Rigidity and flexibility in p-adic symplectic geometry

2025/05/12 by Crespo, Luis, Pelayo, Álvaro · 3 citations
#37P05 #53D35 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Symplectic Geometry (math.SG)

paper · doi:10.48550/arxiv.2505.07663

Abstract

Let 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 (ℚ_p)²ⁿ is symplectomorphic to the p-adic cylinder Z_p²ⁿ(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.

Cited by

• Group actions on p-adic symplectic manifolds
• p-adic angular momentum coupling in symplectic geometry
• Darboux's Theorem in p-adic symplectic geometry

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