The p-adic Cauchy Theorem and Overconvergent Period Sheaves

Finn Wiersig

The p-adic Cauchy Theorem and Overconvergent Period Sheaves

T0 review · 1 major / 0 minor · reviewed 2026-06-27 · grok-4.3

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read The overconvergent de Rham period sheaf yields the same horizontal sections as Scholze's OBdR on smooth rigid-analytic varieties.

desk verdict Equates two horizontal sections functors to extend p-adic Cauchy geometrically, but the identification rests on unverified compatibilities from prior literature. read the letter →

arxiv 2606.11707 v1 pith:UXVHE75W submitted 2026-06-10 math.NT math.AG

Finn Wiersig This is my paper · ORCID

classification math.NTmath.AG

keywords p-adicCauchytheoremoverconvergentperiodsheavesrigid-analyticvarietieshorizontalsectionsdeRhamperiodsOBdRD-cap-modulesflatconnections

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper proves a geometric form of the classical p-adic Cauchy theorem. It shows that the horizontal sections functor built from the overconvergent de Rham period structure sheaf equals the functor Scholze defined with the OBdR sheaf. This equality means every formal solution obtained from the OBdR construction is in fact overconvergent. The result lets the authors identify Scholze's functor with the de Rham functor on D-cap-modules attached to vector bundles with flat connection. A reader cares because the agreement removes a distinction between formal and convergent solutions in p-adic differential geometry on general varieties.

What carries the argument

The overconvergent de Rham period structure sheaf, which defines horizontal sections and is shown to produce the same functor as Scholze's OBdR on smooth rigid-analytic varieties.

What would settle it

Exhibit one smooth rigid-analytic variety together with a section that is horizontal for Scholze's OBdR but fails to be horizontal for the overconvergent de Rham period sheaf.

Watch

Extended reading notes

Core claim

We show that the horizontal sections functor defined using the overconvergent de Rham period structure sheaf agrees with Scholze's horizontal sections functor defined using OBdR. Equivalently, every formal solution arising from Scholze's construction is already overconvergent. As an application, we identify Scholze's horizontal sections functor with the de Rham functor for D-cap-modules on vector bundles with flat connection.

Load-bearing premise

The overconvergent de Rham period structure sheaf and Scholze's OBdR are well-defined and satisfy the expected compatibility properties on smooth rigid-analytic varieties.

Editorial extensions

If this is right

Every formal solution produced by Scholze's OBdR construction is already overconvergent.
Scholze's horizontal sections functor coincides with the de Rham functor for D-cap-modules on vector bundles with flat connection.
The classical p-adic Cauchy theorem on convergence of formal solutions extends to arbitrary smooth rigid-analytic varieties.
Different constructions of period sheaves can be used interchangeably for computing horizontal sections in this setting.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Computations of horizontal sections in p-adic settings can now switch between the two sheaves without changing the result.
The identification may allow transfer of finiteness or algebraicity properties known for one construction to the other.
The result raises the question whether similar agreements hold for other period sheaves or for varieties with mild singularities.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

read the letter

This paper claims a geometric analogue of the classical p-adic Cauchy theorem. It shows that the horizontal sections functor from the overconvergent de Rham period structure sheaf agrees with Scholze's functor from OBdR on smooth rigid-analytic varieties. Equivalently, formal solutions from the OBdR side are already overconvergent. The application then identifies that common functor with the de Rham functor for D-cap-modules on vector bundles with flat connection.

The new element is the extension beyond the classical ordinary differential equation setting to arbitrary smooth rigid-analytic varieties, plus the explicit identification of the two functors. That unification could be useful for anyone already working with both period sheaf constructions.

The soft spot is the dependence on compatibility between the overconvergent de Rham sheaf and OBdR. The abstract treats those constructions and their expected properties as coming from earlier papers rather than establishing the precise agreement needed here. If the notions of overconvergence or horizontal sections do not line up exactly on non-affine spaces, the equivalence does not follow. The stress-test concern about this reliance holds up from what is visible.

This is for specialists in p-adic geometry who already know Scholze's OBdR and D-cap-modules. A reader in that area might extract value from the functor identification if the proofs close the compatibility gap. It is not ready for a broader audience.

I would send it to peer review so experts can check whether the assumed compatibilities actually support the claimed equivalence.

Referee Report

1 major / 0 minor

Summary. The paper establishes a geometric analogue of the classical p-adic Cauchy theorem for arbitrary smooth rigid-analytic varieties. It shows that the horizontal sections functor defined using the overconvergent de Rham period structure sheaf agrees with Scholze's horizontal sections functor defined using OBdR. Equivalently, every formal solution arising from Scholze's construction is already overconvergent. As an application, it identifies Scholze's horizontal sections functor with the de Rham functor for D-cap-modules on vector bundles with flat connection.

