., On Dwork cohomology and algebraic D-modules , in Geometric aspects of Dwork theory (Walter de Gruyter, Berlin, 2004), 245–253.CrossRefGoogle Scholar
Berthelot, P., Cohomologie rigide et théorie de Dwork: le cas des sommes exponentielles, p-adic Cohomol. 119–120 (1984), 17–49.Google Scholar
Berthelot, P., D-modules arithmétiques. I. Opérateurs différentiels de niveau fini , Ann. Sci. École Norm. Sup. (4) 29 (1996), 185–272.CrossRefGoogle Scholar
Bombieri, E., On exponential sums in finite fields , Amer. J. Math. 88 (1966), 71–105.CrossRefGoogle Scholar
Bourgeois, P., Annulation et pureté des groupes de cohomologie rigide associés à des sommes exponentielles , C. R. Acad. Sci. Paris Sér. I Math. 328 (1999), 681–686.CrossRefGoogle Scholar
Caro, D., D-modules arithmétiques surcohérents. Application aux fonctions L , Ann. Inst. Fourier (Grenoble) 54 (2004), 1943–1996.CrossRefGoogle Scholar
Caro, D., F-isocristaux surconvergents et surcohérence différentielle , Invent. Math. 170 (2007), 507–539.CrossRefGoogle Scholar
Caro, D., $\mathscr{D}$ -modules arithmétiques surholonomes, Ann. Sci. Éc. Norm. Supér. (4) 42 (2009), 141–192.CrossRefGoogle Scholar
Caro, D., Le formalisme des six opérations de grothendieck en cohomologie p-adique, Preprint (2012), https://arxiv.org/abs/1209.4020arXiv:1209.4020.Google Scholar
Caro, D. and Tsuzuki, N., Overholonomicity of overconvergent F-isocrystals over smooth varieties , Ann. Math. (2) 176 (2012), 747–813.CrossRefGoogle Scholar
de Jong, A. J., Smoothness, semi-stability and alterations , Inst. Hautes Études Sci. Publ. Math. 83 (1996), 51–93.CrossRefGoogle Scholar
Deligne, P., La conjecture de Weil. I , Inst. Hautes Études Sci. Publ. Math. 43 (1974), 273–307.CrossRefGoogle Scholar
Deligne, P., Cohomologie étale: Séminaire de Géométrie Algébrique du Bois-Marie SGA 4 1/2, Lecture Notes in Mathematics, vol. 569 (Springer-Verlag, Berlin–New York, 1977), avec la collaboration Boutot, de J. F., Grothendieck, A., Illusie, L. et Verdier, J. L..CrossRefGoogle Scholar
Deligne, P. and Katz, N. M., Groupes de monodromie en géométrie algébrique: Séminaire de Géométrie Algébrique du Bois-Marie 1967–1969 (SGA 7 II), Lecture Notes in Mathematics, vol. 340 (Springer-Verlag, Berlin–New York, 1973), Deligne, dirigé par P. et Katz, N..CrossRefGoogle Scholar
Dimca, A., Maaref, F., Sabbah, C. and Saito, M., Dwork cohomology and algebraic D-modules , Math. Ann. 318 (2000), 107–125.CrossRefGoogle Scholar
Dwork, B., On the rationality of the zeta function of an algebraic variety , Amer. J. Math. 82 (1960), 631–648.CrossRefGoogle Scholar
Dwork, B., On the zeta function of a hypersurface , Inst. Hautes Études Sci. Publ. Math. 12 (1962), 5–68.CrossRefGoogle Scholar
Dwork, B., On the zeta function of a hypersurface . II, Ann. Math. (2) 80 (1964), 227–299.CrossRefGoogle Scholar
Esnault, H. and Katz, N. M., Cohomological divisibility and point count divisibility , Compos. Math. 141 (2005), 93–100.CrossRefGoogle Scholar
Esnault, H. and Wan, D., Divisibility of Frobenius eigenvalues on $\ell$ -adic cohomology, Proc. Indian Acad. Sci. Math. Sci. 132 (2022), 60.CrossRefGoogle Scholar
Étesse, J.-Y. and Le Stum, B., Fonctions L associées aux F-isocristaux surconvergents. I. Interprétation cohomologique, Math. Ann. 296 (1993), 557–576.CrossRefGoogle Scholar
Katz, N. M., On the differential equations satisfied by period matrices , Inst. Hautes Études Sci. Publ. Math. 35 (1968), 223–258.CrossRefGoogle Scholar
Katz, N. M., On a theorem of Ax , Amer. J. Math. 93 (1971), 485–499.CrossRefGoogle Scholar
Katz, N. M. and Messing, W., Some consequences of the Riemann hypothesis for varieties over finite fields , Invent. Math. 23 (1974), 73–77.CrossRefGoogle Scholar
Kedlaya, K. S., Finiteness of rigid cohomology with coefficients , Duke Math. J. 134 (2006), 15–97.CrossRefGoogle Scholar
Kiehl, R. and Weissauer, R., Weil conjectures, perverse sheaves and l’adic Fourier transform, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge/A Series of Modern Surveys in Mathematics, 3rd Series, vol. 42 (Springer-Verlag, Berlin, 2001).Google Scholar
Le Stum, B., Rigid cohomology , Cambridge Tracts in Mathematics, vol. 172 (Cambridge University Press, Cambridge, 2007).Google Scholar
Li, P., Exponential sums and rigid cohomology , Finite Fields Appl. 81 (2022), 102031.CrossRefGoogle Scholar
Monsky, P., Formal cohomology. III. Fixed point theorems , Ann. Math. (2) 93 (1971), 315–343.CrossRefGoogle Scholar
Pulita, A., Rank one solvable p-adic differential equations and finite abelian characters via Lubin-Tate groups , Math. Ann. 337 (2007), 489–555.CrossRefGoogle Scholar
Rai, A. and Shuddhodan, K. V., Perverse filtrations via Brylinski-Radon transformations, Preprint (2023), arXiv:2309.13973.Google Scholar
Robert, A. M., A course in p-adic analysis, Graduate Texts in Mathematics, vol. 198 (Springer-Verlag, New York, 2000).Google Scholar
Serre, J.-P., Endomorphismes complètement continus des espaces de Banach p-adiques , Inst. Hautes Études Sci. Publ. Math. 12 (1962), 69–85.CrossRefGoogle Scholar
Tsuzuki, N., Cohomological descent of rigid cohomology for proper coverings , Invent. Math. 151 (2003), 101–133.CrossRefGoogle Scholar
Wan, D., Poles of zeta functions of complete intersections , Chinese Ann. Math. Ser. B 21 (2000), 187–200.CrossRefGoogle Scholar
Wan, D. and Zhang, D., Hodge and Frobenius colevels of algebraic varieties , J. London Math. Soc. (2) 111 (2025), e70160.CrossRefGoogle Scholar