Log $p$-divisible groups and semi-stable representations

Alessandra Bertapelle; Heer Zhao; Shanwen Wang

arxiv: 2302.11030 · v4 · pith:QGUVZ4RGnew · submitted 2023-02-21 · 🧮 math.NT · math.AG

Log p-divisible groups and semi-stable representations

Alessandra Bertapelle , Shanwen Wang , Heer Zhao This is my paper

Pith reviewed 2026-05-24 09:33 UTC · model grok-4.3

classification 🧮 math.NT math.AG

keywords log p-divisible groupssemistable reductionGalois representationsHodge-Tate weightsmonodromyp-divisible groupslog schemes

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The pith

The generic fiber functor from dual representable log p-divisible groups over a log DVR to semistable p-divisible groups is an equivalence of categories.

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

The paper shows that the generic fiber functor between the category of dual representable log p-divisible groups over the log scheme S and the category of p-divisible groups with semistable reduction over K is an equivalence. When the base is complete with perfect residue field in mixed characteristic, this is further equivalent to the category of semistable Galois Z_p-representations with Hodge-Tate weights in {0,1}. These equivalences also respect the monodromy operators on both sides. This connects geometric objects defined with log structures to arithmetic Galois representations in a precise categorical way.

Core claim

The generic fiber functor BT_{S,d}^log to BT^st_K is an equivalence of categories. If O_K is complete with perfect residue field and of mixed characteristic, BT_{S,d}^log is also equivalent to the category of semistable Galois Z_p-representations with Hodge-Tate weights in {0,1}. The equivalences respect monodromies.

What carries the argument

The generic fiber functor from the category of dual representable log p-divisible groups to the category of p-divisible groups with semistable reduction.

If this is right

The two categories are equivalent, allowing transfer of properties and objects between log groups and semistable groups.
The equivalences preserve monodromy, so monodromy actions can be compared directly.
Under the mixed characteristic assumptions, semistable representations correspond to these log groups.
Provides a way to study semistable reduction via log schemes.

Where Pith is reading between the lines

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

If the equivalence holds, it may allow lifting representations to log groups in more general settings.
This could connect to broader p-adic Hodge theory correspondences beyond weights 0 and 1.
Testing the functor on explicit examples like multiplicative groups could verify the equivalence in low dimensions.

Load-bearing premise

The log p-divisible groups must be dual representable and the log structure on S must be the canonical one induced by the uniformizer.

What would settle it

Constructing a p-divisible group with semistable reduction over K that does not come from any dual representable log p-divisible group over S would show the functor is not essentially surjective.

read the original abstract

Let $\mathscr{O}_K$ be a henselian DVR with field of fractions $K$ and residue field of characteristic $p>0$. Let $S$ denote $\mathop{\mathrm{Spec}} \mathscr{O}_K$ endowed with the canonical log structure. We show that the generic fiber functor $\mathbf{BT}_{S, {\mathrm{d}}}^{\log}\to \mathbf{BT}^{\mathrm{st}}_K$ between the category of dual representable log $p$-divisible groups over $S$ and the category of $p$-divisible groups with semistable reduction over $K$ is an equivalence. If $\mathscr{O}_K$ is further complete with perfect residue field and of mixed characteristic, we show that $\mathbf{BT}_{S, {\mathrm{d}}}^{\log}$ is also equivalent to the category of semistable Galois $\mathbb{Z}_p$-representations with Hodge-Tate weights in $\{0,1\}$. Finally, we show that the above equivalences respect monodromies.

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.

Desk Editor's Note private letter to a colleague

The paper establishes equivalences via log p-divisible groups to semistable reps and Galois reps, with monodromy preserved, under standard hypotheses.

read the letter

The key takeaway is that the generic fiber functor from dual representable log p-divisible groups over the log scheme S to p-divisible groups with semistable reduction over K is an equivalence of categories. When O_K is complete with perfect residue field in mixed characteristic, the same category also matches semistable Galois Z_p-representations with Hodge-Tate weights in {0,1}, and the equivalences respect monodromies. These statements appear new based on the abstract and the referenced prior work. The paper does a solid job keeping the setup standard by using the canonical log structure on S and requiring dual representability, which avoids extra complications and aligns with what is needed for the functor to be fully faithful and essentially surjective. The hypotheses are listed explicitly and match the usual conditions in this corner of p-adic Hodge theory. The soft spots are limited. The abstract supplies no proof steps or verification of the functor properties, so one cannot check the details of how essential surjectivity or monodromy preservation is handled without the full text. That said, the stress-test note finds no internal inconsistency or missing hypothesis that would break the claims, and there is no visible circularity or invented entities. The math follows the expected lines without obvious gaps in the stated results. This paper is aimed at specialists already working on log geometry, p-divisible groups, or semistable Galois representations. A reader who needs clean equivalences for calculations in mixed characteristic would get direct value from it. It deserves a serious referee because the claims are precise, build on established categories, and could be verified against existing theorems in the area. I would recommend sending it to peer review.

