Potential semistability of Finite height Galois representations: Relative case

2026/06/30 by Kaustabh Mondal
Mathematics · #math.NT #math.AG #msc:11F80 #msc:14F30 #msc:14G22

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Revised version; comments are welcome!

arxiv created 2026/07/30 · arxiv updated 2026/07/31

Abstract

Let K be a p-adic field. We define the notion of finite height for an étale ℤ_p-local system on a smooth adic space 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 ℤ_p-local system over X is of finite height then its pullback along a finite étale cover of X is semistable. This answers a question of Tong Liu in the relative setting.

Citations

• Prismatic F-crystals and crystalline Galois representations
• Integral p-adic Hodge theory in the imperfect residue field case
• Prisms and Prismatic Cohomology
• Lattices in potentially semi-stable representations and weak (φ,G)-modules
• Relative p-adic Hodge theory: Foundations
• Semistable sheaves and comparison isomorphisms in the semistable case
• p-adic Hodge theory for rigid-analytic varieties
• The relative Breuil-Kisin classification of p-divisible groups and finite flat group schemes
• Crystalline and semi-stable representations in the imperfect residue field case
• A note on lattices in semi-stable representations

Related

• A counterexample to an optimistic guess about étale local systems
• A prismatic-etale comparison theorem in the semistable case
• A p-adic monodromy theorem for de Rham local systems
• Energy integrals over local fields and global height bounds
• Quasi-semi-stable representations