Japanese Journal of Mathematics

, Volume 1, Issue 1, pp 107–136 | Cite as

Miscellany on traces in ℓ-adic cohomology: a survey



We discuss classical questions concerning traces of elements of Galois groups or correspondences in ℓ-adic cohomology, mostly over finite or local fields, such as rationality and independence of ℓ, integrality, congruences modulo powers of ℓ or p. We report on the progress that has been made on this topic during the past ten years.

Keywords and phrases.

ℓ-adic cohomology independence of ℓ Grothendieck’s trace formula Lefschetz trace formula zeta functions over finite fields Euler–Poincaré characteristic Betti number Bloch’s conductor conjecture intersection cohomology Grothendieck’s six operations intermediate extension Weil conjectures Hodge polygon Newton polygon crystalline cohomology Hodge filtration coniveau filtration alteration Fano variety rationally connected Weil group Swan conductor wild ramification Brauer trace log scheme logarithmic differential forms Čebotarev’s density theorem semisimple group Fatou’s lemma 

Mathematics Subject Classification (2000).

14F20 (primary) 14F10 20G15 (secondary) 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. A.
    A. Abbes, The Grothendieck–Ogg–Shafarevich formula for arithmetic surfaces, J. Algebraic Geom., 9 (2000), 529–576.MATHMathSciNetGoogle Scholar
  2. AS.
    A. Abbes and T. Saito, Analyse micro-locale ℓ-adique en caractéristique p>0: Le cas d’un trait, math.AG/0602285.Google Scholar
  3. Ax.
    J. Ax, Zeroes of polynomials over finite fields, Amer. J. Math., 86 (1964), 264–261.MathSciNetGoogle Scholar
  4. BBD.
    A. Beilinson, J. Bernstein and P. Deligne, Faisceaux pervers, Astérisque, 100 (1982).Google Scholar
  5. BO.
    P. Berthelot and A. Ogus, Notes on crystalline cohomology, Math. Notes, 21, Princeton Univ. Press, 1978.Google Scholar
  6. B1.
    S. Bloch, Lectures on Algebraic Cycles, Duke Univ. Math. Ser. IV, 1980.Google Scholar
  7. B2.
    S. Bloch, Cycles on arithmetic schemes and Euler characteristics of curves, In: Algebraic Geometry, Bowdoin, 1985, Proc. Sympos. Pure Math., 46, Part 2, Amer. Math. Soc., Providence, RI, 1987, pp.421–450.Google Scholar
  8. BE.
    S. Bloch and H. Esnault, Künneth projectors for open varieties, preprint, 2005.Google Scholar
  9. dJ.
    A.J. de Jong, Smoothness, semi-stability and alterations, Publ. Math. Inst. Hautes Études Sci., 83 (1996), 51–93.MATHGoogle Scholar
  10. dA.
    P.-L. del Angel, A remark on the Hodge type of projective varieties of low degree, J. Reine Angew. Math., 449 (1994), 173–177.MathSciNetGoogle Scholar
  11. D1.
    P. Deligne, La conjecture de Weil I, Publ. Math. Inst. Hautes Études Sci., 43 (1974), 273–307.MathSciNetGoogle Scholar
  12. D2.
    P. Deligne, La conjecture de Weil II, Publ. Math. Inst. Hautes Études Sci., 52 (1980), 137–252.MATHMathSciNetGoogle Scholar
  13. D3.
    P. Deligne, Théorie de Hodge II, Publ. Math. Inst. Hautes Études Sci., 40 (1972), 5–57.Google Scholar
  14. DD.
    P. Deligne and A. Dimca, Filtration de Hodge et par l’ordre du pôle pour les hypersurfaces singulières, Ann. Sci. École Norm. Sup. (4), 23 (1990), 645–656.MathSciNetGoogle Scholar
  15. DE.
    P. Deligne and H. Esnault, appendix to [E2].Google Scholar
  16. E1.
    H. Esnault, Hodge type of subvarieties of \(\mathbb{P}^n\) of small degrees, Math. Ann., 288 (1990), 549–551.CrossRefMATHMathSciNetGoogle Scholar
  17. E2.
    H. Esnault, Varieties over a finite field with trivial Chow group of 0-cycles have a rational point, Invent. Math., 151 (2003), 317–320.CrossRefMathSciNetGoogle Scholar
