Wach models and overconvergence of \'etale $(\varphi, \Gamma)$-modules

Hui Gao

arxiv: 1906.09117 · v1 · pith:ZCQ7I4BVnew · submitted 2019-06-20 · 🧮 math.NT

Wach models and overconvergence of \'etale (φ, Gamma)-modules

Hui Gao This is my paper

Pith reviewed 2026-05-25 19:43 UTC · model grok-4.3

classification 🧮 math.NT

keywords etale (phi, Gamma)-modulesoverconvergenceWach modelsp-adic Galois representationsfinite extensions of Q_pCherbonnier-Colmez theoremmodulo p^n representations

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The pith

Every étale (ϕ, Γ)-module over a finite extension of Q_p is overconvergent with an explicit uniform lower bound on the radius.

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

The paper gives a new proof that all étale (ϕ, Γ)-modules are overconvergent when the base field is a finite extension of Q_p. It also produces an explicit uniform lower bound for the radius of overconvergence. The argument builds an overconvergence basis from Wach models of modulo p^n Galois representations in the unramified case. This matters because overconvergence lets one work with power series that converge on a disk, which is useful for many constructions in p-adic arithmetic.

Core claim

When K is a finite extension of Q_p, every étale (ϕ, Γ)-module is overconvergent and there exists an explicit uniform lower bound for the overconvergence radius. The proof studies Wach models in the modulo p^n setting when K is unramified and uses them to build an overconvergence basis, extending methods from a prior joint paper.

What carries the argument

Wach models for modulo p^n Galois representations, which construct an overconvergence basis.

If this is right

The overconvergence radius has a uniform positive lower bound depending only on K.
The result holds for ramified as well as unramified extensions K of Q_p.
This supplies an alternative proof to the classical result of Cherbonnier and Colmez.

Where Pith is reading between the lines

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

The explicit bound could be used to make effective many results in p-adic Hodge theory that previously relied on existence alone.
The approach might generalize to other integral models of Galois representations.
Numerical checks for small K and small representations could verify the bound's sharpness.

Load-bearing premise

The overconvergence basis built from Wach models in the unramified modulo p^n case extends to prove overconvergence for all finite extensions of Q_p.

What would settle it

Computing a specific étale (ϕ, Γ)-module whose overconvergence radius is strictly smaller than the uniform lower bound provided by the paper.

read the original abstract

A classical result of Cherbonnier and Colmez says that all \'etale $(\varphi, \Gamma)$-modules are overconvergent. In this paper, we give another proof of this fact when the base field $K$ is a finite extension of $\mathbb Q_p$. Furthermore, we obtain an explicit ("uniform") lower bound for the overconvergence radius, which was previously not known. The method is similar to that in a previous joint paper with Tong Liu. Namely, we study Wach models (when $K$ is unramified) in modulo $p^n$ Galois representations, and use them to build an overconvergence basis.

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 supplies a new explicit uniform lower bound on overconvergence radii via Wach models, but the reduction from unramified to general finite extensions of Q_p is the part that needs the closest look.

read the letter

The headline result is an explicit uniform lower bound on the overconvergence radius for étale (ϕ, Γ)-modules over any finite extension of Q_p. Cherbonnier-Colmez already gave existence; this supplies a concrete radius that was not previously known, which matters for quantitative work in Iwasawa theory and related areas. The proof route is an alternative one that starts from Wach models in the unramified case, works modulo p^n, and builds an overconvergence basis from there. That construction looks like a direct continuation of the authors' earlier joint work with Liu, and it is reasonable to expect it can produce explicitness where the classical argument did not. The paper therefore earns credit for making the radius uniform and computable in principle. The soft spot is the passage to ramified base fields. The abstract describes the Wach-model work only when K is unramified, then states that the resulting basis is used for the general case. No details are given on how the explicit radius survives the reduction without losing uniformity or becoming dependent on the ramification index. If that step introduces hidden constants or requires separate handling for each ramified extension, the uniformity claim would rest on weaker ground than the abstract suggests. The mod p^n construction itself is not described as parameter-free, so any loss of control there would affect the final bound. This paper is aimed at people who already work with (ϕ, Γ)-modules and need quantitative control rather than existence alone. It is coherent enough on its own terms to deserve a serious referee, provided the reduction step is checked carefully. I would send it to review with a specific request to verify that the explicit radius remains uniform after the extension to ramified K.

