Zero Cycles of Degree One on Principal Homogeneous Spaces

TL;DR

This paper proves that under certain conditions, principal homogeneous spaces with a zero cycle of degree one over specific algebraic groups necessarily have a rational point, advancing understanding in algebraic geometry and group theory.

Contribution

It establishes that for certain semisimple algebraic groups, zero cycles of degree one imply the existence of rational points on principal homogeneous spaces.

Findings

Zero cycles of degree one imply rational points for specified algebraic groups.

Conditions exclude groups with E8 factors and non-quasisplit exceptions.

Results apply over fields of characteristic not 2.

Abstract

Let $k$ be a field of characteristic different from 2. Let $G$ be a simply connected or adjoint semisimple algebraic $k$-group which does not contain a simple factor of type $E_8$ and such that every exceptional simple factor of type other than $G_2$ is quasisplit. We show that if a principal homogeneous space under $G$ over $k$ admits a zero cycle of degree 1 then it has a $k$-rational point.