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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 ℂp, where G is a compact p-adic Lie group. We construct an affinoid perfectoid ℤp2-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.
Related
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BibTeX
@misc{qiu2026surjectivity,
title = {{On the Surjectivity of the Geometric Sen Map of Perfectoid Torsors}},
author = {Tian Qiu and Jiahong Yu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.08287},
eprint = {2609.08287},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.08287},
}
RIS
TY - UNPB
TI - On the Surjectivity of the Geometric Sen Map of Perfectoid Torsors
AU - Tian Qiu
AU - Jiahong Yu
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.08287
UR - https://arxiv.org/abs/2609.08287
AB - We study surjectivity of the geometric Sen map for perfectoid G-torsors over smooth rigid analytic spaces over ℂ_p, where G is a compact p-adic Lie group. We construct an affinoid perfectoid 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.
ER -