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

Abstract

Let $p$ be a prime number and $n$ a positive integer. The study of normal forms of $p$-adic analytic integrable systems $F=(f_1,\ldots,f_n):(M,ω)\to(\mathbb{Q}_p)^n$ 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.