Rigidity and a Riemann-Hilbert correspondence for \(p\)-adic local systems. (English) Zbl 1375.14090

Liu, Ruochuan; Zhu, Xinwen

Invent. Math. 207, No. 1, 291-343 (2017).

The subject of this paper is relative \(p\)-adic Hodge theory. The main result (theorem 1.4) is the construction of a functor from the category of \(p\)-adic étale local systems on a smooth rigid analytic variety over a \(p\)-adic field, to the category of vector bundles with an integrable connection on the “base change to \(\mathrm{B}_{dR}\)” of the variety. This construction can be seen as a kind of Riemann-Hilbert correspondence.
A consequence of this construction is the fact (Theorem 1.1) that if the stalk of such a local system at one point is a de Rham \(p\)-adic Galois representation, then the stalk at every point is de Rham. In particular, geometric families of \(p\)-adic Galois representations are much more rigid than their arithmetic counterparts. A corollary of this rigidity result (Theorem 1.2) is that some representations coming from Shimura data are geometric in the sense of Fontaine-Mazur. The paper also contains (Theorem 1.6) an application of these constructions to the \(p\)-adic Simpson correspondence.
There is a detailed informative introduction that clearly explains the main results of the paper.

Reviewer: Laurent Berger (Lyon)

MSC:

14G22	Rigid analytic geometry
14G35	Modular and Shimura varieties
14J60	Vector bundles on surfaces and higher-dimensional varieties, and their moduli
11S15	Ramification and extension theory

Keywords:

rigid analytic variety; Riemann-Hilbert correspondence; \(p\)-adic local system; relative \(p\)-adic Hodge theory; de Rham representation; \(p\)-adic Simpson correspondence

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