Our main goal in this paper is to extend mathematical ideas of Gromov concerning the phenomenon of symplectic squeezing from real to p-adic geometry. Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$n\geqslant 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>⩾</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> 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 <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$(\mathbb {Q}_p)^{2n}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mo stretchy="false">(</mo> <msub> <mi mathvariant="double-struck">Q</mi> <mi>p</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>2</mn> <mi>n</mi> </mrow> </msup> </math></EquationSource> </InlineEquation> is symplectomorphic to the p-adic cylinder <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\textrm{Z}_p^{2n}(R)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mtext>Z</mtext> <mi>p</mi> <mrow> <mn>2</mn> <mi>n</mi> </mrow> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> 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 nonlinear 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

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2026年4月11日

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Luis Crespo,
Álvaro Pelayo

摘要

Our main goal in this paper is to extend mathematical ideas of Gromov concerning the phenomenon of symplectic squeezing from real to p-adic geometry. Let $n\geqslant 2$ $n ⩾ 2$ 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 $(\mathbb {Q}_p)^{2n}$ ${(Q_{p})}^{2 n}$ is symplectomorphic to the p-adic cylinder $\textrm{Z}_p^{2n}(R)$ $Z_{p}^{2 n} (R)$ 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 nonlinear 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.