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A p-adic Riemann--Hilbert correspondence over Robba rings

  • Zekun Chen
  • Ruochuan Liu
  • Yupeng Wang
  • Jiahong Yu
  • Xinwen Zhu
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Abstract

We construct overconvergent period sheaves 𝒪ℂ̃I,† and 𝒪𝔹dR†,+ on the v-site of a liftable semi-stable formal scheme XZ. By establishing a decompletion theorem for small representations, we prove a p-adic Riemann-Hilbert correspondence relating small ℂ̃I-local systems (resp. 𝔹dR+-local systems) on the generic fiber XZ to small flat connections on the associated period ringed spaces. We further analyze the local Frobenius action and show that this correspondence is compatible with the Frobenius structure. As an application, we prove a comparison theorem between the cohomology of these period sheaves and the rigid cohomology of the special fiber, refining results of Le Bras and Colmez--Nizioł.

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Cite

BibTeXRIS
BibTeX
@misc{chen2026padic,
  title = {{A $p$-adic Riemann--Hilbert correspondence over Robba rings}},
  author = {Zekun Chen and Ruochuan Liu and Yupeng Wang and Jiahong Yu and Xinwen Zhu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.21065},
  eprint = {2610.21065},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.21065},
}
RIS
TY  - UNPB
TI  - A p-adic Riemann--Hilbert correspondence over Robba rings
AU  - Zekun Chen
AU  - Ruochuan Liu
AU  - Yupeng Wang
AU  - Jiahong Yu
AU  - Xinwen Zhu
PY  - 2026
DA  - 2026/10/15
PB  - arXiv
DO  - 10.48550/arxiv.2610.21065
UR  - https://arxiv.org/abs/2610.21065
AB  - We construct overconvergent period sheaves 𝒪ℂ̃^I,† and 𝒪𝔹_dR^†,+ on the v-site of a liftable semi-stable formal scheme X_Z. By establishing a decompletion theorem for small representations, we prove a p-adic Riemann-Hilbert correspondence relating small ℂ̃^I-local systems (resp. 𝔹_dR^+-local systems) on the generic fiber X_Z to small flat connections on the associated period ringed spaces. We further analyze the local Frobenius action and show that this correspondence is compatible with the Frobenius structure. As an application, we prove a comparison theorem between the cohomology of these period sheaves and the rigid cohomology of the special fiber, refining results of Le Bras and Colmez--Nizioł.
ER  -