$p$-adic angular momentum coupling in symplectic geometry

TL;DR

This work constructs a $p$-adic analog of the coupled angular momentum on $S^2_p imes S^2_p$ and performs a detailed symplectic singularity classification of its critical points. Using a $p$-adic Darboux framework and Williamson-type analysis, it derives explicit normal forms for rank-$0$ and rank-$1$ singularities across outer, inner, and limit regions, with precise dependence on the prime $p$ (notably counts that differ for $pmod 4$ and for $p=2$). The main novelty is the richer variety of non-equivalent symplectic normal forms in the $p$-adic setting—up to thirteen—versus the real case, highlighting intricate $p$-adic phenomena in integrable systems. These results advance $p$-adic symplectic geometry and open avenues for further study of $p$-adic analogs of classical integrable systems in mathematical physics.

Abstract

The coupled angular momentum is an integrable system with two degrees of freedom which is fundamental in physics and the theory of integrable systems. It is obtained by coupling two angular momenta. We construct a $p$-adic analog of this system for any prime number $p$ and describe its symplectic normal forms at the critical points. This analog has a rich singularity theory with up to thirteen non-equivalent symplectic normal forms, which stands in contrast with the real case where there are exactly three normal forms.

Figures (8)

• Figure 1: In the real case, the coupled angular momentum has four rank zero non-degenerate critical points corresponding to pairing the north and south poles of each of the two spheres. Three of these points are of elliptic-elliptic type, and the remaining one is of focus-focus type.
• Figure 2: Symbolic representation of $\mathrm{S}^2_p\times\mathrm{S}^2_p$, the phase space of the $p$-adic coupled angular momentum, for $p=3$ and $p=7$ (see Definition \ref{['def:angular']} for the definition of $\mathrm{S}^2_p$). The grey points represent $(\mathbb{Z}_p)^3$, and the blue points are those being in $\mathrm{S}^2_p$. Each point has radius $1/9$ above and $1/7$ below.
• Figure 3: The fiber of a focus-focus critical value around the singularity in the real case (above) and the $p$-adic case (below, with the critical point in red). One of the rank $0$ critical points of the (real) coupled angular momentum system has this type for certain values of the parameter $t$; its $p$-adic equivalent $S=(1,0,0,-1,0,0)$ may or may not have the same type depending on the parameters.
• Figure 4: The outer (blue), inner (red) and limit (purple) regions of Definition \ref{['def:regions']} for $p=3$. The $x$-axis is the parameter $t$ where each point corresponds to a radius $1/27$, and the $y$-axis is the parameter $k$ where each point corresponds to a radius $1$. The bottom row of points is actually not a valid choice of parameters, because $k$ must not be in $\mathbb{Z}_p$. The region in which the parameters are has an effect on the normal forms: in the inner and limit regions, according to Sections \ref{['sec:ST-inner']} and \ref{['sec:ST-limit']} respectively, $S$ and $T$ can have up to $5$ different forms, while in the outer region, according to Section \ref{['sec:ST-outer']}, they can only have one or two forms.
• Figure 5: Representation of the orders of the eigenvalues $\lambda$ and $\mu$ (first graphic) and of $\lambda+\mu$ and $\lambda-\mu$ (second graphic) in terms of the order of $2t-1$. The part at the left of $-\mathop{\mathrm{ord}}\nolimits(k)/2$ is the outer region, and the part at the right is the inner region. At the points where one of the graphs changes slope, the orders may not be correct, because there may be a cancellation between terms in a sum with the same order, resulting in a total with greater order than the terms (this is what happens in the limit region).

Theorems & Definitions (69)

• Definition 1.1: $p$-adic coupled angular momentum
• Definition 1.2: $p$-adic non-degenerate critical point CrePel-williamson
• Theorem A: Basic properties of the critical points of the $p$-adic coupled angular momentum
• Definition 1.3: Class (1), (2) or (3) normal forms
• Definition 1.4: Class R3 and I3$(c)$ normal forms
• Definition 1.5: $p$-adic local symplectomorphism CrePel-williamson
• Theorem B: Number of normal forms at rank $0$ singularities
• Theorem C: Description of linear normal forms at rank $0$ singularities
• Theorem D: Number of normal forms at rank $1$ singularities
• Theorem E: Description of normal forms at rank $1$ singularities