Abelian Galois cohomology of quasi-connected reductive groups

Source abstract

Abstract In 1999 Labesse introduced quasi-connected reductive groups and investigated their abelian Galois cohomology over local and global fields of characteristic 0. We (1) generalize some of the constructions of Labesse from quasi-connected reductive groups to arbitrary reductive groups, not necessarily connected or quasi-connected; (2) generalize results of Labesse on the abelian Galois cohomology of quasi-connected reductive groups to the case of local and global fields of arbitrary characteristic; and (3) investigate the functoriality properties of the abelian Galois cohomology. In particular, we introduce the notion of a principal homomorphism of quasi-connected reductive groups, and show that if $G$ is a quasi-connected reductive group over a local or global field $k$ of positive characteristic, then the first Galois cohomology set $H^{1}(k,G)$ has a canonical abelian group structure, which is functorial with respect to principal homomorphisms.