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

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• 作者：Ruochuan Liu ; Xinwen Zhu
• 刊名：Inventiones mathematicae
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：207
• 期：1
• 页码：291-343
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics, general;
• 出版者：Springer Berlin Heidelberg
• ISSN：1432-1297
• 卷排序：207

文摘

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.