p-adic symplectic geometry is four years old and has an author. Luis Crespo and Álvaro Pelayo have built it in a short sequence: the p-adic Jaynes–Cummings model (2024), integrable systems and Weierstrass–Williamson theory (January 2025), Darboux’s theorem (August 2025), and group actions (December 2025) CHECKED [1]–[4].
Three things in that work matter here. Their Theorem A is a p-adic Moser’s path method, and it needs a hypothesis the real one does not: the vector field’s components must vanish on the relevant compact set together with their partial derivatives. The real trick does not transfer unchanged. Their Theorem B is the local Darboux normal form. And their Theorem E is the one worth staring at — every second-countable 2n-dimensional p-adic analytic symplectic manifold is symplectomorphic to a union of balls determined entirely by volume. That is a global rigidity with no real analogue at all.
What is absent, across all four papers and Pelayo’s publication list, is any mention of contact structures, Reeb fields, Legendrian submanifolds or odd dimensions CHECKED. The programme is stated as even-dimensional and is carried out that way.
The RH paper’s §9 question 3 says the construction of αarith on the adele class space is “to our knowledge, new”, and asks whether the space admits a contact topology — tight versus overtwisted. That question cannot be assessed while it is unknown whether a contact form over ℚp carries any arithmetic at all. This note settles the prior question and, in doing so, weakens the later one. That is the honest order.
The prototype of the RH paper §2 is
and the standard model is α = dz − y dx, with α ∧ dα = dx ∧ dy ∧ dz. Both coefficients are nonzero rationals, so both are nonzero in every ℚp COMPUTED. This is not a coincidence of these two forms: α ∧ (dα)n ≠ 0 is an open algebraic condition, and a form that satisfies it over ℚ satisfies it over every completion. No place is excluded, and there is no p at which the prototype fails to be contact.
The Reeb field is defined by α(R) = 1 and ιR dα = 0. That is a linear system, and linear systems do not care what field they are over. For the prototype it returns R = ∂/∂z COMPUTED — the same answer Book 7 Ch Fy computes in the smooth case.
The next step in the real theory does not come with it, and this is the more useful half of the section. Over ℝ the Reeb field integrates to a flow, and the subject is closed orbits: Weinstein’s conjecture, Taubes’ theorem in dimension three. ℚp has no ℝ acting on it, and a p-adic analytic vector field integrates only on a ball of bounded radius. So “closed Reeb orbit” has no p-adic meaning as stated OPEN.
The sentence above says that a question is not yet askable, which is weaker than saying the answer is no. Ch Fy already showed the smooth prototype has no closed Reeb orbit at all, so the analogy was not going to import anything even if it typechecked. Both facts point the same way and neither is a proof about αarith.
The natural hope is that a contact structure over ℚp splits into classes the way a quadratic form does. Over ℚp the square classes are finite and few — four for odd p, eight for p = 2 COMPUTED — and they separate ⟨1,1⟩ from ⟨1,u⟩ for u a non-square. That is why p-adic orthogonal geometry is arithmetically rich.
Alternating forms have no such invariant, and the failure is explicit rather than an absence of proof. For a contact form the structure on H = ker α is a symplectic form up to scale, so the question is whether the multiplier is an invariant. It is not: for every λ there is a similitude realising it, and one can write it down.
Checked for n = 1, 2, 3 against every square class at p = 2, 3, 5, 7 — 60 cases, all solved COMPUTED. Scaling one Lagrangian half and leaving the other alone is available over any field, which is exactly why the multiplier carries no information.
p-adic contact linear algebra is as rigid as real contact linear algebra. Any hope that a contact structure over ℚp is arithmetically interesting at a point ends here, permanently. This is a negative result and it is the most load-bearing paragraph in the note, because it is what stops the next section from being read as more than it is.
Arithmetic appears one level up. Take L a free ℤp-lattice model of rank 2n+1 and α a contact form on it. On H = ker α the form dα|H is alternating, and an alternating form over ℤp has elementary divisors, in equal pairs d1 | d1 | d2 | d2 | … This is Shimura’s Arithmetic of alternating forms, 1963 CHECKED [5]. A ℤp-analytic coordinate change has unit Jacobian and cannot move them; rescaling α by a constant c ∈ ℚp× shifts every valuation by vp(c) simultaneously.
So the invariant of the pair (lattice model, contact structure) is the multiset of valuations of those elementary divisors, taken modulo the simultaneous shift. Its coarse shadow is a single number, and it is the one worth carrying:
Well defined for the same two reasons: the top form scales by a unit determinant under a ℤp coordinate change, and by cn+1 under rescaling. On a rank-5 example — α = dz + (y1dx1 − x1dy1) + 3(y2dx2 − x2dy2) — it takes the values δ2 = 2 and δ3 = 1 COMPUTED, so it is not a parity and the group ℤ/(n+1) is genuinely used.
| form | elem. divisor valuations, p = 2 | p odd | δ2 | δp, p odd |
|---|---|---|---|---|
| dz − y dx (standard) | (0, 0) | (0, 0) | 0 | 0 |
| dz − r² dθ (prototype) | (1, 1) | (0, 0) | 1 | 0 |
The two rows differ at p = 2 and agree at 3, 5, 7, 11 COMPUTED. So the RH paper’s prototype is ℤp-isomorphic to the standard contact model at every odd place and is not at p = 2.