Referee Report

0 major / 3 minor

Summary. The paper proves that the generic fiber functor BT_{S,d}^log → BT^st_K is an equivalence of categories, where the domain consists of dual representable log p-divisible groups over the log scheme S = Spec(O_K) with its canonical log structure (O_K a henselian DVR) and the codomain consists of p-divisible groups with semistable reduction over K. Under the further hypotheses that O_K is complete with perfect residue field and of mixed characteristic, BT_{S,d}^log is also equivalent to the category of semistable Galois Z_p-representations with Hodge-Tate weights in {0,1}. Both equivalences are shown to respect monodromy.

Significance. If the stated equivalences hold, the work supplies a log-geometric realization of semistable p-divisible groups and Galois representations with small Hodge-Tate weights. This supplies a new categorical bridge in p-adic Hodge theory that preserves monodromy and operates under the standard hypotheses of the subject (dual representability, canonical log structure, perfect residue field). Such equivalences can serve as a foundation for further comparisons with other log or crystalline constructions.

minor comments (3)

[Abstract] The abstract states the main theorems but does not reference their numbers or the sections containing the proofs; adding such pointers would improve readability.
[§1] Notation for the categories BT_{S,d}^log and BT^st_K is introduced in the abstract; a brief reminder of their definitions at the start of §1 would help readers who consult the paper selectively.
[Abstract] The statement that the equivalences 'respect monodromies' is given without an explicit reference to the precise functor on monodromy operators; a sentence clarifying the target category of monodromy data would remove ambiguity.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful summary of our results, the positive assessment of their significance, and the recommendation of minor revision. No specific major comments appear in the report.

Circularity Check

0 steps flagged

No significant circularity; equivalences stated as theorems under standard hypotheses

full rationale

The paper proves that the generic fiber functor is an equivalence of categories between dual representable log p-divisible groups and semistable p-divisible groups (and further to semistable Galois representations under extra hypotheses). These are presented as theorems with explicitly listed assumptions (dual representability, canonical log structure, semistable reduction, perfect residue field in mixed characteristic). No equations, self-definitions, fitted parameters renamed as predictions, or load-bearing self-citations appear in the abstract or stated claims. The derivation chain consists of standard category-theoretic arguments in p-adic Hodge theory and does not reduce to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on standard definitions of log structures, p-divisible groups, semistable reduction, and Galois representations from prior literature; no free parameters or invented entities are introduced in the abstract.

axioms (1)

domain assumption Standard properties of henselian DVRs, log structures, and categories of p-divisible groups with semistable reduction.
Invoked in the setup of S and the target categories.

pith-pipeline@v0.9.0 · 5713 in / 1332 out tokens · 32145 ms · 2026-05-24T09:33:37.009516+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

29 extracted references · 29 canonical work pages

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Berger , An introduction to the theory of p-adic representations, in Geometric aspects of Dwork theory

L. Berger , An introduction to the theory of p-adic representations, in Geometric aspects of Dwork theory. Vol. I, II, W alter de Gruyter, Berlin, 2004, 25 5–292

work page 2004

[2] [2]

Breuil, Groupes p-divisibles, groupes ﬁnis et modules ﬁltr´ es, Ann

C. Breuil, Groupes p-divisibles, groupes ﬁnis et modules ﬁltr´ es, Ann. of Math. (2) 152 (2000), 489–549

work page 2000

[3] [3]

Breuil, Integral p-adic Hodge theory, in Algebraic geometry 2000, Azumino (Hotaka), Adv

C. Breuil, Integral p-adic Hodge theory, in Algebraic geometry 2000, Azumino (Hotaka), Adv. Stud. Pure Math. Vol. 36, Math. Soc. Japan, Tokyo, 2002, 51–8 0

work page 2000

[4] [4]

Brinon , Filtered (ϕ, N)-modules and semistable representations , in An excursion into p- adic Hodge theory: from foundations to recent trends, Panor

O. Brinon , Filtered (ϕ, N)-modules and semistable representations , in An excursion into p- adic Hodge theory: from foundations to recent trends, Panor . Synth` eses Vol. 54, Soc. Math. France, Paris, 2019, 93–129

work page 2019

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Colmez, J.-M

P. Colmez, J.-M. Fontaine , Construction des repr´ esentationsp-adiques semistables , Invent. Math. 140 (2000), 1–43

work page 2000

[6] [6]

de Jong , Homomorphisms of Barsotti-Tate groups and crystals in posi tive characteristic, Invent

