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

TL;DR

This paper develops a functor linking p-adic etale local systems to vector bundles with integrable connections, establishing a p-adic Riemann-Hilbert correspondence and demonstrating rigidity properties for p-adic local systems with applications to Shimura varieties.

Contribution

It constructs a functor from p-adic local systems to vector bundles with connections, advancing the p-adic Riemann-Hilbert correspondence and proving a rigidity theorem.

Findings

Established a functor from p-adic local systems to vector bundles with connections.

Proved a rigidity theorem for de Rham p-adic local systems.

Applied results to Shimura varieties.

Abstract

We construct a functor from the category of p-adic etale local systems on a smooth rigid analytic variety X over a p-adic field to the category of vector bundles with an integrable connection over its "base change to B_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.