Ruochuan Liu

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

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

Slope filtrations in families

{\mathrm {B}}_{{\text {dR}}}

Triangulation of refined families

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.

Relative p-adic Hodge theory, II: Imperfect period rings

We prove that the eigencurve associated to a definite quaternion algebra over

Eigencurve over the boundary of the weight space

\mathbb Q

On families of phi, Gamma-modules

satisfies the following properties, as conjectured by Coleman-Mazur and Buzzard-Kilford: (a) over the boundary annuli of the weight space, the eigencurve is a disjoint union of (countably) infinitely many connected components each finite and flat over the weight annuli, (b) the

Relative p-adic Hodge theory, II: (phi, Gamma)-modules

U_p

Finiteness of cohomology of local systems on rigid analytic spaces

-slopes of points on each fixed connected component are proportional to the