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

Abstract

We prove that any geometrically irreducible $\overline{\mathbb{Q}}_p$-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 $\overline{\mathbb{Q}}_p$-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-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 pro-algebraic completion of the fundamental group is de Rham.