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A p-adic Eliasson–Vey linearization theorem

Determine necessary and sufficient conditions on the Williamson type of a non-degenerate critical point m of a p-adic analytic integrable system F:(M,\omega)→(Q)^n that guarantee the existence of open sets U⊂M and V⊂(Q)^{2n}, a p-adic analytic symplectomorphism \phi:V→U with \phi(0)=m, and a local diffeomorphism \varphi of (Q)^n such that (F−F(m))∘\phi=\varphi∘(g_1,…,g_n), where (g_1,…,g_n) is the Williamson normal form of F at m.

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Background

In the real analytic case, the Eliasson–Vey theorem provides linearization of integrable systems near non-degenerate singularities under certain conditions (e.g., absence of hyperbolic components).

The authors pose the p-adic analogue, asking for precise conditions under which a local p-adic analytic symplectomorphism and a local diffeomorphism achieve the Williamson normal form at a non-degenerate critical point, highlighting the lack of a p-adic counterpart to the classical result.

References

The p-adic equivalent of this theorem is well beyond the scope of this paper, and we state it as a question. Question [A p-adic Eliasson-Vey's theorem?] Let n be a positive integer. Let p be a prime number. Given a 2n-dimensional p-adic analytic symplectic manifold (M,\omega), an integrable system F:(M,\omega)\to (Q)n and a non-degenerate critical point m of F, determine under which conditions on the Williamson type of the critical point m there are open sets U\subset M and V\subset (Q){2n}, a p-adic analytic symplectomorphism \phi:V\to U and a local diffeomorphism \varphi of (Q)n such that \phi(0)=m and [(F-F(m))\circ \phi=\varphi\circ(g_1,\ldots,g_n)], where (g_1,\ldots,g_n) is the Williamson normal form of F in m. (In the real case it is enough that there are no hyperbolic blocks.)

— $p$-adic symplectic geometry of integrable systems and Weierstrass-Williamson theory (2501.14444 - Crespo et al., 24 Jan 2025) in Question [A p-adic Eliasson-Vey's theorem?], Section 12 (Application to normal forms of singularities of integrable systems)