Cohomologie galoisienne des groupes quasi-déployés sur des corps de dimension cohomologique [less-than-or-equal]2; Galois cohomology of quasi-split groups over fields of cohomological dimension [less-than-or-equal] 2
Let k be a perfect field with cohomological dimension [less-than-or-eq, slant] 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) [rightward arrow] 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.
Key Words: Galois cohomology; semisimple groups; Bruhat–Tits theory; Hasse principle.