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

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 ¹A_n 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.