Implications of the Hasse Principle for Zero Cycles of Degree One on Principal Homogeneous Spaces

TL;DR

This paper proves that for certain algebraic groups over perfect fields with specific cohomological properties, the existence of a zero cycle of degree one on a principal homogeneous space guarantees a rational point.

Contribution

It establishes a link between zero cycles of degree one and rational points for principal homogeneous spaces under specific algebraic groups over fields with cohomological constraints.

Findings

Zero cycle of degree one implies rational point under given conditions

Hasse principle for the simply connected cover $G^{sc}$ is crucial

Results apply to perfect fields with virtual cohomological dimension ≤ 2

Abstract

Let $k$ be a perfect field of virtual cohomological dimension $\leq 2$. Let $G$ be a connected linear algebraic group over $k$ such that $G^{sc}$ satisfies a Hasse principle over $k$. Let $X$ be a principal homogeneous space under $G$ over $k$. We show that if $X$ admits a zero cycle of degree one, then $X$ has a $k$-rational point.