A Rigid Analytic Curve Can Have a Dense $p$-adic Orbit

Abstract

Let $\mathcal D$ be a product of Drinfeld upper half spaces and let $J$ be the corresponding product of projective linear groups acting on $\mathcal D$. We prove that if $Y$ is a reduced, quasi-compact separated strict $\mathbb C_p$-analytic space and $f: Y\to\mathcal D$ is analytic with $\dim Y<\dim\mathcal D$, then $J f(Y(\mathbb C_p))$ is closed and nowhere dense in $\mathcal D(\mathbb C_p)$. The quasi-compactness assumption is necessary even when the source is a strict rigid analytic curve, while a quasi-compact perfectoid source can still have an orbit closure with nonempty interior. For every finite extension $F/\mathbb Q_p$ and $d\geq3$, we construct a closed smooth irreducible non-quasi-compact curve in $Ω_F^{d-1}$ with dense orbit under a fixed diagonal one-parameter subgroup. We also construct a $\mathbb Q_p$-defined analytic map from the open unit disc to $\mathbf P^2$ with dense $\operatorname{PGL}_3(\mathbb Q_p)$-orbit, and an analytic map from a one-dimensional affinoid perfectoid torus to $\mathbf P^2$ whose $\operatorname{PGL}_3(\mathbb Q_p)$-orbit closure has nonempty interior.