$p$-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.