Appears in collection : Franco-Asian Summer School on Arithmetic Geometry

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 $L|_{X_{\overline{K}}}$ embeds into $M|_{X_{\overline{K}}}$. 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 $X_{\overline{K}}$ 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 $\pi_1$ 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 $Gal(\bar{Q}/Q)$ 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.