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Rigidity and a Riemann–Hilbert correspondence for p-adic local systems

Abstract

We construct a functor from the category of p-adic étale local systems on a smooth rigid analytic variety X over a p-adic field to the category of vector bundles with an integrable connection on its “base change to \({\mathrm {B}}_{{\text {dR}}}\)”, which can be regarded as a first step towards the sought-after p-adic Riemann–Hilbert correspondence. As a consequence, we obtain the following rigidity theorem for p-adic local systems on a connected rigid analytic variety: if the stalk of such a local system at one point, regarded as a p-adic Galois representation, is de Rham in the sense of Fontaine, then the stalk at every point is de Rham. Along the way, we also establish some basic properties of the p-adic Simpson correspondence. Finally, we give an application of our results to Shimura varieties.

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Notes

  1. 1.

    It is pointed out by Abbes the subtlety to compare our construction with the construction of Abbes-Gros’ in [2], although we believe that they are essentially the same.

  2. 2.

    We use Z instead of Y as in the statement of the theorem since Y has another meaning according to our previous convention.

  3. 3.

    In relative p-adic Hodge theory, the \({\tilde{\mathbf{C }}}\) and \(\mathbf C \) types of rings refer to relative version of the perfect and imperfect Robba rings in classical p-adic Hodge theory respectively.

  4. 4.

    We learned that L. Fargues independently observed this.

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Acknowledgments

The authors thank A. Abbes, P. Colmez, K. S. Kedlaya, K.-W. Lan, P. Scholze and Y. Tian for valuable discussions, and Koji Shimizu for very careful reading of this paper and useful comments. Parts of the work were done while the first author was staying at California Institute of Technology and the second author was staying at Beijing International Center for Mathematical Research. The authors would like to thank these institutions for their hospitality.

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Correspondence to Xinwen Zhu.

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R. Liu is partially supported by NSFC-11571017 and the Recruitment Program of Global Experts of China. X. Zhu is partially supported by NSF DMS-1303296/1535464 and a Sloan Fellowship.

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Liu, R., Zhu, X. Rigidity and a Riemann–Hilbert correspondence for p-adic local systems. Invent. math. 207, 291–343 (2017). https://doi.org/10.1007/s00222-016-0671-7

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