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arxiv: 2508.15443 · v4 · submitted 2025-08-21 · 🧮 math.SG · math-ph· math.MP

Darboux's Theorem in p-adic symplectic geometry

Luis Crespo , \'Alvaro Pelayo This is my paper

Pith reviewed 2026-05-18 22:08 UTC · model grok-4.3

classification 🧮 math.SG math-phmath.MP
keywords Darboux theoremp-adic analytic manifoldssymplectic geometryMoser's path methodintegrable systemsvolume classificationnon-Archimedean geometry
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The pith

Any two symplectic forms on a p-adic analytic manifold are locally isomorphic.

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

The paper proves a non-Archimedean version of Darboux's theorem. Any two symplectic forms on a p-adic analytic manifold can be matched locally by a coordinate change. This coordinate change comes from the flow of a vector field, and the flow is a power series with positive radius of convergence. If this holds, p-adic phase spaces are locally standard, so attention shifts from the geometry of the space to the equations of the dynamics. The result also yields a classification of second-countable p-adic analytic symplectic manifolds by their p-adic volume.

Core claim

We prove that any two symplectic forms on a p-adic analytic manifold are locally isomorphic. The proof uses a non-Archimedean version of Moser's path method to push one form onto the other by a flow. The central technical step shows that this flow is given by a power series with non-zero radius of convergence, using geometric analytic estimates rather than algebraic arguments alone. As a global consequence, second-countable p-adic analytic symplectic manifolds are classified up to isomorphism by their p-adic volume.

What carries the argument

Non-Archimedean Moser's path method that produces a vector field whose flow is a power series with positive radius of convergence.

If this is right

  • The phase space defined by a p-adic manifold is locally standard.
  • Attention can be concentrated on the equations defining the dynamics rather than on the space itself.
  • Local problems such as the existence of flows or the normalization of singularities in integrable systems can be studied in standard coordinates.
  • Second-countable p-adic analytic symplectic manifolds are classified in terms of p-adic volume.

Where Pith is reading between the lines

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

  • Other local normal-form results from symplectic geometry may extend to the p-adic setting if analogous convergence estimates can be established.
  • The theorem suggests that p-adic versions of Hamiltonian mechanics could use the same local coordinates as in the real case for small-scale analysis.
  • Explicit low-dimensional examples, such as p-adic tori with standard symplectic forms, could be used to compute and verify the actual radius of convergence.

Load-bearing premise

The vector field from the non-Archimedean Moser's method generates a flow that is a power series with positive radius of convergence, relying on geometric analytic estimates.

What would settle it

A concrete p-adic analytic manifold together with two symplectic forms for which no local diffeomorphism equating the forms has coordinate functions given by a power series with positive radius of convergence.

Figures

Figures reproduced from arXiv: 2508.15443 by \'Alvaro Pelayo, Luis Crespo.

Figure 1
Figure 1. Figure 1: Illustration of the p-adic analytic Darboux’s Theorem (Theo￾rem B). It follows from Theorem B that the only local invariant of p-adic analytic symplectic manifolds is the dimension. See [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: p-adic tubular neighborhoods of dimension 0, 1 and 2 subman￾ifolds of a dimension 3 manifold as in Definition 2.6. Definition 2.6 (p-adic Tubular Neighborhood). Let ℓ be a positive integer. Let p be a prime number. Let M be an ℓ-dimensional p-adic analytic manifold and Q a compact submanifold of M. Let N(Q) be the normal bundle of Q: N(Q) = n (x, v) ∈ TM : v Tu = 0 for all u ∈ TxQ o . For k ∈ N ∪ {0}, let … view at source ↗
Figure 3
Figure 3. Figure 3: Illustration of the proof of Theorem 4.1. First figure: the form α = ω1 −ω0 is exact on U, that is, it is dβ for some 1-form β, and this form is given by a power series in a subset U2 ⊂ U. Second figure: we determine S as a subset of U2 × T and a subset U3 ⊂ U2 that avoids the points in S. Third figure: the vector field Xt on U3. Fourth figure: the resulting sets U0 and U1. Proof of Theorem C. The proof we… view at source ↗
Figure 4
Figure 4. Figure 4: Steps of the proof of Theorem C. First figure: the manifold M, the compact submanifold Q, and a tubular neighborhood U of Q. Second figure: open subsets U1, U2, U3, U4 covering Q such that ω0 and β are given as power series. Third figure: we reduce Ui to U ′ i to avoid the points in Si . Fourth figure: the vector field Xi,t in U ′ i and the resulting set U ′′ i . Fifth figure: the set U0 is the union of al… view at source ↗
Figure 5
Figure 5. Figure 5: The balls Bki, for p = 3, d = 2, 0 ⩽ k ⩽ 2, and 1 ⩽ i ⩽ 8. The ball Bki has radius p k and a color which depends on i. In order to prove Theorem E, the starting point is to use Theorem B to write a p-adic analytic symplectic manifold as a disjoint union of balls. Lemma 6.1. Let p be a prime number. Let (M, ω) be a second countable p-adic analytic symplectic manifold. Then (M, ω) is symplectomorphic to a di… view at source ↗
Figure 6
Figure 6. Figure 6: An example of the proof of Theorem E. The colored balls form a p-adic analytic symplectic manifold in (Q3) 2 . Those with radius 3 can be mapped to B11, B12, B13 and B14 according to their colors, while those of radius 1 are mapped to B01, B02, B03, B04, B05, and the remaining nine to B15. A generalization of this model changes the symplectic form to ω = iX j∈J dψj ∧ dψ¯ j 1 + µ|ψj | 2 and takes as Hamilto… view at source ↗
Figure 7
Figure 7. Figure 7: A representation of the ball B(0, 25) in (Q5) 2 . Each point is a ball of radius 1. The red point in the center is B(0, 1) and the green points form B(0, 5) [PITH_FULL_IMAGE:figures/full_fig_p027_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Left: a representation of the ball B(0, 25) in (Q5) 3 . Each square is a ball of radius 5. Right: the ball B(0, 5), where each point is a ball of radius 1 and the red point in the center is B(0, 1). 27 [PITH_FULL_IMAGE:figures/full_fig_p027_8.png] view at source ↗
read the original abstract

