Relative $(φ, Γ)$-modules and p-adic differential equations
Abstract
Let X be an affinoid rigid analytic space over a p-adic field, equipped with a suitable étale map to a unit polydisk. We provide a formalism of \emph{imperfect relative period rings} over X, which lie inside the corresponding perfect relative period rings constructed by Kedlaya--Liu. Then we establish a relative version of the Fontaine--Cherbonnier--Colmez equivalence between p-adic local systems on X and étale $(φ, Γ)$-modules over such imperfect relative period rings, generalizing previous works of Andreatta--Brinon and others. Using this equivalence, we construct p-adic differential equations attached to de Rham local systems, carrying both geometric and arithmetic differential operators. This generalizes the work of Berger to the relative geometric setting. Along the way, we study a relative Fontaine--Sen theory on the decompletion of Γ-modules over the relative BdR+-period rings.
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BibTeX
@misc{diao2026relative,
title = {{Relative $(φ, Γ)$-modules and $p$-adic differential equations}},
author = {Hansheng Diao and Yong Suk Moon and Zijian Yao},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.08179},
eprint = {2609.08179},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.08179},
}
RIS
TY - UNPB
TI - Relative $(φ, Γ)$-modules and p-adic differential equations
AU - Hansheng Diao
AU - Yong Suk Moon
AU - Zijian Yao
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2609.08179
UR - https://arxiv.org/abs/2609.08179
AB - Let X be an affinoid rigid analytic space over a p-adic field, equipped with a suitable étale map to a unit polydisk. We provide a formalism of \emph{imperfect relative period rings} over X, which lie inside the corresponding perfect relative period rings constructed by Kedlaya--Liu. Then we establish a relative version of the Fontaine--Cherbonnier--Colmez equivalence between p-adic local systems on X and étale $(φ, Γ)$-modules over such imperfect relative period rings, generalizing previous works of Andreatta--Brinon and others. Using this equivalence, we construct p-adic differential equations attached to de Rham local systems, carrying both geometric and arithmetic differential operators. This generalizes the work of Berger to the relative geometric setting. Along the way, we study a relative Fontaine--Sen theory on the decompletion of Γ-modules over the relative B_dR^+-period rings.
ER -