  18. E3.
    H. Esnault, Deligne’s integrality theorem in unequal characteristic and rational points over finite fields, preprint, 2004, to appear in Ann. of Math.Google Scholar
  19. EK.
    H. Esnault and N. Katz, Cohomological divisibility and point count divisibility, Compos. Math., 141 (2005), 93–100.MathSciNetGoogle Scholar
  20. ENS.
    H. Esnault, M. Nori and V. Srinivas, Hodge type of projective varieties of low degree, Math. Ann., 293 (1992), 1–6.CrossRefMathSciNetGoogle Scholar
  21. EW.
    H. Esnault and D. Wan, Hodge type of the exotic cohomology of complete intersections, C. R. Acad. Sci. Paris, Série I, 336 (2003), 153–157.MathSciNetGoogle Scholar
  22. F1.
    K. Fujiwara, Rigid geometry, Lefschetz–Verdier trace formula and Deligne’s conjecture, Invent. Math., 127 (1997), 489–533.CrossRefMATHMathSciNetGoogle Scholar
  23. F2.
    K. Fujiwara, Independence of ℓ for Intersection Cohomology (after Gabber), In: Algebraic Geometry 2000, Azumino, Adv. Stud. Pure Math., 36, 2002, pp. 145–151.Google Scholar
  24. Gr.
    A. Grothendieck, Le groupe de Brauer III : Exemples et compléments, In: Dix Exposés sur la Cohomologie des Schémas, Masson et Cie, North-Holland Pub. Comp., (eds. A. Grothendieck and N. Kuiper), Adv. Stud. Pure Math., 1968, pp. 88–188.Google Scholar
  25. I1.
    L. Illusie, Complexe Cotangent et Déformations, Lecture Notes in Math., 239, Springer-Verlag, 1971.Google Scholar
  26. I2.
    L. Illusie, Théorie de Brauer et caractéristique d’Euler-Poincaré, d’après P. Deligne, Astérisque, 82-83 (1978-79), 161–172.Google Scholar
  27. I3.
    L. Illusie, On semistable reduction and the calculation of nearby cycles, In: Geometric Aspects of Dwork Theory, (eds. A. Adolphson, F. Baldassarri, P. Berthelot, N. Katz, and F. Loeser), Walter de Gruyter, 2004, pp. 785–803.Google Scholar
  28. KS1.
    K. Kato and T. Saito, On the conductor formula of Bloch, Publ. Math. Inst. Hautes Études Sci., 100 (2004), 5–151.MathSciNetGoogle Scholar
  29. KS2.
    K. Kato and T. Saito, Ramification theory for varieties over a perfect field, math.AG/0402010.Google Scholar
  30. K1.
    N. Katz, On a theorem of Ax, Amer. J. Math., 93 (1971), 485–499.MATHMathSciNetGoogle Scholar
  31. K2.
    N. Katz, Gauss Sums, Kloosterman Sums, and Monodromy Groups, Ann. of Math. Stud., 116 (1988).Google Scholar
  32. K3.
    N. Katz, Affine cohomological transforms, perversity, and monodromy, J. Amer. Math. Soc., 6 (1993), 149–222.MATHMathSciNetGoogle Scholar
  33. K4.
    N. Katz, Review of ℓ-adic Cohomology, Proc. Sympos. Pure Math., 55 (1994), 21–30.MATHGoogle Scholar
  34. K5.
    N. Katz, Independence of ℓ and weak Lefschetz, Proc. Sympos. Pure Math., 55 (1994), 101–114.MATHGoogle Scholar
  35. KM.
    N. Katz and W. Messing, Some consequences of the Riemann hypothesis for varieties over finite fields, Invent. Math., 23 (1974), 73–77.CrossRefMathSciNetGoogle Scholar
  36. Kl.
    S. Kleiman, Algebraic cycles and the Weil conjectures, In: Dix Exposés sur la Cohomologie des Schémas, Masson et Cie, North-Holland Pub. Comp., 1968, 359–386.Google Scholar
  37. Ko.
    J. Kollár, Rational Curves on Algebraic Varieties, Ergeb. Math. Grenzgeb. (3), 32, Springer-Verlag, 1996.Google Scholar
  38. La.
    G. Laumon, Comparaison de caractéristiques d’Euler-Poincaré en cohomologie ℓ-adique, C. R. Acad. Sci. Paris, Série I, 292 (1981), 209–212.MATHMathSciNetGoogle Scholar
  39. McP.
    R. MacPherson, Chern classes for singular varieties, Ann. of Math., 100 (1974), 423–432.MATHMathSciNetGoogle Scholar
  40. O.