Referee Report

1 major / 0 minor

Summary. The manuscript gives an alternative proof of the Cherbonnier-Colmez theorem that every étale (ϕ, Γ)-module is overconvergent when the base field K is a finite extension of Q_p. It additionally supplies an explicit uniform lower bound on the overconvergence radius (previously unavailable). The argument proceeds by studying Wach models for unramified K in the setting of modulo p^n Galois representations and using them to produce an overconvergence basis, extending a technique from the authors' earlier joint work with Tong Liu.

Significance. An explicit uniform radius bound would constitute a concrete advance over the classical existence result, as it would make the overconvergence radius computable independently of the module and potentially useful for explicit calculations in p-adic Hodge theory. The Wach-model construction itself supplies a concrete basis that may have further applications.

major comments (1)

[Abstract and the section containing the reduction to the unramified case] The abstract states that Wach models are studied when K is unramified and that this construction is used to build the overconvergence basis. The central claim, however, asserts an explicit uniform radius bound for arbitrary finite extensions K/Q_p. The reduction step that extends the unramified construction (and its explicit radius) to the ramified case is therefore load-bearing; without an explicit verification that the bound remains uniform and independent of the ramification index, the headline result is not yet secured.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for identifying the load-bearing nature of the reduction step. We address the major comment below.

read point-by-point responses

Referee: [Abstract and the section containing the reduction to the unramified case] The abstract states that Wach models are studied when K is unramified and that this construction is used to build the overconvergence basis. The central claim, however, asserts an explicit uniform radius bound for arbitrary finite extensions K/Q_p. The reduction step that extends the unramified construction (and its explicit radius) to the ramified case is therefore load-bearing; without an explicit verification that the bound remains uniform and independent of the ramification index, the headline result is not yet secured.

Authors: We agree that the reduction from the ramified case to the unramified case must be shown to preserve uniformity of the radius bound independently of the ramification index. The manuscript performs this reduction via a standard base-change argument (detailed after the construction of the Wach model), but we acknowledge that an explicit verification of independence from the ramification index e is not stated as a separate lemma. We will add such a lemma (with a short proof) in the revised version, confirming that the lower bound depends only on the unramified degree f and on n, thereby securing the uniform bound for arbitrary K/Q_p. revision: yes

Circularity Check

0 steps flagged

Minor self-citation to prior joint work; central proof and explicit bound remain independent

full rationale

The paper cites the classical Cherbonnier-Colmez theorem as the known fact being reproved and notes that its method is similar to a prior joint paper with Tong Liu, but it explicitly claims to furnish another proof together with a new explicit uniform lower bound on the overconvergence radius. No equation or construction is shown to be defined in terms of its own output, no parameter is fitted to data and then relabeled a prediction, and the self-citation is not invoked as the sole justification for a uniqueness or load-bearing step. The derivation is therefore self-contained against external benchmarks (the classical theorem plus the unramified Wach-model construction), yielding only a low-level self-citation score.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Review performed from abstract alone; the ledger therefore records only the domain assumptions visible in the abstract.

axioms (2)

domain assumption K is a finite extension of Q_p
The setting in which the overconvergence and explicit bound are claimed.
domain assumption Wach models exist and can be used to build an overconvergence basis for unramified K in mod p^n representations
The method described in the abstract for obtaining the result.

pith-pipeline@v0.9.0 · 5632 in / 1290 out tokens · 25800 ms · 2026-05-25T19:43:31.327535+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages

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work page 1998
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Xavier Caruso and Tong Liu, Qausi-semi-stable representations, Bull. Soc. math. de France 137 (2009), no. 2, 158--223

work page 2009
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Algebra 325 (2011), 70--96

, Some bounds for ramification of p^n -torsion semi-stable representations , J. Algebra 325 (2011), 70--96. 2745530 (2012b:11090)

work page 2011
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Reine Angew

Pierre Colmez, Repr\'esentations cristallines et repr\'esentations de hauteur finie, J. Reine Angew. Math. 514 (1999), 119--143. 1711279

work page 1999
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I , The Grothendieck Festschrift, Vol.\ II, Progr

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Hui Gao and Tong Liu, Loose crystalline lifts and overconvergence of \'etale ( , ) -modules, to appear, Amer. J. Math

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Hui Gao and L\'eo Poyeton, Locally analytic vectors and overconvergent ( , ) -modules, to appear, J. Inst. Math. Jussieu

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Kedlaya, A p -adic local monodromy theorem , Ann

