A counterexample to an optimistic guess about étale local systems

2020/02/06 by Brian Lawrence, Lawrence, Brian, Shizhang Li +1 · 1 citation
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Number Theory (math.NT) #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2002.03799

openalex publication_date 2020/02/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In p-adic Hodge theory, it is known that if a Galois representation is de Rham, then it becomes semistable after extension of the base field. Liu and Zhu asked whether a corresponding result holds in the relative setting: given an étale local system on a quasi-compact rigid analytic variety (for example, a projective scheme) over a p-adic field, does it become semistable after a finite extension of the base field? We give a counterexample.

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