The invariant is not what convinces; the witness is. Over ℝ one standardises the prototype with
which the script verifies coefficient by coefficient. That substitution has determinant 2. It lies in GL3(ℤp) for odd p and not in GL3(ℤ2) COMPUTED. Invariant and witness agree, which is the only reason to trust either.
Worth saying plainly: the 2 comes from r²dθ = x dy − y dx being written antisymmetrically, and there is no dm³ content in it. A form that had been written −x dy would have δ2 = 0. That is a fact about the chart, not about the geometry — and it is precisely why δp is an invariant of a model and never of a manifold.
| statement | status |
|---|---|
| Alternating forms over ℤp are classified by elementary divisors | Classical. Shimura, JMSJ 1963 |
| Symplectic similitudes of every multiplier exist over any field | Classical. Immediate from the symplectic basis theorem |
| Square classes of ℚp×; quadratic forms are separated by them | Classical. Serre, A Course in Arithmetic, ch. III–IV |
| p-adic Darboux, Moser and the global ball classification | Classical, 2025. Crespo & Pelayo [3]; even-dimensional only |
| That contact forms over ℚp have been given none of this | Observed. Searched; no counter-instance found |
| §4: no arithmetic invariant at a point, with the similitude as witness | New as written here, and elementary. Anyone who asked would get the same answer in an afternoon |
| §5: δp and the elementary-divisor invariant applied to a contact model | New as written here. The machinery is Shimura’s; the application appears unwritten |
| §6: the prototype is non-standard exactly at p = 2 | New, and about this corpus’s own chart. Not a fact about anyone else’s geometry |
| Contact Darboux over ℚp | OPEN not attempted here |
Nothing above required a new technique. The whole of §4 and §5 is what a specialist would produce on request, and the claim is only that nobody has been asked. “Unwritten” is a statement about a literature search, not about difficulty, and it is the weakest kind of novelty there is. It is still a novelty claim, and this note makes it in the three rows above and nowhere else.
It bears on the RH paper’s §5.2, the local contact forms, and on nothing after it. Three separations, in decreasing order of obviousness.
Category. The adele class space ℂℚ = 𝕊ℚ×/ℚ× is not a p-adic analytic manifold. It is the non-Hausdorff quotient that motivated Connes’ noncommutative treatment in the first place. Crespo and Pelayo’s category does not contain αarith, so their theorems do not apply to it and nothing here is transported to it.
Direction. §9 question 3 hopes that if αarith defines a tight contact structure, that would be a rigidity consistent with RH. If the contact analogue of Crespo–Pelayo’s Theorem E holds — global standardness — then there is no tight/overtwisted dichotomy over ℚp and the hoped-for statement has no content at the finite places. That would be a negative answer to the paper’s own question, and the paper should carry it as such rather than leaving the question open in a form that cannot come back either way OPEN.
Scale. Nothing here bounds anything, computes anything about ζ, or bears on §6 global positivity. The last row of the RH paper’s status table is unchanged by this note, and so is every row above it.
“Contact structure” and “p-adic” can be put in one sentence without either word doing work, and the resulting sentence sounds like a programme. The test applied throughout was whether a computation could come back empty. §4 came back empty and is reported as such; §6 came back with one place and not a family. A construction that produced an interesting invariant at every prime would have been the warning sign, not the result.
[1] L. Crespo & Á. Pelayo, The p-adic Jaynes–Cummings model in symplectic
geometry, arXiv:2406.18415 (2024).
[2] L. Crespo & Á. Pelayo, p-adic symplectic geometry of integrable systems and
Weierstrass–Williamson theory, arXiv:2501.14444 (2025).
[3] L. Crespo & Á. Pelayo, Darboux’s Theorem in p-adic symplectic
geometry, arXiv:2508.15443 (2025). Theorems A, B, C and E as cited in §1.
[4] L. Crespo & Á. Pelayo, Group actions on p-adic symplectic manifolds,
arXiv:2512.15575 (2025).
[5] G. Shimura, Arithmetic of alternating forms and quaternion hermitian forms,
J. Math. Soc. Japan 15 (1963), 33–65.
[6] J.-P. Serre, A Course in Arithmetic, Springer GTM 7, ch. III–IV, for the square
classes of ℚp× and the classification of quadratic forms.
[7] This corpus: RH_arithmetic_contact_structure.md §2, §5.2,
§9 (rendered at book4/rh-paper.html);
book7 Ch Fy for the smooth Reeb computation;
book6/wp96-the-second-instrument.html for the two-column instrument
argument this note reuses in §4.
[8] Every number in this note is regenerated by wp106-verify.py
(sympy 1.14; exact Fraction and Smith normal form; all checks pass).