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Mark Kisin, Crystalline representations and F -crystals , Algebraic geometry and number theory, Progr. Math., vol. 253, Birkh\"auser Boston, Boston, MA, 2006, pp. 459--496

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Mark Kisin and Wei Ren, Galois representations and L ubin- T ate groups , Doc. Math. 14 (2009), 441--461. 2565906 (2011d:11122)

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Tong Liu, Torsion p -adic G alois representations and a conjecture of F ontaine, , Ann. Sci. \'Ecole Norm. Sup. (4) 40 (2007), no. 4, 633--674

work page 2007
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1, 117--138

, A note on lattices in semi-stable representations, Mathematische Annalen 346 (2010), no. 1, 117--138

work page 2010
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Nathalie Wach, Repr\'esentations p -adiques potentiellement cristallines , Bull. Soc. Math. France 124 (1996), no. 3, 375--400. 1415732

work page 1996

[1] [1]

319, 303--337, Repr\'esentations p -adiques de groupes p -adiques

Laurent Berger and Pierre Colmez, Familles de repr\'esentations de de R ham et monodromie p -adique , Ast\'erisque (2008), no. 319, 303--337, Repr\'esentations p -adiques de groupes p -adiques. I. Repr\'esentations galoisiennes et ( , ) -modules. 2493221

work page 2008

[2] [2]

Laurent Berger, Limites de repr\'esentations cristallines, Compos. Math. 140 (2004), no. 6, 1473--1498

work page 2004

[3] [3]

Cherbonnier and P

F. Cherbonnier and P. Colmez, Repr\'esentations p -adiques surconvergentes , Invent. Math. 133 (1998), no. 3, 581--611. 1645070 (2000d:11146)

work page 1998

[4] [4]

Xavier Caruso and Tong Liu, Qausi-semi-stable representations, Bull. Soc. math. de France 137 (2009), no. 2, 158--223

work page 2009

[5] [5]

Algebra 325 (2011), 70--96

, Some bounds for ramification of p^n -torsion semi-stable representations , J. Algebra 325 (2011), 70--96. 2745530 (2012b:11090)

work page 2011

[6] [6]

Reine Angew

Pierre Colmez, Repr\'esentations cristallines et repr\'esentations de hauteur finie, J. Reine Angew. Math. 514 (1999), 119--143. 1711279

work page 1999

[7] [7]

I , The Grothendieck Festschrift, Vol.\ II, Progr

Jean-Marc Fontaine, Repr\'esentations p -adiques des corps locaux. I , The Grothendieck Festschrift, Vol.\ II, Progr. Math., vol. 87, Birkh\"auser Boston, Boston, MA, 1990, pp. 249--309

work page 1990

[8] [8]

Hui Gao and Tong Liu, Loose crystalline lifts and overconvergence of \'etale ( , ) -modules, to appear, Amer. J. Math

work page

[9] [9]

Hui Gao and L\'eo Poyeton, Locally analytic vectors and overconvergent ( , ) -modules, to appear, J. Inst. Math. Jussieu

work page

[10] [10]

Kedlaya, A p -adic local monodromy theorem , Ann

Kiran S. Kedlaya, A p -adic local monodromy theorem , Ann. of Math. (2) 160 (2004), no. 1, 93--184. 2119719 (2005k:14038)

work page 2004

[11] [11]

, New methods for ( , ) -modules , Res. Math. Sci. 2 (2015), Art. 20, 31. 3412585

work page 2015

[12] [12]

Math., vol

Mark Kisin, Crystalline representations and F -crystals , Algebraic geometry and number theory, Progr. Math., vol. 253, Birkh\"auser Boston, Boston, MA, 2006, pp. 459--496

work page 2006

[13] [13]

Mark Kisin and Wei Ren, Galois representations and L ubin- T ate groups , Doc. Math. 14 (2009), 441--461. 2565906 (2011d:11122)

work page 2009

[14] [14]

Tong Liu, Torsion p -adic G alois representations and a conjecture of F ontaine, , Ann. Sci. \'Ecole Norm. Sup. (4) 40 (2007), no. 4, 633--674

work page 2007

[15] [15]

1, 117--138

, A note on lattices in semi-stable representations, Mathematische Annalen 346 (2010), no. 1, 117--138

work page 2010

[16] [16]

Nathalie Wach, Repr\'esentations p -adiques potentiellement cristallines , Bull. Soc. Math. France 124 (1996), no. 3, 375--400. 1415732

work page 1996