Darboux's Theorem in $p$-adic symplectic geometry

TL;DR

This paper establishes a non-Archimedean Darboux's Theorem for $p$-adic symplectic manifolds, showing local standardization of phase spaces and providing a classification based on $p$-adic volume.

Contribution

It introduces a non-Archimedean version of Moser's Path Method with convergence guarantees and classifies $p$-adic symplectic manifolds via $p$-adic volume.

Findings

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

The flow used in the proof is given by a convergent power series.

Classifies $p$-adic symplectic manifolds in terms of $p$-adic volume.

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.