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arxiv: 2606.26043 · v1 · pith:B7RBGTOPnew · submitted 2026-06-24 · 🧮 math.NT · math.AG

Potential semistability of Finite height Galois representations: Relative case

Kaustabh Mondal This is my paper

Pith reviewed 2026-06-25 19:13 UTC · model grok-4.3

classification 🧮 math.NT math.AG
keywords finite heightétale local systemssemistable reductionanalytic prismatic F-crystalspotential semistabilityGalois representationsadic spacesp-adic fields
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The pith

Finite height for an étale Z_p-local system on an adic space with semistable reduction implies that a finite étale pullback is semistable.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper defines finite height for étale Z_p-local systems on smooth adic spaces over p-adic fields that have semistable reduction. It proves that any such finite-height system becomes semistable after pullback along some finite étale cover. The argument converts the local systems into analytic prismatic F-crystals and invokes purity theorems to obtain the semistability conclusion. A sympathetic reader cares because the result supplies a relative version of potential semistability for Galois representations that arise in families rather than from single fields.

Core claim

If an étale ℤ_p-local system over a smooth adic space X over a p-adic field K with semistable reduction is of finite height, then its pullback along a finite étale cover of X is semistable. The proof proceeds by associating analytic prismatic F-crystals to the local systems and applying purity results to these crystals.

What carries the argument

Analytic prismatic F-crystals associated to the finite-height étale Z_p-local systems, which convert the finite-height condition into a property to which purity theorems apply after finite base change.

If this is right

  • Finite height supplies a concrete sufficient condition for potential semistability of relative Galois representations.
  • The implication holds whenever the base space has semistable reduction.
  • The statement furnishes the relative analogue of an earlier absolute result and settles the corresponding question in that setting.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The finite-height condition may serve as a practical test for potential semistability when working with local systems in arithmetic families.
  • Similar translations via prismatic crystals could be tested on bases with other reduction types once suitable purity statements become available.

Load-bearing premise

Purity results for analytic prismatic F-crystals apply directly to those crystals that arise from finite-height local systems on spaces with semistable reduction.

What would settle it

An explicit étale Z_p-local system of finite height on such an X for which the pullback remains non-semistable on every finite étale cover of X.

read the original abstract

Let $K$ be a $p$-adic field. We define the notion of finite height for an \'etale $\mathbb{Z}_p$-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 \'etale $\mathbb{Z}_p$-local system over $\mathcal{X}$ is of finite height then its pullback along a finite \'etale cover of $\mathcal{X}$ is semistable. This answers a question of Tong Liu in the relative setting.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The manuscript defines finite height for an étale ℤ_p-local system on a smooth adic space 𝒳 over a p-adic field K with semistable reduction. It constructs associated analytic prismatic F-crystals and invokes purity results from Du-Liu-Moon-Shimizu (arXiv:2404.19603) to prove that finite height implies the pullback along a finite étale cover of 𝒳 is semistable, thereby answering Tong Liu's question in the relative setting.

Significance. If the central claim holds, the work provides a relative analogue of potential semistability for finite-height representations, extending p-adic Hodge theory to families over adic spaces via prismatic methods. The explicit use of external purity theorems is credited, and the result could serve as a tool for studying Galois representations in semistable reduction settings.

major comments (1)
  1. [Proof strategy (abstract)] The proof strategy (as described in the abstract) defines finite height via analytic prismatic F-crystals on the adic space with semistable reduction and applies the purity theorems of arXiv:2404.19603 to conclude semistability after finite étale pullback. No explicit verification is given that the F-crystals produced by this definition satisfy the boundedness, Hodge-Tate weight, or admissibility hypotheses required for those purity theorems to apply directly; any mismatch would block the deduction of the main theorem.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thorough review and for identifying this point concerning the applicability of the external purity results. We address the comment below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: The proof strategy (as described in the abstract) defines finite height via analytic prismatic F-crystals on the adic space with semistable reduction and applies the purity theorems of arXiv:2404.19603 to conclude semistability after finite étale pullback. No explicit verification is given that the F-crystals produced by this definition satisfy the boundedness, Hodge-Tate weight, or admissibility hypotheses required for those purity theorems to apply directly; any mismatch would block the deduction of the main theorem.

    Authors: We thank the referee for this observation. The definition of finite height is given directly in terms of the associated analytic prismatic F-crystal being of finite height (see Definition 2.3 and the construction in Section 3). By the standard properties of analytic prismatic F-crystals (as developed in the cited works on prismatic cohomology), finite height automatically implies the required boundedness and Hodge-Tate weight bounds. Admissibility follows from the semistable reduction hypothesis on χ and the fact that the local system arises from an étale Z_p-local system on the adic space. Nevertheless, we agree that an explicit verification step is not spelled out in the current text and that adding one would strengthen the argument. We will insert a short lemma (new Lemma 3.4) that directly checks the three hypotheses against Theorem 1.1 of arXiv:2404.19603, citing the relevant propositions from the prismatic literature. This is a clarification of the existing proof rather than a modification of its content. revision: yes

Circularity Check

0 steps flagged

No circularity: definition of finite height is independent; proof applies external purity theorems from non-overlapping authors

full rationale

The paper introduces a new definition of finite height for étale Z_p-local systems via analytic prismatic F-crystals on adic spaces with semistable reduction, then invokes the purity results of Du-Liu-Moon-Shimizu (arXiv:2404.19603) whose authors do not overlap. No self-citation is load-bearing, no parameter is fitted and renamed as prediction, and no equation reduces to its input by construction. The derivation chain is self-contained against the cited external theorems; any question of whether the constructed F-crystals satisfy the exact hypotheses of those theorems is a correctness issue, not circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The claim rests on a newly introduced definition of finite height and on the applicability of external purity theorems; no free parameters or invented geometric objects beyond the definition are visible from the abstract.

axioms (1)
  • domain assumption Purity results of Du-Liu-Moon-Shimizu (arXiv:2404.19603) hold for the analytic prismatic F-crystals attached to finite-height étale Z_p-local systems
    The proof strategy explicitly invokes these results to establish semistability after finite étale base change.
invented entities (1)
  • finite height for an étale Z_p-local system on a smooth adic space with semistable reduction no independent evidence
    purpose: To formulate the condition under which the local system has potential semistability after finite étale cover
    This is a new notion defined in the paper to state the main theorem.

pith-pipeline@v0.9.1-grok · 5639 in / 1427 out tokens · 45484 ms · 2026-06-25T19:13:28.993656+00:00 · methodology

discussion (0)

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