arXiv subjects

Luis Crespo

Publications and source records attributed to Luis Crespo.

9 recordsLinked to original sources

arXiv2026

$p$-adic integrable systems: from biquadratic equations to local models

Let $p$ be a prime number and $n$ a positive integer. The study of normal forms of $p$-adic analytic integrable systems $F=(f_1,\ldots,f_n):(M,\omega)\to(\mathbb{Q}_p)^n$ is essential to understand their geometrical and dynamical properties. Even though in some cases, such as dimension $4$, there is a classification of the local normal forms, it can be a challenge to determine them explicitly. Our goal in this paper is to introduce techniques to compute information about these local normal forms. We then explain how this is useful for instance to study the $p$-coupled angular momentum. The techniques we introduce cover all cases in dimension $4$ and require solving biquadratic equations. Along the way we define two new notions: almost eigenvectors and aligned symplectic coordinates. They are useful to prove our results but also of independent interest. The proofs use our previous classification of normal forms and rely on a combination of analytic estimates and Galois theory of $p$-adic extension fields. However, the statements of the main results are essentially self-contained and do not require prior knowledge of $p$-adic integrable systems or $p$-adic symplectic geometry.

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arXiv2026

$p$-adic symplectic geometry of integrable systems and Weierstrass-Williamson theory II

This paper is a sequel to arXiv:2501.14444, in which we shall give proofs of several results stated in arXiv:2501.14444 (Theorems D--L) which, for brevity and clarity, we postponed to this sequel paper. These results were the following: for any prime number $p$, first we show that every $2$-by-$2$ symmetric matrix with coefficients in $\mathbb{Q}_p$ can be reduced to a canonical form, and we give the exact numbers of families of normal forms with one parameter and of isolated normal forms, which depend on $p$. Then we make the same analysis for $4$-by-$4$ matrices. We also prove that, for higher size, the number of families of normal forms of matrices, even in the non-degenerate case, grows almost exponentially with the size. The paper can be read independently of arXiv:2501.14444 as we recall the statements of arXiv:2501.14444 that we shall prove here. The statements and proofs of the present paper are of an algebraic and arithmetical nature, and rely mainly on Galois theory of $p$-adic extension fields.

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arXiv2025

Group actions on $p$-adic symplectic manifolds

Let $p$ be a prime number. We introduce symplectic actions of $p$-adic analytic Lie groups on $p$-adic symplectic manifolds. Then we show that any $p$-adic symplectic action $G\times(M,\omega)\to(M,\omega)$ has a momentum map $\mu:M\to\mathfrak{g}^*$, and that a proper $p$-adic symplectic action is Hamiltonian if and only if every orbit is isotropic. We conclude by defining $p$-adic symplectic toric manifolds, by analogy with the real case.

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arXiv2025

$p$-adic angular momentum coupling in symplectic geometry

The coupled angular momentum is an integrable system with two degrees of freedom which is fundamental in physics and the theory of integrable systems. It is obtained by coupling two angular momenta. We construct a $p$-adic analog of this system for any prime number $p$ and describe its symplectic normal forms at the critical points. This analog has a rich singularity theory with up to thirteen non-equivalent symplectic normal forms, which stands in contrast with the real case where there are exactly three normal forms.

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arXiv2025

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

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.

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arXiv2025

Rigidity and flexibility in $p$-adic symplectic geometry

Let $n\ge 2$ be an integer and let $p$ be a prime number. We prove that the analog of Gromov's non-squeezing theorem does not hold for $p$-adic embeddings: for any $p$-adic absolute value $R$, the entire $p$-adic space $(\mathbb{Q}_p)^{2n}$ is symplectomorphic to the $p$-adic cylinder $\mathrm{Z}_p^{2n}(R)$ of radius $R$, showing a degree of flexibility which stands in contrast with the real case. However, some rigidity remains: we prove that the $p$-adic affine analog of Gromov's result still holds. We will also show that in the non-linear situation, if the $p$-adic embeddings are equivariant with respect to a torus action, then non-squeezing holds, which generalizes a recent result by Figalli, Palmer and the second author. This allows us to introduce equivariant $p$-adic analytic symplectic capacities, of which the $p$-adic equivariant Gromov width is an example.

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arXiv2025

$p$-adic symplectic geometry of integrable systems and Weierstrass-Williamson theory I

The local symplectic theory of integrable systems is fundamental to understand their global theory, as well as the behavior near singularities of fundamental models from classical and quantum mechanics which are known to be integrable, such as the Jaynes-Cummings model and the coupled angular momenta. We establish the foundations of the local symplectic geometry of $p$-adic integrable systems on $4$-dimensional $p$-adic analytic symplectic manifolds, by classifiying all their possible local models. In order to do this we develop a new approach, of independent interest, to the theory of Weierstrass and Williamson concerning the diagonalization of real matrices by real symplectic matrices. We show that this approach can be generalized to $p$-adic matrices, leading to a classification of real $(2n)$-by-$(2n)$ matrices and of $p$-adic $2$-by-$2$ and $4$-by-$4$ matrix normal forms, including, up to dimension $4$, the classification in the degenerate case, for which the literature is limited even in the real case. A combination of these results and the Hardy-Ramanujan formula shows that both the number of $p$-adic matrix normal forms and the number of local models of $p$-adic integrable systems grow almost exponentially with their dimensions, in strong contrast with the real case. These results fit in a program, proposed a decade ago by Voevodsky, Warren and the second author, to develop a $p$-adic theory of integrable systems with the goal of later implementing it using proof assistants. In the present paper we shall prove all of the statements concerning integrable systems; their proofs rely on the $p$-adic analog of the Weierstrass and Williamson theory of matrices. The proofs concerning this new $p$-adic theory of matrices are of an algebraic and arithmetical nature, and we give them in the sequel paper, part II.

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arXiv2024

The $p$-adic Jaynes-Cummings model in symplectic geometry

The notion of classical $p$-adic integrable system on a $p$-adic symplectic manifold was proposed by Voevodsky, Warren and the second author a decade ago in analogy with the real case. In the present paper we introduce and study, from the viewpoint of symplectic geometry and topology, the basic properties of the $p$-adic version of the classical Jaynes-Cummings model. The Jaynes-Cummings model is a fundamental example of integrable system going back to the work of Jaynes and Cummings in the 1960s, and which applies to many physical situations, for instance in quantum optics and quantum information theory. Several of our results depend on the value of $p$: the structure of the model depends on the class of the prime $p$ modulo $4$ and $p=2$ requires special treatment.

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arXiv2023

Ewald's Conjecture and integer points in algebraic and symplectic toric geometry

We solve several open problems concerning integer points of polytopes arising in symplectic and algebraic geometry. In this direction we give the first proof of a broad case of Ewald's Conjecture (1988) concerning symmetric integral points of monotone lattice polytopes in arbitrary dimension. We also include an asymptotic quantitative study of the set of points appearing in Ewald's Conjecture. Then we relate this work to the problem of displaceability of orbits in symplectic toric geometry. We conclude with a proof for the $2$-dimensional case, and for a number of cases in higher dimensions, of Nill's Conjecture (2009), which is a generalization of Ewald's conjecture to smooth lattice polytopes. Along the way the paper introduces two new classes of polytopes which arise naturally in the study of Ewald's Conjecture and symplectic displaceability: neat polytopes, which are related to Oda's Conjecture, and deeply monotone polytopes.

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