Trialitarian groups and the Hasse principle

摘要

Let F be a field of characteristic not=2 such that F((-1)~(1/2)) is of cohomological 2- and 3-dimension <=2. For G a simply connected group of type ~3D_4 or ~6D_4 over F, we show that the natural map H~1(F, G)-> #PI# v implied by #OMEGA#_F H~1 (F_v, G), where #OMEGA#_F is the set of orderings of F and F_v denotes the completion of F at v, restricts to be injective on the image of H~1 (F, Z(G)) in H~1 (F, G). For F not formally real, this implies that Serre's "Conjecture II" [Ser94, III.3.1] holds for such groups if and only if trialitarian groups are classified by their Tits algebras over F.