Title: Separable Pseudo-reductive Bands with Applications to Rational Points Show Chinese title

Title: 可分伪约化带及其在有理点中的应用

Abstract: We extend the Galois-theoretic Borovoi-Springer interpretation of algebraic bands to a class of \'etale-locally represented bands on the fppf site of an arbitrary field $k$, which we call separable bands. Next, a band represented \'etale-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 the Hasse principle for 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. Show Chinese abstract

Abstract: 我们将Galois理论的Borovoi-Springer对代数带的解释扩展到一个在任意域的fppf层上的étale局部表示的带类$k$，我们称之为可分带。 接下来，由伪约化群在$k$上局部表示的带被证明在$[k : k^p] = p$时是全局可表示的，一般情况下存在反例。 当$k$是一个全局或局部域时，我们推导出Borovoi的阿贝尔化理论到由光滑连通代数群表示的可分带的一般推广。 作为应用，我们证明对于具有光滑连通几何稳定子的伪约化群（更一般地，具有分裂不可约余-radical的光滑连通仿射代数群）的齐次空间，布劳尔-马宁障碍是哈塞原理唯一的障碍。