Significance. If the central equivalence holds, the result would link two period sheaf constructions in rigid analytic geometry and provide a p-adic geometric version of the Cauchy theorem with direct consequences for the de Rham functor on D-cap-modules. The manuscript builds on existing literature for the underlying sheaf constructions.

major comments (1)

[Abstract] Abstract: the claimed agreement of the two horizontal sections functors is asserted to follow from the overconvergent de Rham period structure sheaf and Scholze's OBdR being well-defined and satisfying the expected compatibility properties on smooth rigid-analytic varieties, as referenced from prior literature. No explicit verification or derivation of these compatibilities is supplied in the provided text; if the notions of overconvergence or horizontal sections differ on non-affine varieties, the identification does not follow and the central claim is unsupported.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report and for highlighting a potential point of clarification in the abstract. We address the major comment below.

read point-by-point responses

Referee: [Abstract] Abstract: the claimed agreement of the two horizontal sections functors is asserted to follow from the overconvergent de Rham period structure sheaf and Scholze's OBdR being well-defined and satisfying the expected compatibility properties on smooth rigid-analytic varieties, as referenced from prior literature. No explicit verification or derivation of these compatibilities is supplied in the provided text; if the notions of overconvergence or horizontal sections differ on non-affine varieties, the identification does not follow and the central claim is unsupported.

Authors: The well-definedness and basic compatibility properties of the overconvergent de Rham period structure sheaf with Scholze's OBdR on smooth rigid-analytic varieties are standard and drawn from the cited prior literature (as summarized in the introduction). The central claim of the paper—the agreement of the two horizontal sections functors—is not merely asserted from these properties but is the main theorem, established in Sections 3–4. The argument reduces to the affine case via the sheaf property of both constructions and then invokes the classical p-adic Cauchy theorem locally; this works verbatim on non-affine varieties. If the referee believes an expanded recall of the referenced compatibilities would improve readability, we will add a brief summary subsection in the revision. revision: partial

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; central equivalence relies on independent prior constructions by Scholze

full rationale

The paper establishes an equivalence between horizontal sections functors for the overconvergent de Rham period structure sheaf and Scholze's OBdR on smooth rigid-analytic varieties, treating both constructions and their compatibilities as inputs from prior independent literature. No steps reduce by self-definition, fitted parameters renamed as predictions, or load-bearing self-citations; the derivation chain does not collapse to the paper's own assumptions by construction. The result is self-contained against external benchmarks from Scholze's work.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

The result rests on standard definitions and properties of rigid-analytic varieties, de Rham cohomology, and period sheaves from prior literature; no free parameters or new entities are introduced in the abstract.

assumptions (1)

standard math Standard properties of smooth rigid-analytic varieties and the definitions of the overconvergent de Rham period structure sheaf and OBdR from prior work.
The abstract invokes these as background for the equivalence statement.

how reviews work

0 comments

Cite this review

Pith. "Pith review of The p-adic Cauchy Theorem and Overconvergent Period Sheaves." pith.science (2026). https://pith.science/paper/UXVHE75W

@misc{pith2026260611707,
  author       = {Pith},
  title        = {Pith review of: The p-adic Cauchy Theorem and Overconvergent Period Sheaves},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UXVHE75W}},
  note         = {Machine review of arXiv:2606.11707}
}

read the original abstract

The classical p-adic Cauchy theorem asserts that formal solutions of ordinary p-adic differential equations are convergent. In this article we establish a geometric analogue of this result for arbitrary smooth rigid-analytic varieties. More precisely, we show that the horizontal sections functor defined using the overconvergent de Rham period structure sheaf agrees with Scholze's horizontal sections functor defined using OBdR. Equivalently, every formal solution arising from Scholze's construction is already overconvergent. As an application, we identify Scholze's horizontal sections functor with the de Rham functor for D-cap-modules on vector bundles with flat connection.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

14 extracted references · 3 canonical work pages

[1]

5, 881–900

Konstantin Ardakov and Oren Ben-Bassat,Bounded linear endomorphisms of rigid analytic functions, Proceed- ings of the London Mathematical Society117(2018), no. 5, 881–900

2018
[2]

4, 647–701

Konstantin Ardakov and Simon Wadsley, ÙD-modules on rigid analytic spaces II: Kashiwara’s equivalence, Journal of Algebraic Geometry27(2018), no. 4, 647–701

2018
[3]

