Japanese Journal of Mathematics

, Volume 5, Issue 1, pp 73–102 | Cite as

Weights in arithmetic geometry



The concept of weights on the cohomology of algebraic varieties was initiated by fundamental ideas and work of A. Grothendieck and P. Deligne. It is deeply connected with the concept of motives and appeared first on the singular cohomology as the weights of (possibly mixed) Hodge structures and on the etale cohomology as the weights of eigenvalues of Frobenius. But weights also appear on algebraic fundamental groups and in p-adic Hodge theory, where they become only visible after applying the comparison functors of Fontaine. After rehearsing various versions of weights, we explain some more recent applications of weights, e.g. to Hasse principles and the computation of motivic cohomology, and discuss some open questions.

Keywords and phrases:

weights étale cohomology Hasse principles 

Mathematics Subject Classification (2010):

11G25 11G35 14F20 14F30 


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  1. Be.
    Berthelot, Finitude et pureté cohomologique en cohomologie rigide. With an appendix by A.J. de Jong, Invent. Math., 128 (1997), 329–377.Google Scholar
  2. BK.
    S. Bloch and K. Kato, p-adic étale cohomology, Inst. Hautes Études Sci. Publ. Math., 63 (1986), 107–152.Google Scholar
  3. BO.
    S. Bloch and A. Ogus, Gersten’s conjecture and the homology of schemes, Ann. Sci. Ecole Norm. Sup. (4), 7 (1974), 181–202.Google Scholar
  4. CT.
    J.-L. Colliot-Thélène, On the reciprocity sequence in the higher class field theory of function fields, In: Algebraic K-Theory and Algebraic Topology, Lake Louise, AB, 1991, (eds. J.F. Jardine and V.P. Snaith), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 407, Kluwer Acad. Publ., 1993, pp. 35–55.Google Scholar
  5. CTJ.
    J.-L. Colliot-Thélène and U. Jannsen, Sommes de carrés dans les corps de fonctions, C. R. Acad. Sci. Paris Sér. I Math., 312 (1991), 759–762.Google Scholar
  6. CTSS.
    J.-L. Colliot-Thélène, J.-J. Sansuc and C. Soulé, Torsion dans le groupe de Chow de codimension deux, Duke Math. J., 50 (1983), 763–801.Google Scholar
  7. CP1.
    V. Cossart and O. Piltant, Resolution of singularities of threefolds in positive characteristic. I. Reduction to local uniformization on Artin–Schreier and purely inseparable coverings, J. Algebra, 320 (2008), 1051–1082.MATHCrossRefMathSciNetGoogle Scholar
  8. CP2.
    V. Cossart and O. Piltant, Resolution of singularities of threefolds in positive characteristic. II, J. Algebra, 321 (2009), 1836–1976.MATHCrossRefMathSciNetGoogle Scholar
  9. CJS.
    V. Cossart, U. Jannsen and S. Saito, Canonical embedded and non-embedded resolution of singularities for excellent two-dimensional schemes, preprint, 2009, arXiv:0905.2191[math.AG].Google Scholar
  10. De1.
    P. Deligne, Théorie de Hodge. I, In: Actes du Congrès International des Mathématiciens, Nice, 1970, Tome 1, Gauthier-Villars, Paris, 1971, pp. 425–430.Google Scholar
  11. De2.
    P. Deligne, Théorie de Hodge. II, Inst. Hautes Études Sci. Publ. Math., 40 (1971), 5–57.Google Scholar
  12. De3.
    P. Deligne, Théorie de Hodge. III, Inst. Hautes Études Sci. Publ. Math., 44 (1974), 5–77.Google Scholar
  13. De4.
    P. Deligne, La conjecture de Weil. I, Inst. Hautes Études Sci. Publ. Math., 43 (1974), 273–307.Google Scholar
  14. De5.
    P. Deligne, Poids dans la cohomologie des variétés algébriques, In: Proceedings of the International Congress of Mathematicians, Vancouver, BC, 1974, Vol. 1, Canad. Math. Congress, Montreal, Que., 1975, pp. 79–85.Google Scholar
  15. De6.
    P. Deligne, La conjecture de Weil. II, Inst. Hautes Études Sci. Publ. Math., 52 (1980), 137–252.Google Scholar
  16. deJ.
    A.J. de Jong, Smoothness, semi-stability and alterations, Inst. Hautes Études Sci. Publ. Math., 83 (1996), 51–93.Google Scholar
  17. Fa1.
    G. Faltings, Crystalline cohomology and p-adic Galois-representations, In: Algebraic Analysis, Geometry, and Number Theory, Baltimore, MD, 1988, Johns Hopkins Univ. Press, Baltimore, MD, 1989, pp. 25–80.Google Scholar
