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

Abstract

We establish the foundations of the local linear symplectic geometry of $p$-adic integrable systems on $p$-adic analytic symplectic $4$\--dimensional manifolds, by classifiying all their possible local linear models. In order to do this we develop a new approach, of independent interest, to the theory of Weierstrass and Williamson concerning the diagonalization of real matrices by real symplectic matrices. We show that this approach can be generalized to $p$\--adic matrices, leading to a classification of real $(2n)$-by-$(2n)$ matrices and of $p$-adic $2$-by-$2$ and $4$-by-$4$ matrix normal forms, including, up to dimension $4$, the classification in the degenerate case, for which the literature is limited even in the real case. A combination of these results and the Hardy-Ramanujan formula shows that both the number of $p$-adic matrix normal forms and the number of local linear models of $p$-adic integrable systems grow almost exponentially with their dimensions, in strong contrast with the real case. The paper also includes a number of results concerning symplectic linear algebra over arbitrary fields in arbitrary dimensions as well as applications to $p$-adic mechanical systems and singularity theory for $p$-adic analytic maps on $4$-manifolds. These results fit in a program, proposed a decade ago by Voevodsky, Warren and the second author, to develop a $p$-adic theory of integrable systems with the goal of later implementing it using proof assistants.

Figures (18)

• Figure 1: Normal forms of regular and critical points of elliptic-elliptic, focus-focus and elliptic-regular type of an integrable system $F:\mathbb{R}^4\to\mathbb{R}^2$. Some of these can be normal forms of Theorem \ref{['thm:integrable']} (see Remark \ref{['rem:elliptic']}).
• Figure 2: Symbolic representation of $2$-dimensional fiber of focus-focus model if $p\equiv 1\mod 4$, as a case of point (1) of Theorem \ref{['thm:integrable']}, which coincides with the elliptic-elliptic model. The four "cones" are $2$-dimensional planes in $4$-dimensional space.
• Figure 3: Symbolic representation of $2$-dimensional fiber of focus-focus model if $p\not\equiv 1\mod 4$, as a case of point (2) of Theorem \ref{['thm:integrable']}, which coincides with the fiber in the real case. The two "cones" are actually $2$-dimensional planes in $4$-dimensional space that meet at a point.
• Figure 4: A diagram of the normal forms of Theorem \ref{['thm:williamson4']} for $p\equiv 1\mod 4$ (first row), $p\equiv 3\mod 4$ (second row) and $p=2$ (third to fifth row). Each point (if it is in a single cell) or line (if it is in two cells) represents a normal form. The numbers below the tables represent the first extension of $\mathbb{Q}_p$ (the one containing the squares of the eigenvalues of $\Omega_0^{-1}M$) and those in the rows and columns represent the second extension (for the eigenvalues themselves). In the first table in each block, there are two such extensions corresponding to the row and the column; in the other ones, there is one extension obtained multiplying the numbers in the row and column.
• Figure 5: 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 $0$.

Theorems & Definitions (165)

• Definition 1.1: Non-residue sets and coefficient functions
• Theorem A: $p$-adic integrable local linear models in dimension $4$
• Remark 1.2
• Remark 1.3
• Remark 1.4
• Theorem B: Number of $p$-adic integrable local linear models, in dimension $4$
• Remark 1.5
• Remark 1.6
• Definition 1.7
• Theorem C: Number of $p$-adic integrable local linear models, arbitrary dimension