$p$-adic symplectic geometry of integrable systems and Weierstrass-Williamson theory II

TL;DR

This work delivers a comprehensive algebraic/arithmetic treatment of the Weierstrass-Williamson-type classification for $p$-adic symmetric matrices under congruence by symplectic transformations. By lifting problems to algebraically closed fields, exploiting Jordan bases and DSq/Hilbert-symbol machinery, and then constraining back to $\mathbb{Q}_p$, the authors obtain explicit normal forms in dimensions 2 and 4, with detailed case distinctions for $p\neq2$ and $p=2$, and extend the framework to higher dimensions. The results underpin the $p$-adic local models of integrable systems (CrePel-integrable) and their real counterparts, yielding exact counts of infinite and isolated normal form families, asymptotic growth in dimension, and concrete applications to classical mechanical systems such as the Jaynes-Cummings model. The approach reveals both similarities and crucial differences with the real theory, highlights the role of field extensions in the $p$-adic setting, and provides a scalable blueprint for analyzing $p$-adic integrable systems through their linearized (quadratic) constituents.

Abstract

This paper is a sequel to arXiv:2501.14444, in which we shall give proofs of several results stated in arXiv:2501.14444 (Theorems D--L) which, for brevity and clarity, we postponed to this sequel paper. These results were the following: for any prime number $p$, first we show that every $2$-by-$2$ symmetric matrix with coefficients in $\mathbb{Q}_p$ can be reduced to a canonical form, and we give the exact numbers of families of normal forms with one parameter and of isolated normal forms, which depend on $p$. Then we make the same analysis for $4$-by-$4$ matrices. We also prove that, for higher size, the number of families of normal forms of matrices, even in the non-degenerate case, grows almost exponentially with the size. The paper can be read independently of arXiv:2501.14444 as we recall the statements of arXiv:2501.14444 that we shall prove here. The statements and proofs of the present paper are of an algebraic and arithmetical nature, and rely mainly on Galois theory of $p$-adic extension fields.

Figures (9)

• Figure 1: Top: $\mathop{\mathrm{DSq}}\nolimits(\mathbb{Q}_p,c)$ for $c\in\mathbb{Q}_p$ and $p\ne 2$. In each group of four circles, the upper circles represent even order numbers and the lower circles odd order, and the right circles represent square leading digits and the left circles non-square digits. Bottom: these four classes depicted for $p=3$. Each circle "contains" the points with the same color and the black point at the right is $0$.
• Figure 2: Top: $\mathop{\mathrm{DSq}}\nolimits(\mathbb{Q}_2,c)$ for $c\in\mathbb{Q}_2$. In each group of eight circles, the upper circles represent even order numbers and the lower circles odd order, the two rightmost circles in the row represent a $0$ as second digit and the two leftmost circles a $1$, and in each pair of circles, the rightmost one has $0$ as third digit and the leftmost one has $1$. Bottom: a depiction of the eight classes. Each circle "contains" the points with the same color and the black point in the lower left is $0$.
• Figure 3: The $16$ classes of Table \ref{['table:-1']}. Each class contains the points $x+y\mathrm{i}$ with a given symbol, where $x$ and $y$ are the horizontal and vertical coordinates. The circles, triangles, squares and diamonds correspond to the four values of the first index (here $1$, $1+\mathrm{i}$, $1+2\mathrm{i}$ and $-1+3\mathrm{i}$), and the colors red, green, purple and blue to the four values of the second index (here $1$, $2$, $3$ and $6$).
• Figure 4: The $16$ classes of Table \ref{['table:2']}.
• Figure 5: The $16$ classes of Table \ref{['table:-2']}.

Theorems & Definitions (103)

• Lemma 2.1: CrePel-integrable
• Lemma 2.2: CrePel-integrable
• Definition 2.3
• Theorem 2.4
• Definition 2.5
• Lemma 2.6
• proof
• Corollary 2.7
• proof
• Corollary 2.8