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Rigidity and flexibility in $p$-adic symplectic geometry

Luis Crespo, Álvaro Pelayo

TL;DR

This work develops a comprehensive picture of rigidity and flexibility in $p$-adic symplectic geometry. It proves a linear non-squeezing analogue for $p$-adic affine embeddings while revealing striking nonlinear flexibility: the whole $p$-adic space $(\mathbb{Q}_p)^{2n}$ can be symplectically squeezed into a thin cylinder via a $p$-adic analytic symplectomorphism. Introducing a polar-coordinate framework and symmetry considerations, the authors establish equivariant squeezing results and construct $p$-adic analytic symplectic capacities, including $(\mathrm{G}_p)^n$-capacities, while proving the nonexistence of a global capacity in the general ($n\ge 2$) setting. The results illuminate a nuanced contrast between real and $p$-adic symplectic topology, with implications for how symmetry and arithmetic structure govern rigidity and embedding phenomena in non-Archimedean contexts.

Abstract

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.

Rigidity and flexibility in $p$-adic symplectic geometry

TL;DR

This work develops a comprehensive picture of rigidity and flexibility in -adic symplectic geometry. It proves a linear non-squeezing analogue for -adic affine embeddings while revealing striking nonlinear flexibility: the whole -adic space can be symplectically squeezed into a thin cylinder via a -adic analytic symplectomorphism. Introducing a polar-coordinate framework and symmetry considerations, the authors establish equivariant squeezing results and construct -adic analytic symplectic capacities, including -capacities, while proving the nonexistence of a global capacity in the general () setting. The results illuminate a nuanced contrast between real and -adic symplectic topology, with implications for how symmetry and arithmetic structure govern rigidity and embedding phenomena in non-Archimedean contexts.

Abstract

Let be an integer and let be a prime number. We prove that the analog of Gromov's non-squeezing theorem does not hold for -adic embeddings: for any -adic absolute value , the entire -adic space is symplectomorphic to the -adic cylinder of radius , showing a degree of flexibility which stands in contrast with the real case. However, some rigidity remains: we prove that the -adic affine analog of Gromov's result still holds. We will also show that in the non-linear situation, if the -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 -adic analytic symplectic capacities, of which the -adic equivariant Gromov width is an example.
Paper Structure (21 sections, 38 theorems, 196 equations, 16 figures)

This paper contains 21 sections, 38 theorems, 196 equations, 16 figures.

Table of Contents

  1. Introduction
  2. $p$-adic linear non-squeezing and $p$-adic non-linear squeezing
  3. $p$-adic squeezing and non-squeezing while preserving symmetries
  4. Works on $p$-adic geometry and symplectic geometry
  5. $p$-adic balls and $p$-adic cylinders
  6. $p$-adic affine Gromov's non-squeezing
  7. $p$-adic non-linear symplectic squeezing
  8. Characterization of the $p$-adic linear squeezing property
  9. $p$-adic linear symplectic width and the maps which preserve it
  10. General definitions and results
  11. $p$-adic linear symplectic width of $p$-adic ellipsoids
  12. A construction of $p$-adic polar coordinates
  13. Nonlinear $p$-adic equivariant squeezing and nonlinear $p$-adic equivariant non-squeezing
  14. $p$-adic analytic symplectic capacities and $p$-adic analytic symplectic $G$-capacities
  15. Examples
  16. ...and 6 more sections

Key Result

Theorem A

Let $n$ be an integer with $n\geqslant 2$. Let $p$ be a prime number. Let $r$ be a $p$-adic absolute value. There exists a $p$-adic affine symplectic embedding $f : \mathrm{B}_p^{2n}(r) \hookrightarrow \mathrm{Z}_p^{2n} (1)$ if and only if $r \leqslant 1$.

Figures (16)

  • Figure 1: Each one of the four figures is a symbolic representation of $\mathrm{B}_2^m(r_0)=\{(v_1,v_2,\ldots,v_m)\in(\mathbb{Q}_2)^{m}:\|(v_1,v_2,\ldots,v_m)\|_2<2r_0\},$as it appears in expression \ref{['eq:ball']}, for some $m$ ($2$ in the upper figures and $3$ in the lower figures) and $r_0$ ($16$ and $8$ respectively). Each dot is a $2$-adic ball of radius $1$, and the $2$-adic balls that are closer in the representation are also $2$-adically closer. The dots of the same color are those $2$-adic balls that are contained in $\mathrm{B}_2^s(r)\times(\mathbb{Q}_2)^{m-s},$where $s$ is $1$, $2$ or $3$ depending on the figure and $r$ is $1$ for the red dots, $2$ for orange, $4$ for yellow, $8$ for green and $16$ for blue.
  • Figure 2: Each one of the four figures is a symbolic representation of $\mathrm{B}_3^m(r_0)=\{(v_1,v_2,\ldots,v_m)\in(\mathbb{Q}_3)^{m}:\|(v_1,v_2,\ldots,v_m)\|_3<3r_0\},$as it appears in expression \ref{['eq:ball']}, for some $m$ ($2$ in the upper figures and $3$ in the lower figures) and $r_0$ ($27$ and $9$ respectively). Each dot is a $3$-adic ball of radius $1$, and the $3$-adic balls that are closer in the representation are also $3$-adically closer. The dots of the same color are those $3$-adic balls that are contained in $\mathrm{B}_3^s(r)\times(\mathbb{Q}_3)^{m-s},$where $s$ is $1$, $2$ or $3$ depending on the figure and $r$ is $1$ for the red dots, $3$ for orange, $9$ for green and $27$ for blue.
  • Figure 3: Illustration of the proof of Theorem \ref{['thm:linear']}, or more precisely of its real analog. If $u_2=S^{\mathrm{T}} \mathrm{e}_2$ is longer than $\mathrm{e}_2$ and the cylinder is narrower than the ball, $f$ would send out of the cylinder a vector as long as possible in any direction close to $u_2$.
  • Figure 4: Gromov's non-squeezing theorem tells us that it is not possible to embed a ball inside a cylinder narrower than the ball while preserving the symplectic form. Theorem \ref{['thm:total-embedding']} shows that the situation is different for $p$-adic manifolds: the entire $p$-adic $2n$-dimensional space $(\mathbb{Q}_p)^{2n}$ is symplectomorphic to any thin $p$-adic cylinder of the same dimension.
  • Figure 5: A representation of the $p$-adic analytic symplectomorphism of Theorem \ref{['thm:total-embedding']} for $p=3$. The horizontal and vertical directions represent $x_1$ and $x_2$. The embedding is continuous in the $p$-adic case because the balls of radius $1$ (represented here by squares) are at a fixed distance from each other. In the real case the balls would share common boundaries, and the embedding is discontinuous.
  • ...and 11 more figures

Theorems & Definitions (103)

  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Definition 3.1
  • Definition 3.2: $p$-adic linear/affine symplectomorphism of $(\mathbb{Q}_p)^{2n}$
  • Definition 3.3: $p$-adic affine symplectic embedding between open subsets of $(\mathbb{Q}_p)^{2n}$
  • Theorem 3.4: $p$-adic analog of the affine Gromov's non-squeezing theorem
  • proof
  • Definition 4.1: $p$-adic analytic symplectic manifolds and embeddings
  • ...and 93 more