GEOMETRICALLY IRREDUCIBLE p-ADIC LOCAL SYSTEMS ARE DE RHAM UP TO A TWIST

We prove that any geometrically irreducible Qp-local system on a smooth alge-braic variety over a p-adic field K becomes de Rham after a twist by a character of the Galois group of K. In particular, for any geometrically irreducible Qp-local system on a smooth variety over a number field the associated projective representa-tion of the fundamental group automatically satisfies the assumptions of the relative Fontaine-Mazur conjecture. The proof uses p-adic Simpson and Riemann-Hilbert correspondences of Diao, Lan, Liu, and Zhu and the Sen operator on the decomple-tions of those developed by Shimizu. Along the way, we observe that a p-adic local system on a smooth geometrically connected algebraic variety over K is Hodge-Tate if its stalk at one closed point is a Hodge-Tate Galois representation. Moreover, we prove a version of the main theorem for local systems with arbitrary geometric monodromy, which allows us to conclude that the Galois action on the proalgebraic completion of net 1 is de Rham.