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

Abstract: Let $K$ be a p-adic field. Although there are plenty of non-de Rham representations of the Galois group of $K$, it turns out that for any smooth variety $X$ over $K$ and a $\overline{\mathbf{Q}}_p$-local system $L$ on $X$ such that the restriction of $L$ to $X_{\overline{K}}$ is irreducible, there exists a character of the Galois group of $K$ such that twisting by this character turns $L$ into a de Rham local system. In particular, for a geometrically irreducible $\overline{\mathbf{Q}}_p$-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 Riemann-Hilbert correspondence in the form constructed by Liu and Zhu as well as its logarithmic version constructed by Diao-Lan-Liu-Zhu and their decompletions developed by Shimizu.

commutative algebraalgebraic geometrynumber theory

Audience: researchers in the topic

Recent Advances in Modern p-Adic Geometry (RAMpAGe)

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