Totaro’s question for adjoint groups of types 𝐴₁ and 𝐴_{2𝑛}

Abstract

Let G G be a smooth connected linear algebraic group over a field k k , and let X X be a G G -torsor. Totaro asked: if X X admits a zero-cycle of degree d ≥ 1 d \geq 1 , then does X X have a closed étalé point of degree dividing d d ? We give an affirmative answer for absolutely simple classical adjoint groups of types A 1 A_1 and A 2 n A_{2n} over fields of characteristic ≠ 2 \neq 2 .