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    Garibaldi, S. and Semenov, N. 2010. Degree 5 Invariant of E8. International Mathematics Research Notices,


    Gille, Philippe and Semenov, Nikita 2010. Zero-cycles on projective varieties and the norm principle. Compositio Mathematica, Vol. 146, Issue. 02, p. 457.


    Berhuy, Grégory Frings, Christoph and Tignol, Jean-Pierre 2007. Galois cohomology of the classical groups over imperfect fields. Journal of Pure and Applied Algebra, Vol. 211, Issue. 2, p. 307.


    Chernousov, V. Gille, P. and Reichstein, Z. 2006. Resolving G-torsors by abelian base extensions. Journal of Algebra, Vol. 296, Issue. 2, p. 561.


    Colliot-Th�l�ne, Jean-Louis 2005. Un th�or�me de finitude pour le groupe de Chow des z�ro-cycles d?un groupe alg�brique lin�aire sur un corps p-adique. Inventiones mathematicae, Vol. 159, Issue. 3, p. 589.


    Borovoi, Mikhail Kunyavskiı̆, Boris and Gille, Philippe 2004. Arithmetical birational invariants of linear algebraic groups over two-dimensional geometric fields. Journal of Algebra, Vol. 276, Issue. 1, p. 292.


    Colliot-Thélène, Jean-Louis 2004. Résolutions flasques des groupes réductifs connexes. Comptes Rendus Mathematique, Vol. 339, Issue. 5, p. 331.


    Colliot-Thélène, Jean-Louis Gille, Philippe and Parimala, Raman 2001. Arithmétique des groupes algébriques linéaires sur certains corps géométriques de dimension deux. Comptes Rendus de l'Académie des Sciences - Series I - Mathematics, Vol. 333, Issue. 9, p. 827.


    Reichstein, Z. and Youssin, B. 2001. On a property of special groups. Transformation Groups, Vol. 6, Issue. 3, p. 261.


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Cohomologie galoisienne des groupes quasi-déployés sur des corps de dimension cohomologique ≤2; Galois cohomology of quasi-split groups over fields of cohomological dimension ≤ 2

  • Philippe Gille (a1)
  • DOI: http://dx.doi.org/10.1023/A:1002473132282
  • Published online: 01 February 2001
Abstract

Let k be a perfect field with cohomological dimension [les ] 2. Serre's conjecture II claims that the Galois cohomology set H1(k,G) is trivial for any simply connected semi-simple algebraic G/k and this conjecture is known for groups of type 1An after Merkurjev–Suslin and for classical groups and groups of type F4 and G2 after Bayer–Parimala. For any maximal torus T of G/k, we study the map H1(k, T) → H1(k, G) using an induction process on the type of the groups, and it yields conjecture II for all quasi-split simply connected absolutely almost k-simple groups with type distinct from E8. We also have partial results for E8 and for some twisted forms of simply connected quasi-split groups. In particular, this method gives a new proof of Hasse principle for quasi-split groups over number fields including the E8-case, which is based on the Galois cohomology of maximal tori of such groups.

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Compositio Mathematica
  • ISSN: 0010-437X
  • EISSN: 1570-5846
  • URL: /core/journals/compositio-mathematica
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