Duke Mathematical Journal

On the characteristic polynomial of the Frobenius on étale cohomology

Andreas-Stephan Elsenhans and Jörg Jahnel

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

Let X be a smooth proper variety of even dimension d over a finite field. We establish a restriction on the value at (1) of the characteristic polynomial of the Frobenius on the middle-dimensional étale cohomology of X with coefficients in Ql(d/2).

Article information

Source
Duke Math. J., Volume 164, Number 11 (2015), 2161-2184.

Dates
Received: 27 November 2013
Revised: 11 September 2014
First available in Project Euclid: 13 August 2015

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1439470581

Digital Object Identifier
doi:10.1215/00127094-3129381

Mathematical Reviews number (MathSciNet)
MR3385131

Zentralblatt MATH identifier
1348.14056

Subjects
Primary: 14F20: Étale and other Grothendieck topologies and (co)homologies
Secondary: 14G15: Finite ground fields 14J20: Arithmetic ground fields [See also 11Dxx, 11G25, 11G35, 14Gxx] 14J35: $4$-folds

Keywords
Proper variety over finite field Middle cohomology Frobenius Characteristic polynomial Artin-Tate formula

Citation

Elsenhans, Andreas-Stephan; Jahnel, Jörg. On the characteristic polynomial of the Frobenius on étale cohomology. Duke Math. J. 164 (2015), no. 11, 2161--2184. doi:10.1215/00127094-3129381. https://projecteuclid.org/euclid.dmj/1439470581


