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

TL;DR

This paper analyzes the classification of symmetric matrices over p-adic fields, providing canonical forms and counting the number of normal form families, revealing exponential growth with matrix size.

Contribution

It extends previous work by explicitly classifying symmetric matrices over -fields, counting normal form families, and analyzing growth patterns using Galois theory.

Findings

Classification of 2x2 symmetric matrices over -fields

Counting of normal form families depending on prime p

Exponential growth of normal forms with matrix size

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.