  18. Fo1.
    J.-M. Fontaine, Sur certains types de représentations p-adiques du groupe de Galois d’un corps local; construction d’un anneau de Barsotti–Tate. (French), Ann. of Math. (2), 115 (1982), 529–577.Google Scholar
  19. Fo2.
    Letter to Jannsen, dated November 26, 1987.Google Scholar
  20. FM.
    J.-M. Fontaine and W. Messing, p-adic periods and p-adic étale cohomology, In: Current Trends in Arithmetical Algebraic Geometry, Arcata, CA, 1985, Contemp. Math., 67, Amer. Math. Soc., Providence, RI, 1987, pp. 179–207.Google Scholar
  21. FuG.
    K. Fujiwara, A proof of the absolute purity conjecture (after Gabber), In: Algebraic Geometry 2000, Azumino, Hotaka, Adv. Stud. Pure Math., 36, Math. Soc. Japan, Tokyo, 2002, pp. 153–183.Google Scholar
  22. Ge.
    T. Geisser, Arithmetic homology and an integral version of Kato’s conjecture, preprint, arXiv:0704.1192v2[math.KT].Google Scholar
  23. Gs.
    H. Gillet and C. Soulé, Descent, motives and K-theory, J. Reine Angew. Math., 478 (1996), 127–176.MATHMathSciNetGoogle Scholar
  24. Gr.
    M. Gros, Classes de Chern et Classes de Cycles en Cohomologie de Hodge–Witt Logarithmique. (French), Mém. Soc. Math. France (N.S.), 21 (1985), 87 p.Google Scholar
  25. Grs.
    M. Gros and N. Suwa, Application d’Abel–Jacobi p-adique et cycles algébriques, Duke Math. J., 57 (1998), 579–613CrossRefMathSciNetGoogle Scholar
  26. Hi.
    H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero, Ann. of Math. (2), 79 (1964), 109–326.CrossRefMathSciNetGoogle Scholar
  27. HyKa.
    O. Hyodo and K. Kato, Semi-stable reduction and crystalline cohomology with logarithmic poles, In: Périodes p-adiques, Bures-sur-Yvette, 1988, Astérisque, 223, Soc. Math. France, Paris, 1994, pp. 221–268.Google Scholar
  28. Il.
    L. Illusie, Complexe de de Rham–Witt et cohomologie cristalline, Ann. Sci. École Norm Sup. (4), 12 (1979), 501–661.MathSciNetGoogle Scholar
  29. It.
    T. Ito, Weight-monodromy conjecture over equal characteristic local fields, Amer. J. Math., 127 (2005), 647–658.MATHCrossRefMathSciNetGoogle Scholar
  30. Ja1.
    U. Jannsen, On the Galois cohomology of l-adic representations attached to varieties over local or global fields, In: Séminaire de Théorie des Nombres, Paris, 1986–87, Progr. Math., 75, Birkhäuser Boston, Boston, MA, 1988, pp. 165–182.Google Scholar
  31. Ja2.
    U. Jannsen, On the l-adic cohomology of varieties over number fields and its Galois cohomology, In: Galois Groups Over Q, Berkeley, CA, 1987, Math. Sci. Res. Inst. Publ., 16, Springer-Verlag, New York, 1989, pp. 315–360.Google Scholar
  32. Ja3.
    U. Jannsen, Mixed Motives and Algebraic K-theory. With Appendices by S. Bloch and C. Schoen, Lecture Notes in Math., 1400, Springer-Verlag, Berlin, 1990.Google Scholar
  33. Ja4.
    U. Jannsen, On finite-dimensional motives and Murre’s conjecture, In: Algebraic Cycles and Motives. Vol. 2, London Math. Soc. Lecture Note Ser., 344, Cambridge Univ. Press, Cambridge, 2007, pp. 112–142.Google Scholar
  34. Ja5.
    U. Jannsen, Hasse principles for higher-dimensional fields, preprint, 2009, arXiv:0910.2803[math.AG].Google Scholar
  35. JS1.
    U. Jannsen and S. Saito, Kato homology of arithmetic schemes and higher class field theory over local fields, Doc. Math., Extra vol.: K. Kato’s Fiftieth Birthday (2003), 479–538 (electronic).Google Scholar
  36. JS2.
    U. Jannsen and S. Saito, Kato conjecture and motivic cohomology over finite fields, preprint, 2009, arXiv:0910.2815[math.AG].Google Scholar
  37. Ka1.
    K. Kato, A Hasse principle for two-dimensional global fields. With an appendix by J.-L. Colliot-Théléne, J. Reine Angew. Math., 366 (1986), 142--183.Google Scholar
  38. Ka2.
    K. Kato, Semi-stable reduction and p-adic étale cohomology, In: Périodes p-adiques, Bures-sur-Yvette, 1988, Astérisque, 223, Soc. Math. France, Paris, 1994, pp. 269– 293.Google Scholar
  39. KM.
