Separable Pseudo-reductive Bands with Applications to Rational Points

Abstract

We extend the Galois-theoretic Borovoi-Springer interpretation of algebraic bands to a class of étale-locally represented bands on the fppf site of an arbitrary field $k$, which we call separable bands. Next, a band represented étale-locally over $k$ by a pseudo-reductive group is shown to be globally representable when $[k : k^p] = p$, with counterexamples in general. When $k$ is a global or local field, we deduce a generalization of Borovoi's abelianization theory to separable bands represented by smooth connected algebraic groups. As an application, we prove that the Brauer-Manin obstruction is the only obstruction to both the Hasse principle and weak approximation on a homogeneous space of a pseudo-reductive group (more generally, of a smooth connected affine algebraic group with split unipotent radical) having a smooth connected geometric stabilizer.