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

The local symplectic theory of integrable systems is fundamental to understand their global theory, as well as the behavior near singularities of fundamental models from classical and quantum mechanics which are known to be integrable, such as the Jaynes-Cummings model and the coupled angular momenta. We establish the foundations of the local symplectic geometry of $p$-adic integrable systems on $4$-dimensional $p$-adic analytic symplectic manifolds, by classifiying all their possible local 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 models of $p$-adic integrable systems grow almost exponentially with their dimensions, in strong contrast with the real case. 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. In the present paper we shall prove all of the statements concerning integrable systems; their proofs rely on the $p$-adic analog of the Weierstrass and Williamson theory of matrices. The proofs concerning this new $p$-adic theory of matrices are of an algebraic and arithmetical nature, and we give them in the sequel paper, part II.

💡 Research Summary

The paper develops a comprehensive local symplectic theory for integrable systems defined over p‑adic analytic symplectic manifolds, focusing on the four‑dimensional case. The authors first extend the classical Weierstrass‑Williamson classification—originally a real‑matrix diagonalisation result—to the p‑adic setting. By introducing the notions of non‑residue sets (X_{p}) and (Y_{p}) and a family of coefficient functions (C^{i}{k}) and (D^{i}{k}), they obtain normal forms for all (2\times2) and (4\times4) p‑adic matrices under symplectic similarity, including degenerate cases that are scarcely treated in the real literature.

Using this matrix theory, they classify every possible local model of a non‑degenerate critical point of a p‑adic integrable system (F=(f_{1},f_{2})\colon (M,\omega)\to\mathbb Q_{p}^{2}) on a 4‑dimensional p‑adic analytic symplectic manifold. In suitable local symplectic coordinates ((x,\xi,y,\eta)) and after an invertible linear change (B\in M_{2}(\mathbb Q_{p})), the map can be written as \