We prove a non Archimedean Darboux's Theorem: any two symplectic forms on a $p$-adic analytic manifold are locally isomorphic. Understanding local problems such as the existence of flows or the normalization of singularities in the theory of integrable systems, is essential to understand the physics behind these systems. Our result tells us that the phase space defined by a $p$-adic manifold is locally standard, allowing us to concentrate on the equations defining the dynamics rather than on the space itself. Our proof uses a non Archimedean version of Moser's Path Method to push one symplectic form onto another one by a flow. A central technical contribution of the paper is the proof that the flow is given by a power series with \emph{non zero radius of convergence}, which requires geometric analytic estimates and does not follow from algebraic considerations. As a global application, we derive a classification of second-countable $p$-adic analytic symplectic manifolds in terms of $p$-adic volume, which generalizes a classical theorem of J-P. Serre.

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 / 1 minor

Summary. The paper proves a non-Archimedean Darboux theorem asserting that any two symplectic forms on a p-adic analytic manifold are locally isomorphic. The proof adapts Moser's path method to construct a time-dependent vector field whose flow realizes the isomorphism between the forms. A central technical contribution is the demonstration, via geometric analytic estimates rather than purely algebraic arguments, that this flow is given by a power series possessing non-zero radius of convergence. The local result is applied to obtain a classification of second-countable p-adic analytic symplectic manifolds in terms of p-adic volume, generalizing a theorem of Serre.

Significance. If the radius-of-convergence estimates hold, the result supplies a local normal-form theorem in p-adic symplectic geometry that reduces local questions about integrable systems and dynamics to the standard model. The explicit flow construction and the global classification by volume furnish both constructive and topological content that parallels classical statements in real or complex symplectic geometry.

major comments (1)
  1. The geometric analytic estimates establishing that the time-1 map of the Moser flow is a power series with positive radius of convergence are load-bearing for the local-isomorphism claim. The manuscript must show explicitly that these estimates control the p-adic valuation growth of all higher-order coefficients (i.e., that lim sup |a_k|^{1/k} remains finite in the non-Archimedean norm) arising from the Lie derivatives along the time-dependent vector field; without such control the radius may vanish for some symplectic forms, as noted in the stress-test concern.
minor comments (1)
  1. The statement of the main theorem should specify the dimension of the manifold and the precise p-adic field (or ring of integers) under consideration.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for emphasizing the centrality of the radius-of-convergence estimates. We address the major comment below and have revised the manuscript to make the control on coefficient growth fully explicit.

read point-by-point responses

  1. Referee: The geometric analytic estimates establishing that the time-1 map of the Moser flow is a power series with positive radius of convergence are load-bearing for the local-isomorphism claim. The manuscript must show explicitly that these estimates control the p-adic valuation growth of all higher-order coefficients (i.e., that lim sup |a_k|^{1/k} remains finite in the non-Archimedean norm) arising from the Lie derivatives along the time-dependent vector field; without such control the radius may vanish for some symplectic forms, as noted in the stress-test concern.

    Authors: We agree that explicit verification of the growth control is necessary for clarity. The geometric analytic estimates in Section 3 bound the p-adic norms of all iterated Lie derivatives of the symplectic forms along the Moser vector field. These bounds are uniform for analytic forms and directly imply that the coefficients a_k of the time-1 flow satisfy v_p(a_k) >= c k - d for constants c, d > 0, which forces lim sup |a_k|^{1/k} to be finite (in fact bounded by an explicit constant depending only on the radius of the initial forms). We will add a short new lemma (Lemma 3.7) that extracts this lim-sup bound from the existing estimates and confirms the radius remains positive. The stress-test concern does not arise under our hypotheses because the estimates hold uniformly on compact p-adic analytic charts; we have revised the manuscript to include this explicit step. revision: yes

Circularity Check

0 steps flagged

No circularity: explicit Moser flow construction with independent analytic estimates

full rationale

The derivation proceeds by constructing a time-dependent vector field via the non-Archimedean Moser equation and then proving that its flow is a convergent power series on a p-adic ball. This convergence is established by geometric analytic estimates on the Lie derivatives and higher-order terms, which the paper explicitly states do not follow from algebraic considerations alone. No quantity is defined in terms of the target result, no parameter is fitted to the output, and no load-bearing step reduces to a self-citation or prior ansatz by the same authors. The central claim is therefore an independent existence proof rather than a tautology or renaming.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The proof relies on standard properties of p-adic analytic manifolds and the existence of a non-Archimedean flow; no new free parameters or invented entities are introduced.

axioms (2)
  • domain assumption p-adic analytic manifolds admit a well-defined notion of symplectic form and local coordinates in which the standard form can be written
    Invoked when stating the local isomorphism target.
  • domain assumption The non-Archimedean version of Moser's path method is well-defined on p-adic manifolds
    Central technical step of the proof.

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