TOTARO’S QUESTION ON ZERO-CYCLES ON G2, F4 AND E6 TORSORS

In a 2004 paper, Totaro asked whether a G-torsor X that has a zero-cycle of degree d > 0 will necessarily have a closed ´etale point of degree dividing d, where G is a connected algebraic group. This question is closely related to several conjectures regarding exceptional algebraic groups. Totaro gave a positive answer to his question in the following cases: G simple, split, and of type G2, type F4, or simply connected of type E6. We extend the list of cases where the answer is ‘yes’ to all groups of type G2 and some nonsplit groups of type F4 and E6. No assumption on the characteristic of the base field is made. The key tool is a lemma regarding linkage of Pfister forms.