    T. Ochiai, ℓ-independence of the trace of monodromy, Math. Ann., 315 (1999), 321–340.MATHMathSciNetGoogle Scholar
  41. P1.
    R. Pink, On the calculation of local terms in the Lefschetz-Verdier trace formula and its application to a conjecture of Deligne, Ann. of Math., 135 (1992), 483–525.MATHMathSciNetGoogle Scholar
  42. P2.
    R. Pink, The Mumford-Tate conjecture for Drinfeld-modules, Publ. Res. Inst. Math. Sci., 33 (1997), 393–425.MATHMathSciNetGoogle Scholar
  43. R.
    M. Raynaud, Caractéristique d’Euler–Poincaré d’un faisceau et cohomologie des variétés abéliennes, Sém. Bourbaki 1964/65, In: Dix Exposés sur la Cohomologie des Schémas, North-Holland Pub. Comp., Amsterdam, Masson et Cie, Paris, 286, 1968.Google Scholar
  44. S1.
    T. Saito, Self-intersection 0-cycles and coherent sheaves on arithmetic schemes, Duke Math. J., 57 (1988), 555–578.MATHMathSciNetGoogle Scholar
  45. S2.
    T. Saito, Parity in Bloch’s conductor formula in even dimension, J. Théor. Nombres Bordeaux, 16 (2004), 403–421.MATHMathSciNetGoogle Scholar
  46. S3.
    T. Saito, Weight spectral sequences and independence of ℓ, J. Inst. Math. Jussieu, 2 (2003), 583–634.MATHMathSciNetGoogle Scholar
  47. Se1.
    J-P. Serre, Zeta and L functions, In: Arithmetical Algebraic Geometry, Proc. of a Conference held at Purdue Univ., Dec. 5-7, 1963, (ed. O. Schilling), Harper and Row, 1965, 82–92 (= [Se Oe, 64]).Google Scholar
  48. Se2.
    J-P. Serre, Représentations Linéaires des Groupes Finis, 3ème édition corrigée, Hermann, 1978.Google Scholar
  49. Se3.
    J-P. Serre, Arithmetic Groups, In: Homological Group Theory, C.T.C. Wall edit., London Math. Soc. Lecture Notes Ser., 36, Cambridge Univ. Press, 1979, pp. 77–169 (= [Se Oe,120]).Google Scholar
  50. Se4.
    J-P. Serre, Bounds for the orders of finite subgroups of reductive groups, in Group Representation Theory, (eds. M. Geck, D. Testerman and J. Thévenaz.), EPFL Press, Lausanne, to appear.Google Scholar
  51. SeOe.
    J-P. Serre, Oeuvres (collected papers), I, II and III (1986), IV (2000), Springer-Verlag.Google Scholar
  52. V.
    Y. Varshavsky, Lefschetz–Verdier trace formula and a generalization of a theorem of Fujiwara, May 2005, math.AG/0505564.Google Scholar
  53. Vi1.
    I. Vidal, Théorie de Brauer et conducteur de Swan, J. Algebraic. Geom., 13 (2004), 349–391.MATHMathSciNetGoogle Scholar
  54. Vi2.
    I. Vidal, Courbes nodales et ramification sauvage virtuelle, Manuscripta Math., 118 (2005), 43–70.CrossRefMATHMathSciNetGoogle Scholar
  55. W.
    E. Warning, Bemerkung zur vorstehenden Arbeit von Herrn Chevalley, Abh. Math. Sem. Univ. Hamburg, 11 (1936), 76–83.Google Scholar
  56. EGAII.
    A. Grothendieck avec J. Dieudonné, Éléments de Géométrie Algébrique: II, Étude globale élémentaire de quelques classes de morphismes, Publ. Math. Inst. Hautes Études Sci., 8 (1961).Google Scholar
  57. SGA4 ½.
    P. Deligne, Cohomologie Étale, Lecture Notes in Math., 569, Springer-Verlag, 1977.Google Scholar
  58. SGA5.
    A. Grothendieck, Cohomologie l-adique et Fonctions L, Séminaire de géométrie algébrique du Bois-Marie 1965-66, Lecture Notes in Math., 589, Springer-Verlag, 1977.Google Scholar
  59. SGA7.
    A. Grothendieck, P. Deligne et N. Katz, Groupes de Monodromie en Géométrie Algébrique, Séminaire de géométrie algébrique du Bois-Marie 1967-1969, I, II, Lecture Notes in Math., 288, 340, Springer-Verlag, 1972-1973.Google Scholar

Copyright information

© The Mathematical Society of Japan and Springer-Verlag 2006

Authors and Affiliations

  1. 1.Département de MathématiqueUniversité de Paris-SudOrsay CedexFrance

Personalised recommendations