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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)
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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