Given a smooth scheme $X$ over the ring of integers of a $p$-adic field, we introduce the notion of a relative Breuil-Kisin-Fargues module $M$ on $X$. Each such $M$ simultaneously encodes the data of a lisse étale sheaf, a module with flat connection, and a crystal, whose cohomologies are then intertwined by a relative form of the $A_{inf}$ cohomology introduced in "Integral $p$-adic Hodge theory" by Bhatt-M-Scholze. They are moreover closely related to other work in relative $p$-adic Hodge theory, notably Faltings small generalised representations and his relative Fontaine Lafaille theory. Joint with Takeshi Tsuji.[-]

Given a $p$-adic local system $L$ on a smooth algebraic variety $X$ over a finite extension $K$ of $Q_{p}$, it is always possible to find a de Rham local system $M$ on $X$ such that the underlying local system $\left.L\right|_{X_{\bar{K}}}$ embeds into $\left.M\right|_{X_{\bar{K}}}$. I will outline the proof that relies on the $p$-adic Riemann-Hilbert correspondence of Diao-Lan-Liu-Zhu. As a consequence, the action of the Galois group $G_{K}$ on the pro-algebraic completion of the étale fundamental group of $X_{\bar{K}}$ is de Rham, in the sense that every finite-dimensional subrepresentation of the ring of regular functions on that group scheme is de Rham. This implies that every finite-dimensional subrepresentation of the ring of regular functions on the pro-algebraic completion of the geometric pi $i_{1}$ of a smooth variety over a number field satisfies the assumptions of the Fontaine-Mazur conjecture. Complementing this result, I will sketch a proof of the fact that every semi-simple representation of $G a l(\bar{Q} / Q)$ arising from geometry is a subquotient of the ring of regular functions on the pro-algebraic completion of the fundamental group of the projective line with 3 punctures.[-]