A Perfectoid Pro-Étale Lie Torsor Can Have a Non-Surjective Sen Map

Source abstract

We construct a smooth connected rigid analytic curve over $\mathbb C_p$ with an affinoid perfectoid profinite étale $\mathbb Z_p^2$-torsor whose geometric Sen morphism is not surjective, thereby disproving Rodr\'ıguez Camargo's conjectured implication from perfectoidness to surjectivity. The construction relies on a pullback theorem showing that perfectoid profinite étale towers are preserved under universally injective morphisms of rigid analytic spaces.