Potential semistability of Finite height Galois representations: Relative case

Abstract: Let $K$ be a $p$-adic field. We define the notion of finite height for an étale $\mathbb{Z}_p$-local system on a smooth adic space $\mathcal{X}$ over $K$ with semistable reduction. Using analytic prismatic $F$-crystals and purity results of Du-Liu-Moon-Shimizu (arXiv:2404.19603), we prove that if an étale $\mathbb{Z}_p$-local system over $\mathcal{X}$ is of finite height then its pullback along a finite étale cover of $\mathcal{X}$ is semistable. This answers a question of Tong Liu in the relative setting.