$p$-adic symplectic geometry of integrable systems and Weierstrass-Williamson theory II
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 , first we show that every -by- symmetric matrix with coefficients in 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 . Then we make the same analysis for -by- 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 -adic extension fields.
Keywords
Cite
@article{arxiv.2602.09898,
title = {$p$-adic symplectic geometry of integrable systems and Weierstrass-Williamson theory II},
author = {Luis Crespo and Álvaro Pelayo},
journal= {arXiv preprint arXiv:2602.09898},
year = {2026}
}
Comments
62 Pages, 9 Figures. This paper is a sequel (part II), to arXiv:2501.14444 (part I). Part I contained classification statements concerning p-adic integrable systems and p-adic matrices. In Part I we gave proofs of all statements concerning p-adic integrable systems. In part II we provide proofs of all statements (which we repeat to be as self contained as possible) concerning p-adic matrices