$p$-adic symplectic geometry of integrable systems and Weierstrass-Williamson theory II
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914319453650944 |
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| author | Crespo, Luis Pelayo, Álvaro |
| author_facet | Crespo, Luis Pelayo, Álvaro |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_09898 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | $p$-adic symplectic geometry of integrable systems and Weierstrass-Williamson theory II Crespo, Luis Pelayo, Álvaro Symplectic Geometry Number Theory 11C20, 11E95, 37J06, 22E35, 53D05, 12F05 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. |
| title | $p$-adic symplectic geometry of integrable systems and Weierstrass-Williamson theory II |
| topic | Symplectic Geometry Number Theory 11C20, 11E95, 37J06, 22E35, 53D05, 12F05 |
| url | https://arxiv.org/abs/2602.09898 |