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

By Ruochuan Liu, Xinwen Zhu. 2016

113 citations recorded by the source corpus.

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 $${\mathrm {B}}_{{\text {dR}}}$$BdR”, 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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/9bd1c466173ee8a32cbb8a9ab12010048660f4f4

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

On locally analytic vectors of the completed cohomology of modular curves (2020, methodology): “on, we will show that taking locally analytic vectors is exact under certain conditions. Results of this section are closely related to the work of Faltings [Fal05], Abbes-Gros-Tsuji [AGT16], Liu-Zhu [LZ17] on p-adic Simpson correspondences. See Remark 3.1.7 below. 3.1. Statement of the main result.”
Logarithmic Riemann-Hilbert correspondences for rigid varieties. (2018, methodology): “ically isomorphic. In addition, (Dalg dR(L|XE v ) ⊗Ev,ι C,Fil •) is a complex variation of Hodge structures. This is closely related to a relative version of the Fontaine–Mazur conjecture proposed in [LZ17], but it might be more approachable because it is stated purely in terms of sheaves. Even so, ”