Export citation

References

  • [1] M. Artin, Supersingular $K3$ surfaces, Ann. Sci. Éc. Norm. Supér. (4) 7 (1974), 543–567.
  • [2] M. Artin, P. Deligne, A. Grothendieck, B. Saint-Donat, and J.-L. Verdier, Théorie des topos et cohomologie étale des schémas, Séminaire de Géométrie Algébrique du Bois-Marie (SGA 4), Lecture Notes in Math. 305, Springer, Berlin, 1973.
  • [3] A. Beauville, Complex Algebraic Surfaces, London Math. Soc. Lecture Note Ser. 68, Cambridge Univ. Press, Cambridge, 1983.
  • [4] P. Berthelot, Cohomologie cristalline des schémas de caractéristique $p>0$, Lecture Notes in Math. 407, Springer, Berlin, 1974.
  • [5] P. Berthelot and A. Ogus, Notes on Crystalline Cohomology, Princeton Univ. Press, Princeton, 1978.
  • [6] J. W. S. Cassels, Arithmetic on curves of genus 1, VIII: On the conjectures of Birch and Swinnerton-Dyer, J. Reine Angew. Math. 217 (1965), 180–189.
  • [7] J. W. S. Cassels, “Global fields” in Algebraic Number Theory (Brighton, 1965), Thompson, Washington, 1967, 42–84.
  • [8] A. Chambert-LiOr, Cohomologie cristalline: un survol, Expo. Math. 16 (1998), 333–382.
  • [9] F. Charles, The Tate conjecture for K3 surfaces over finite fields, Invent. Math. 194 (2013), 119–145.
  • [10] P. Deligne, La conjecture de Weil, I, Publ. Math. Inst. Hautes Études Sci. 43 (1974), 273–307.
  • [11] P. Deligne, Cohomologie Étale, Séminaire de Géométrie Algébrique du Bois Marie (SGA 4$\frac{1}{2}$), Lecture Notes in Math. 569, Springer, Berlin, 1977.
  • [12] P. Deligne, La conjecture de Weil, II, Publ. Math. Inst. Hautes Études Sci. 52 (1980), 137–252.
  • [13] P. Deligne, “Relèvement des surfaces $K3$ en caractéristique nulle” in Algebraic Surfaces (Orsay, 1976–1978), Lecture Notes in Math. 868, Springer, Berlin, 1981, 58–79.
  • [14] P. Deligne and L. Illusie, Relèvements modulo $p^{2}$ et décomposition du complexe de de Rham, Invent. Math. 89 (1987), 247–270.
  • [15] P. Deligne and N. Katz, Groupes de monodromie en géométrie algébrique, II, Séminaire de Géométrie Algébrique du Bois Marie (SGA 7), Lecture Notes in Math. 340, Springer, Berlin, 1973.
  • [16] J. Dieudonné, Eléments d’analyse, Tome II: Chapitres XII et XV, Cahiers Sci. Fasc. XXXI, Gauthier-Villars, Paris, 1968.
  • [17] M. Eichler, Zur Zahlentheorie der Quaternionen-Algebren, J. Reine Angew. Math. 195 (1955), 127–151.
  • [18] A.-S. Elsenhans and J. Jahnel, “On Weil polynomials of $K3$ surfaces” in Algorithmic Number Theory (Nancy, 2010), Lecture Notes in Comput. Sci. 6197, Springer, Berlin, 2010, 126–141.
  • [19] A.-S. Elsenhans and J. Jahnel, On the computation of the Picard group for $K3$ surfaces, Math. Proc. Cambridge Philos. Soc. 151 (2011), 263–270.
  • [20] A.-S. Elsenhans and J. Jahnel, The Picard group of a $K3$ surface and its reduction modulo $p$, Algebra Number Theory 5 (2011), 1027–1040.
  • [21] B. H. Gross, “Heights and the special values of $L$-series” in Number Theory (Montreal, 1985), CMS Conf. Proc. 7, Amer. Math. Soc., Providence, 1987, 115–187.
  • [22] B. Hassett, Special cubic fourfolds, Compos. Math. 120 (2000), 1–23.
  • [23] T. Honda, Isogeny classes of abelian varieties over finite fields, J. Math. Soc. Japan 20 (1968), 83–95.
  • [24] L. Illusie, Complexe de de Rham-Witt et cohomologie cristalline, Ann. Sci. Éc. Norm. Supér. (4) 12 (1979), 501–661.
  • [25] L. Illusie and M. Raynaud, Les suites spectrales associées au complexe de de Rham-Witt, Publ. Math. Inst. Hautes Études Sci. 57 (1983), 73–212.
  • [26] N. M. Katz and W. Messing, Some consequences of the Riemann hypothesis for varieties over finite fields, Invent. Math. 23 (1974), 73–77.
  • [27] M. Lieblich, D. Maulik, and A. Snowden, Finiteness of $K3$ surfaces and the Tate conjecture, Ann. Sci. Éc. Norm. Supér. (4) 47 (2014), 285–308.
  • [28] Yu. I. Manin, Theory of commutative formal groups over fields of finite characteristic (in Russian), Uspekhi Mat. Nauk 18, no. 6 (1963), 3–90; English translation in Russian Math. Surveys 18, no. 6 (1963), 1–83.
  • [29] B. Mazur, Frobenius and the Hodge filtration, Bull. Amer. Math. Soc. 78 (1972), 653–667.
  • [30] B. Mazur, Frobenius and the Hodge filtration (estimates), Ann. of Math. (2) 98 (1973), 58–95.
  • [31] J. S. Milne, On a conjecture of Artin and Tate, Ann. of Math. (2) 102 (1975), 517–533.
  • [32] J. W. Milnor and J. D. Stasheff, Characteristic Classes, Ann. of Math. Stud. 76, Princeton Univ. Press, Princeton, 1974.
  • [33] K. Motose, On values of cyclotomic polynomials, VIII, Bull. Fac. Sci. Technol. Hirosaki Univ. 9 (2006), 15–27.
  • [34] K. P. Pera, The Tate conjecture for $K3$ surfaces in odd characteristic, preprint, arXiv:1301.6326 [mathNT].
  • [35] J.-P. Serre, “Sur la topologie des variétés algébriques en caractéristique $p$” in Symposium internacional de topologí a algebraica (Mexico City, 1958), Universidad Nacional Autónoma de México, Mexico City, 1958, 24–53.
  • [36] J. H. Silverman, The Arithmetic of Elliptic Curves, Grad. Texts in Math. 106, Springer, New York, 1986.
  • [37] E. H. Spanier, Algebraic Topology, McGraw-Hill, New York, 1966.
  • [38] U. Stuhler, Konjugationsklassen unipotenter Elemente in einfachen algebraischen Gruppen vom Typ $B_{n}$, $C_{n}$, $D_{n}$ und $G_{2}$, Ph.D. dissertation, Georg-August-Universität Göttingen, Göttingen, Germany, 1970.
  • [39] J. Suh, Symmetry and parity in Frobenius action on cohomology, Compos. Math. 148 (2012), 295–303.
  • [40] T. Urabe, The bilinear form of the Brauer group of a surface, Invent. Math. 125 (1996), 557–585.
  • [41] R. van Luijk, $K3$ surfaces with Picard number one and infinitely many rational points, Algebra Number Theory 1 (2007), 1–15.
  • [42] G. E. Wall, On the conjugacy classes in the unitary, symplectic and orthogonal groups, J. Aust. Math. Soc. 3 (1963), 1–62.
  • [43] D. B. Zagier, Zetafunktionen und quadratische Körper, Springer, Berlin, 1981.
  • [44] Yu. G. Zarkhin, The Brauer group of an abelian variety over a finite field (in Russian), Izv. Ross. Akad. Nauk Ser. Mat. 46, no. 2 (1982), 211–243; English translation in Math. USSR Izv. 20, no. 2 (1983), 203–234.