The contrast between real and $p$-adic integrability

Abstract

A fundamental question in $p$-adic symplectic geometry and $p$-adic Hamiltonian mechanics is to what extent we can learn new information about integrable systems with coefficients in the rational numbers $\mathbb{Q}$, if we view them as $p$-adic systems instead of real ones. We combine geometric analytic estimates with Galois theory in order to show that at a local level, these real and $p$-adic viewpoints give completely different information. Then we focus on understanding this local behavior for the $p$-adic case by proving effective methods to compute local normal forms. Our results are motivated by a program outlined by Pelayo-Voevodsky-Warren a decade ago, to develop a $p$-adic theory of integrable systems and symplectic geometry.

arXiv comment22 pages, 2 figures. Presentation reorganized and simplified. Added motivations from physics. Main results unchanged. Added Theorem B. As a result, old Theorems B, C are now Theorems C, D. To make the paper more concise we removed previous sections 6-10 which essentially concerned examples and applications, and which we may include in a separate paper