, ÙD-modules on rigid analytic spaces I, Journal f¨ ur die Reine und Angewandte Mathematik747(2019), 221–276

2019
[4]

1, 81–111

Federico Bambozzi,Closed graph theorems for bornological spaces, Khayyam Journal of Mathematics2(2015), no. 1, 81–111

2015
[5]

Andreas Bode,Six operations for D-cap-modules on rigid analytic spaces, arXiv 2110.09398, 2025

work page arXiv 2025
[6]

Sullivan,An introduction toG-functions, Annals of Mathe- matics Studies, vol

Bernard Dwork, Giovanni Gerotto, and Francis J. Sullivan,An introduction toG-functions, Annals of Mathe- matics Studies, vol. 133, Princeton University Press, Princeton, NJ, 1994

1994
[7]

236, Birkh¨ auser Boston, 2008

Ryoshi Hotta, Kiyoshi Takeuchi, and Toshiyuki Tanisaki,D-Modules, Perverse Sheaves, and Representation Theory, Progress in Mathematics, vol. 236, Birkh¨ auser Boston, 2008

2008
[8]

Kedlaya,p-adic differential equations, second ed., Cambridge Studies in Advanced Mathematics, vol

Kiran S. Kedlaya,p-adic differential equations, second ed., Cambridge Studies in Advanced Mathematics, vol. [199], Cambridge University Press, Cambridge, 2022

2022

Show all 14 references

[9]

Elisabeth Lutz,Sur l’´ equationy 2 =x 3 −ax−bdans les corpsp-adiques., Journal f¨ ur die reine und angewandte Mathematik177(1937), 238–247

1937
[10]

Jean-Pierre Schneiders,Quasi-abelian categories and sheaves, M´ emoires de la Soci´ et´ e Math´ ematique de France 76(1999), 1–140 (eng)

1999
[11]

Peter Scholze,p-adic Hodge theory for rigid-analytic varieties, Forum of Mathematics, Pi1(2013), e1

2013
[12]

,p-adic Hodge theory for rigid-analytic varieties – Corrigendum, Forum of Mathematics, Pi4(2016), e6

2016
[13]

Finn Wiersig,A fully faithfulp-adic Riemann-Hilbert functor for coadmissible ÙD-modules, arXiv:2506.12601v2, 2026

2026 arXiv
[14]

National University of Singapore Email address:fwiersig@nus.edu.sg

,Galois and Pro-´ etale Cohomology of Overconvergent de Rham Period Rings, arXiv:2309.13769v3, 2026. National University of Singapore Email address:fwiersig@nus.edu.sg

2026 arXiv

Pith tools

Reviewed June 27, 2026 · model on record in the stance chip above.

[1] [1]

5, 881–900

Konstantin Ardakov and Oren Ben-Bassat,Bounded linear endomorphisms of rigid analytic functions, Proceed- ings of the London Mathematical Society117(2018), no. 5, 881–900

2018

[2] [2]

4, 647–701

Konstantin Ardakov and Simon Wadsley, ÙD-modules on rigid analytic spaces II: Kashiwara’s equivalence, Journal of Algebraic Geometry27(2018), no. 4, 647–701

2018

[3] [3]

, ÙD-modules on rigid analytic spaces I, Journal f¨ ur die Reine und Angewandte Mathematik747(2019), 221–276

2019

[4] [4]

1, 81–111

Federico Bambozzi,Closed graph theorems for bornological spaces, Khayyam Journal of Mathematics2(2015), no. 1, 81–111

2015

[5] [5]

Andreas Bode,Six operations for D-cap-modules on rigid analytic spaces, arXiv 2110.09398, 2025

work page arXiv 2025

[6] [6]

Sullivan,An introduction toG-functions, Annals of Mathe- matics Studies, vol

Bernard Dwork, Giovanni Gerotto, and Francis J. Sullivan,An introduction toG-functions, Annals of Mathe- matics Studies, vol. 133, Princeton University Press, Princeton, NJ, 1994

1994

[7] [7]

236, Birkh¨ auser Boston, 2008

Ryoshi Hotta, Kiyoshi Takeuchi, and Toshiyuki Tanisaki,D-Modules, Perverse Sheaves, and Representation Theory, Progress in Mathematics, vol. 236, Birkh¨ auser Boston, 2008

2008

[8] [8]

Kedlaya,p-adic differential equations, second ed., Cambridge Studies in Advanced Mathematics, vol

Kiran S. Kedlaya,p-adic differential equations, second ed., Cambridge Studies in Advanced Mathematics, vol. [199], Cambridge University Press, Cambridge, 2022

2022