A counterexample to an optimistic guess about \'etale local systems

TL;DR

This paper provides a counterexample to a conjecture in p-adic Hodge theory, showing that an étale local system on a rigid analytic variety does not necessarily become semistable after a finite extension, contrary to an optimistic guess.

Contribution

The paper constructs a specific counterexample disproving the conjecture that étale local systems become semistable after finite base field extensions in the relative setting.

Findings

Counterexample to the conjecture in p-adic Hodge theory

Étale local systems may not become semistable after finite extension

Challenges assumptions about relative semistability in p-adic geometry

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 \'etale 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.