A Rigid Analytic Curve Can Have a Dense p-adic Orbit
Abstract
Let 𝒟 be a product of Drinfeld upper half spaces and let J be the corresponding product of projective linear groups acting on 𝒟. We prove that if Y is a reduced, quasi-compact separated strict ℂp-analytic space and f: Y→𝒟 is analytic with dim Y<dim𝒟, then J f(Y(ℂp)) is closed and nowhere dense in 𝒟(ℂp). 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/ℚp and d≥3, we construct a closed smooth irreducible non-quasi-compact curve in ΩFd−1 with dense orbit under a fixed diagonal one-parameter subgroup. We also construct a ℚp-defined analytic map from the open unit disc to P2 with dense PGL3(ℚp)-orbit, and an analytic map from a one-dimensional affinoid perfectoid torus to P2 whose PGL3(ℚp)-orbit closure has nonempty interior.
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BibTeX
@misc{fu2026rigid,
title = {{A Rigid Analytic Curve Can Have a Dense $p$-adic Orbit}},
author = {Yu Fu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.12759},
eprint = {2610.12759},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.12759},
}
RIS
TY - UNPB
TI - A Rigid Analytic Curve Can Have a Dense p-adic Orbit
AU - Yu Fu
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.12759
UR - https://arxiv.org/abs/2610.12759
AB - Let 𝒟 be a product of Drinfeld upper half spaces and let J be the corresponding product of projective linear groups acting on 𝒟. We prove that if Y is a reduced, quasi-compact separated strict ℂ_p-analytic space and f: Y→𝒟 is analytic with dim Y<dim𝒟, then J f(Y(ℂ_p)) is closed and nowhere dense in 𝒟(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/ℚ_p and d≥3, 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 ℚ_p-defined analytic map from the open unit disc to P^2 with dense PGL_3(ℚ_p)-orbit, and an analytic map from a one-dimensional affinoid perfectoid torus to P^2 whose PGL_3(ℚ_p)-orbit closure has nonempty interior.
ER -