Geometrically irreducible p-adic local systems are de Rham up to a twist

Abstract

We prove that any geometrically irreducible Q‾p-local system on a smooth algebraic 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 Q‾p-local system on a smooth variety over a number field the associated projective representation 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 decompletions 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 π1ét is de Rham.