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

Abstract This paper is a sequel to [3], in which we shall give proofs of several results stated in [3] (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 $$\mathbb {Q}_p$$ 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. In addition, we give a new proof of the real Weierstrass-Williamson classification theorem. The paper can be read independently of [3] as we recall the statements of [3] 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.