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Exact counting of p-adic matrix normal forms in arbitrary even dimensions

Develop a general method to compute the exact number of families of non-degenerate normal forms of (2n)-by-(2n) symmetric matrices over the p-adic field Q up to multiplication by a symplectic matrix, by performing the required sum over all partitions of n that accounts for the multiplicity of blocks of each size; extend beyond the current lower-bound approach and small-n computations to arbitrary n.

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Background

The paper establishes lower bounds for the number of families of non-degenerate normal forms of (2n)-by-(2n) p-adic symmetric matrices via a construction using partitions of n and block-diagonal matrices M(P,p). These bounds grow at least as e{\pi\sqrt{2n/3}}/(4n\sqrt{3}).

The authors note that the exact count could be strengthened by incorporating the fact that more than one block of each size can appear, but this requires summing over partitions with appropriate multiplicities. While they provide explicit counts for small n, a general method for arbitrary n is not currently available.

References

Theorem \ref{thm:num-forms} could be strengthened by using that there is not only one block of each size, but this would imply making a sum over the partitions. We do not know how to make that for general n, but we can do it for small n, obtaining the results in Table \ref{table:numforms}.

— $p$-adic symplectic geometry of integrable systems and Weierstrass-Williamson theory (2501.14444 - Crespo et al., 24 Jan 2025) in Remarks and applications, Section 9 (Comments on the p-adic classification in higher dimensions and proof of Theorem \ref{thm:num-forms})