Totaro's Question for Tori of Low Rank

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 $d\ge 1$, then does $X$ have a closed étale 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 $\le 2$.

arXiv comment22 pages. Minor updates: abstract, introduction