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

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.

💡 Research Summary

The paper is a sequel to arXiv:2501.14444 and supplies the missing proofs of Theorems D–L announced there. Its central theme is the symplectic classification of symmetric matrices over the p‑adic field ℚₚ, a problem that underlies the p‑adic symplectic geometry of integrable systems introduced in the first part. The authors adopt a two‑step strategy: first they work over an algebraically closed field of characteristic ≠ 2, where the classification is much simpler, and then they descend to ℚₚ (or ℝ) using Galois theory.

In Section 2 they develop the general algebraic machinery. For a symmetric matrix M they study the operator A = Ω₀⁻¹M, where Ω₀ is the standard symplectic form. When the eigenvalues of A are pairwise distinct, Lemma 2.1 provides a symplectic basis consisting of eigenvectors with opposite eigenvalues. When all eigenvalues are zero, they introduce the notion of a “good tuple” K = (k₁,…,k_t) and an involution f_K that pairs odd‑length Jordan blocks. Theorem 2.4 shows that one can choose a basis {u_{i,j}} indexed by K such that Au_{i,j}=u_{i,j‑1} and the symplectic form pairs u_{i,j} only with u_{f_K(i),k_i+1‑j}. The technical lemmas (2.6–2.11) control the inner products of these basis vectors and allow an inductive construction of an “R‑acceptable” basis, which is essential for later descent arguments.

Section 3 specializes to 2 × 2 symmetric matrices over ℚₚ. The characteristic polynomial of A is quadratic, so three cases arise: two distinct p‑adic eigenvalues, a pair of non‑real conjugate eigenvalues (living in a quadratic extension), or a double eigenvalue. By analyzing the action of the Galois group of the relevant extension, the authors obtain a complete list of symplectic normal forms (Theorem 3.10). They also count the families of normal forms that depend on a single parameter and the isolated normal forms, showing that the numbers depend on the congruence class of p modulo 4 (Theorem 3.11).

Sections 4 and 5 treat 4 × 4 matrices. The eigenvalue structure is richer, and the authors separate the non‑degenerate case (all eigenvalues distinct) from the degenerate case (some eigenvalues repeated). In the non‑degenerate situation Theorem 5.11 classifies normal forms into elliptic, hyperbolic, and focus‑focus blocks, each parametrized by a p‑adic invariant r_i. In the degenerate situation Theorem 5.13 shows how Jordan blocks of size >1 intertwine with symplectic constraints, again using the good‑tuple machinery. Theorem 5.15 gives the exact count of normal forms as a function of p, highlighting a dramatic increase in the number of families when p = 2 versus odd primes.

Section 6 revisits the classical Weierstrass‑Williamson theorem for real symmetric matrices. Rather than working directly over ℝ, the authors lift the problem to ℂ (the algebraic closure of ℝ), apply the classification from Section 2, and then descend by imposing invariance under complex conjugation. This yields a new proof of the Williamson normal form (Theorem 6.2) and its extension to the general case (Theorem 6.3). The approach demonstrates that the same “lift‑to‑closure, descend‑via‑Galois” paradigm works uniformly for ℝ and ℚₚ.

Section 7 investigates the asymptotic growth of the number of symplectic normal forms for 2n × 2n matrices as n increases. Theorem 7.2 proves that the count grows essentially exponentially, i.e., like exp(c·n) for a constant c depending on p and on whether the matrices are non‑degenerate. Theorem 7.3 provides explicit lower bounds for n ≤ 10, confirming the rapid growth already at modest dimensions.

Section 8 applies the theory to a concrete p‑adic integrable model, the p‑adic Jaynes‑Cummings system, showing how the normal‑form classification simplifies the analysis of its Hamiltonian matrix. Section 9 supplies numerous explicit examples (for p = 5, 7, etc.) illustrating the theorems.

Overall, the paper makes several substantial contributions:

1. It introduces the concepts of “good tuples” and “R‑acceptable bases,” which allow a systematic handling of Jordan blocks under symplectic congruence.
2. It provides a complete, p‑dependent classification of 2 × 2 and 4 × 4 symmetric matrices over ℚₚ, including exact counts of parameter families and isolated forms.
3. It offers a novel proof of the real Weierstrass‑Williamson theorem via algebraic closure and Galois descent, unifying the real and p‑adic treatments.
4. It establishes that the number of symplectic normal forms grows almost exponentially with matrix size, a quantitative insight relevant for high‑dimensional p‑adic dynamical systems.

These results deepen the understanding of symplectic congruence over non‑Archimedean fields and lay a solid algebraic foundation for future work in p‑adic quantum mechanics, non‑Archimedean integrable systems, and arithmetic symplectic geometry.