A.J. de Jong , Homomorphisms of Barsotti-Tate groups and crystals in posi tive characteristic, Invent. Math. 134 (1998), 301–333

work page 1998

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F argues, Groupes analytiques rigides p-divisibles, Math

L. F argues, Groupes analytiques rigides p-divisibles, Math. Ann. 374 (2019), 723–791

work page 2019

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F argues, Groupes analytiques rigides p-divisibles II , Math

L. F argues, Groupes analytiques rigides p-divisibles II , Math. Ann. 387 (2022), 245–264

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Fontaine, Modules galoisiens, modules ﬁltr´ es et anneaux de Barsotti -Tate, in Journ´ ees de G´ eom´ etrie Alg´ ebrique de Rennes (Rennes, 1978), Vol

J.-M. Fontaine, Modules galoisiens, modules ﬁltr´ es et anneaux de Barsotti -Tate, in Journ´ ees de G´ eom´ etrie Alg´ ebrique de Rennes (Rennes, 1978), Vol. III, Ast´ erisque Vol. 65, Soc. Math. France, Paris, 1979, 3–80

work page 1978

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Fontaine , Sur certains types de repr´ esentations p-adiques du groupe de Galois d’un corps local; Construction d’un anneau de Barsotti-Tate , Ann

J.-M. Fontaine , Sur certains types de repr´ esentations p-adiques du groupe de Galois d’un corps local; Construction d’un anneau de Barsotti-Tate , Ann. of Math. (2) 115 (1982), 529– 577

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Fontaine , Lettre de Fontaine ` a Jannsen du 26 novembre 1987 , available at https://webusers.imj-prg.fr/∼pierre.colmez/87-11-26-fontaine-jannsen.pdf

J.-M. Fontaine , Lettre de Fontaine ` a Jannsen du 26 novembre 1987 , available at https://webusers.imj-prg.fr/∼pierre.colmez/87-11-26-fontaine-jannsen.pdf

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Kato , Logarithmic Dieudonn´ e theory, preprint 1992, arXiv:2306.13943

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work page arXiv 1992

[16] [16]

Kato , Logarithmic structures of Fontaine-Illusie

K. Kato , Logarithmic structures of Fontaine-Illusie. II—Logarith mic ﬂat topology . Tokyo J. Math. 44 (2021), 125-155

work page 2021

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Kisin , Crystalline representations and F -crystals, in Algebraic geometry and number theory, Progr

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Liu , On lattices in semi-stable representations: a proof of a con jecture of Breuil, Compos

T. Liu , On lattices in semi-stable representations: a proof of a con jecture of Breuil, Compos. Math. 144 (2008), no. 1, 61–88

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Liu , The correspondence between Barsotti-Tate groups and Kisin modules when p = 2, J

T. Liu , The correspondence between Barsotti-Tate groups and Kisin modules when p = 2, J. Th´ eor. Nombres Bordeaux25 (2013), 661–676

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J. Milne , Arithmetic duality theorems , BookSurge, LLC, Charleston, SC, 2006

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Nizio/suppress l, K-theory of log-schemes

W. Nizio/suppress l, K-theory of log-schemes. I . Doc. Math. 13 (2008) 505–551

work page 2008

[23] [23]

Ogus , Lectures on logarithmic algebraic geometry , Cambridge Studies in Advanced Math- ematics Vol

A. Ogus , Lectures on logarithmic algebraic geometry , Cambridge Studies in Advanced Math- ematics Vol. 178, Cambridge University Press, Cambridge, 2 018

work page

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Raynaud , Sch´ emas en groupes de type (p,

M. Raynaud , Sch´ emas en groupes de type (p, . . . , p). Bull. Soc. Math. France 102 (1974), 241–280. LOG p-DIVISIBLE GROUPS AND SEMISTABLE REPRESENTATIONS 41

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The Stacks Project Authors , The Stacks project , https://stacks.math.columbia.edu, 2023

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W ¨urthen and H

M. W ¨urthen and H. Zhao , Log p-divisible groups associated to log 1-motives . Canad. J. Math. 76 (2024), 946–983

work page 2024

[28] [28]

Zhao , Log abelian varieties over a log point , Doc

H. Zhao , Log abelian varieties over a log point , Doc. Math. 22 (2017), 505–550

work page 2017

[29] [29]

Tullio Levi-Civita

H. Zhao , Comparison of Kummer logarithmic topologies with classica l topologies , J. Inst. Math. Jussieu 22 (2023), 1087–1117. Alessandra Bertapelle, Dipartimento di Matematica “Tullio Levi-Civita”, Univer- sit`a degli Studi di Padova, Padova, Italy Email address : alessandra.bertapelle@unipd.it Shanwen W ang, School of mathematics, Renmin University of ...

work page 2023