Log-crystalline representations and $(φ, Γ)$-modules
Abstract
Let F be an absolutely unramified mixed characteristic local field with rings of integers OF. For a small affine algebra R over OF, we consider the logarithmic étale fundamental group GR of its generic fibre equipped with a "horizontal" log-structure, in the sense of Fujiwara--Kato. We study p-adic representations of GR and obtain a classification of these representations in terms of étale (φ, ΓR)-modules. Moreover, we define and study the notion of log-crystalline representations of GR, a generalisation of crystalline representations from the non-logarithmic/smooth case. Furthermore, in the logarithmic setting, we show that log-crystalline representations of GR are equivalent to Wach modules for R, extending our previous results from the (non-logarithmic) crystalline case.
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BibTeX
@misc{abhinandan2026logcrystalline,
title = {{Log-crystalline representations and $(φ, Γ)$-modules}},
author = {Abhinandan and Denis Benois},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.11266},
eprint = {2609.11266},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.11266},
}
RIS
TY - UNPB
TI - Log-crystalline representations and $(φ, Γ)$-modules
AU - Abhinandan
AU - Denis Benois
PY - 2026
DA - 2026/09/10
PB - arXiv
DO - 10.48550/arxiv.2609.11266
UR - https://arxiv.org/abs/2609.11266
AB - Let F be an absolutely unramified mixed characteristic local field with rings of integers O_F. For a small affine algebra R over O_F, we consider the logarithmic étale fundamental group G_R of its generic fibre equipped with a "horizontal" log-structure, in the sense of Fujiwara--Kato. We study p-adic representations of G_R and obtain a classification of these representations in terms of étale (φ, Γ_R)-modules. Moreover, we define and study the notion of log-crystalline representations of G_R, a generalisation of crystalline representations from the non-logarithmic/smooth case. Furthermore, in the logarithmic setting, we show that log-crystalline representations of G_R are equivalent to Wach modules for R, extending our previous results from the (non-logarithmic) crystalline case.
ER -