Advertisement

Mathematische Zeitschrift

, Volume 270, Issue 3–4, pp 1057–1065 | Cite as

On Galois cohomology of semisimple groups over local and global fields of positive characteristic, II

  • Nguyêñ Quôć ThǎńgEmail author
Article

Abstract

We show that the recent results of Prasad and Rapinchuk (Adv. Math. 207(2), 646–660, 2006) on the existence and uniqueness of certain global forms of semisimple algebraic groups with given local behaviour in the case of number fields still hold in the case of global function fields.

Keywords

Forms of linear algebraic groups Galois cohomology Flat cohomology 

Mathematics Subject Classification (2000)

Primary 11E72 Secondary 18G50 20G10 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Borel A., Harder G.: Existence of cocompact subgroups of reductive groups over local fields. J. reine und angew. Math. Bd. 298, 53–64 (1978)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Bruhat F., Tits J.: Groupes algébriques sur un corps local. Chapitre III. Compléments et applications à la cohomologie galoisienne. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 34, 671–698 (1987)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Demazure M., Gabriel P.: Groupes algébriques. Tom 1, Paris (1970)zbMATHGoogle Scholar
  4. 4.
    Douai, J.-C.: 2-Cohomologie galoisienne des groupes semi-simples, Thèse, Université des Sciences et Tech. de Lille 1 (1976)Google Scholar
  5. 5.
    Frey G.: Dualitätssatz für endlichen kommutative Gruppenschemata über Kongruenzfunktionenkörpern. J. reine und angew. Math. Bd. 301, 46–58 (1978)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Gille P.: La R-équivalence sur les groupes réductifs définis sur un corps de nombres. Pub. Math. I. H. E. S., t. 86, 199–235 (1997)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Giraud J.: Cohomologie non-abélienne, Grund. des Math. Wiss. Bd, vol. 179. Springer-Verlag, Berlin (1971)Google Scholar
  8. 8.
    Harder G.: Über die Galoiskohomologie halbeinfacher algebraischer Gruppen. III. J. reine angew. Math. Bd. 274/275, 125–138 (1975)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Kneser, M.: Lecture on Galois cohomology of classical groups, Tata Inst. Fund. Res. (1969)Google Scholar
  10. 10.
    Mazur B.: On the passage from local to global in number theory. Bull. Amer. Math. Soc. (N.S.) 29(1), 14–50 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Milne J.: Arithmetic duality theorems, 2nd ed. BookSurge, Charleston (2006)zbMATHGoogle Scholar
  12. 12.
    Neukirch J., Schmidt A., Wingberg K.: Cohomology of number fields. 2nd edn. Grund. der Math. Wiss., Bd, vol. 323. Springer-Verlag, Berlin (2008)Google Scholar
  13. 13.
    Platonov V., Rapinchuk A.: Algebraic groups and Number theory. Academic Press, Dublin (1993)Google Scholar
  14. 14.
    Poitou, G. (ed.): Cohomologie galoisienne des modules finis. Séminaire de l’Institut de Mathématiques de Lille, sous la direction de G. Poitou. Travaux et Recherches Mathématiques, No. 13 Dunod, Paris (1967)Google Scholar
  15. 15.
    Prasad G., Rapinchuk A.: On the existence of isotropic forms of semi-simple algebraic groups over number fields with prescribed local behavior. Adv. Math. 207(2), 646–660 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Serre, J.-P.: Cohomologie galoisienne, Lecture Notes in Math. vol. 5, 5th éd. Springer-Verlag, Berlin (1995)Google Scholar
  17. 17.
    Demazure, M., Grothendieck, A. (éd.): Schémas en groupes, Lecture Notes in Math, vol. 151–153. Springer-Verlag, Berlin (1970)Google Scholar
  18. 18.
    Shatz S.: Cohomology of Artinian group schemes over local fields. Annals of Mathematics 79, 411–449 (1964)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Shatz S.: Profinite groups: arithmetic and geometry, Annals of Math. Studies, vol. 67. Princeton University Press, New Jersy (1972)Google Scholar
  20. 20.
    Thǎńg N.Q.: On Galois cohomology of semisimple groups over local and global fields of positive characteristic. Math. Z. Bd. 259(2), 457–470 (2008)Google Scholar
  21. 21.
    Thǎńg N.Q., Tân N.D.: On the Galois and flat cohomology of unipotent algebraic groups over local and global function fields. I. J. Algebra 319, 4288–4324 (2008)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Tits, J.: Classification of algebraic semisimple groups, in “Algebraic Groups and Discontinuous Subgroups” (Proc. Sympos. Pure Math., vol. 9, Boulder, Colo., 1965) pp. 33–62 Amer. Math. Soc., Providence, R.I., (1966)Google Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Institute of MathematicsHanoiVietnam

Personalised recommendations