On the Surjectivity of the Geometric Sen Map of Perfectoid Torsors

Abstract

We study surjectivity of the geometric Sen map for perfectoid $G$-torsors over smooth rigid analytic spaces over ${\mathbb{C}}_p$, where $G$ is a compact $p$-adic Lie group. We construct an affinoid perfectoid ${\mathbb{Z}}_p^2$-torsor over a smooth curve whose Sen map is not surjective, thereby disproving Rodríguez Camargo's conjectured implication from perfectoidness to surjectivity. We prove that the Sen map of every perfectoid $G$-torsor over a smooth curve is nevertheless surjective on a Zariski open dense subset. Finally, we show that locally analytic decompletion implies Sen surjectivity for diamondian affinoid perfectoid torsors.