$p$-adic Riemann-Hilbert correspondence for local systems

I will explain the construction of $p$-adic Riemann-Hilbert functor by R. Liu and X. Zhu. For a $Q_p$-etale local system on a rigid-analytic variety $X$ over a $p$-adic field K this functor gives a vector bundle with flat connection on a ringed space which can be thought of as the base change of $X$ to the field $B_dR(K^{cyc})$. I will also explain the construction of an operator $\phi$ on this vector bundle due to K. Shimizu which is a relative analogue of the Sen-Fontaine operator. In the first part of the talk I will recall Sen-Fontaine theory and in particular the construction of $\phi$ in the zero-dimensional case.