Totaro's Question for Tori of Low Rank

TL;DR

This paper affirms Totaro's question for algebraic tori of rank at most 2, establishing a link between zero-cycles and étale points in this context.

Contribution

It provides the first positive answer to Totaro's question specifically for low-rank algebraic tori, filling a gap in the literature.

Findings

Confirmed Totaro's question for tori of rank ≤ 2

Established existence of étale points dividing the degree of zero-cycles

Extended understanding of torsor properties in algebraic geometry

Abstract

Let $G$ be a smooth connected linear algebraic group and $X$ be a $G$-torsor. Totaro asked: if $X$ admits a zero-cycle of degree $d \geq 1$, then does $X$ have a closed \'etale point of degree dividing $d$? This question is entirely unexplored in the literature for algebraic tori. We settle Totaro's question affirmatively for algebraic tori of rank $\leq 2$.