    N.M. Katz and W. Messing, Some consequences of the Riemann hypothesis for varieties over finite fields, Invent. Math., 23 (1974), 73–77.MATHCrossRefMathSciNetGoogle Scholar
  40. MS.
    A.S. Merkurjev and A.A. Suslin, K-cohomology of Severi–Brauer Varieties and the norm residue homomorphism, Math. USSR-Izv., 21 (1983), 307–340.Google Scholar
  41. Mi1.
    J.S. Milne, Étale Cohomology, Princeton Math. Ser., 33, Princeton Univ. Press, Princeton, NJ, 1980.Google Scholar
  42. Mi2.
    J.S. Milne, Values of zeta functions of varieties over finite fields, Amer. J. Math., 108 (1986), 297–360.MATHCrossRefMathSciNetGoogle Scholar
  43. Mi.
    J. Milnor, Algebraic K-theory and quadratic forms, Invent. Math., 9 (1970), 318–344.MATHCrossRefMathSciNetGoogle Scholar
  44. NaKK.
    Y. Nakkajima, Weight filtration and slope filtration on the rigid cohomology of a variety in characteristic p > 0, preprint, 180 p.Google Scholar
  45. NS.
    Y. Nakkajima and A. Shiho, Weight Filtrations on Log Crystalline Cohomologies of Families of Open Smooth Varieties, Lecture Notes in Math., 1959, Springer-Verlag, Berlin, 2008.Google Scholar
  46. Pp.
    Périodes p-adiques, In: Papers from the Seminar Held in Bures-sur-Yvette, 1988, Astérisque, 223, Soc. Math. France, Paris, 1994. pp. 1–397.Google Scholar
  47. RZ.
    M. Rapoport and Th. Zink, Über die lokale Zetafunktion von Shimuravarietäten. Monodromiefiltration und verschwindende Zyklen in ungleicher Charakteristik, Invent. Math., 68 (1982), 21–101.MATHCrossRefMathSciNetGoogle Scholar
  48. Ra.
    W. Raskind, Higher l-adic Abel–Jacobi mappings and filtrations on Chow groups, Duke Math. J., 78 (1995), 33–57.MATHCrossRefMathSciNetGoogle Scholar
  49. Ro.
    M. Rost, Norm varieties and algebraic cobordism, In: Proceedings of the International Congress of Mathematicians, Vol. II, Beijing, 2002, Higher Ed. Press, Beijing, 2002, pp. 77–85.Google Scholar
  50. SaM.
    M. Saito, Monodromy filtration and positivity, preprint, June 25, 2000.Google Scholar
  51. SaS.
    S. Saito, Cohomological Hasse principle for a threefold over a finite field, In: Algebraic K-theory and Algebraic Topology, Lake Louise, AB, 1991, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 407, Kluwer Acad. Publ., Dordrecht, 1993, pp. 229– 241.Google Scholar
  52. SaT1.
    T. Saito, Modular forms and p-adic Hodge theory, Invent. Math., 129 (1997), 607–620.Google Scholar
  53. SaT2.
    T. Saito, Weight spectral sequences and independence of l, J. Inst. Math. Jussieu, 2 (2003), 583–634.Google Scholar
  54. Sou.
    C. Soulé, The rank of étale cohomology of varieties over p-adic or number fields, Compositio Math., 53 (1984), 113–131.MATHMathSciNetGoogle Scholar
  55. SJ.
    A. Suslin and S. Joukhovitski, Norm varieties, J. Pure Appl. Algebra, 206 (2006), 245–276.MATHCrossRefMathSciNetGoogle Scholar
  56. Su.
    N. Suwa, A note on Gersten’s conjecture for logarithmic Hodge–Witt sheaves, Ktheory, 9 (1995), 245–271.Google Scholar
  57. Ta.
    J. Tate, Relations between K 2 and Galois cohomology, Invent. Math., 36 (1976), 257–274.MATHCrossRefMathSciNetGoogle Scholar
  58. Tsu.
    T. Tsuji, p-adic étale cohomology and crystalline cohomology in the semi-stable reduction case, Invent. Math., 137 (1999), 233–411.MATHCrossRefMathSciNetGoogle Scholar
  59. V1.
    V. Voevodsky, Motivic cohomology with Z=2-coefficients, Publ. Math. Inst. Hautes Études Sci., 98 (2003), 59–104.Google Scholar
  60. V2.
    V. Voevodsky, On motivic cohomology with \(\mathbb Z/l\) -coefficients, K-theory Preprint Archives, http://www.math.uiuc.edu/K-theory.
  61. Weib.
    C. Weibel, The norm residue isomorphism theorem, J. Topol., 2 (2009), 346--372.MATHCrossRefMathSciNetGoogle Scholar
  62. Weil.
    A. Weil, Numbers of solutions of equations in finite fields, Bull. Amer. Math. Soc., 55 (1949). 497–508.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© The Mathematical Society of Japan and Springer Japan 2010

Authors and Affiliations

  1. 1.NWF I-MathematikUniversität RegensburgRegensburgGermany

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