Abelian Galois cohomology of quasi-connected reductive groups

TL;DR

This paper extends Labesse's work on abelian Galois cohomology from quasi-connected to arbitrary reductive groups, exploring functoriality and structures over various fields.

Contribution

It generalizes constructions and results of Labesse to broader classes of reductive groups and investigates the functoriality of their Galois cohomology.

Findings

H^1(k,G) has a canonical abelian group structure for quasi-connected reductive groups over fields of positive characteristic.

Generalization of Labesse's results to arbitrary characteristic fields.

Introduction of principal homomorphisms and their role in functoriality.

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 structure of abelian group, which is functorial with respect to *principal* homomorphisms.