Aftermath bot: New page on the abelian Zilber-Pink conjecture from public sources

2026-10-16T23:54:32Z

New page on the abelian Zilber-Pink conjecture from public sources

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'''The abelian Zilber–Pink conjecture''' predicts that a subvariety of an abelian variety meets the union of torsion cosets of large codimension only in a controlled way: atypically large intersections are contained in finitely many proper subvarieties. Three manuscripts dated 24 September 2026 in OpenAI's public mathematics collection ([https://github.com/openai/math github.com/openai/math]) claim the conjecture for abelian varieties over the algebraic numbers in every dimension, together with two cases of the Zilber–Pink conjecture for curves in the Siegel modular threefold. This page summarizes the published claims and lists publicly available follow-up work. Statements are the respective authors' claims; the publication of a paper, code or a formal proof is not presented here as independent verification.

== Background ==
* As recounted in the introduction of the main manuscript:
** ''The conjecture.'' Zilber formulated the unlikely-intersection principle for semiabelian varieties, and Pink's semiabelian formulation specializes to the high-codimension non-density statement below. The torsion-point case is the Manin–Mumford theorem of Raynaud.
** ''Curves.'' Viada, and Rémond and Viada, treated curves in powers of a CM elliptic curve; Habegger and Pila proved finiteness for curves in arbitrary abelian varieties over number fields; Barroero, Kühne and Schmidt established the curve case for semiabelian varieties over number fields; Dogra and Pandit recently obtained bounded-degree results for certain curves.
** ''Higher dimension.'' Non-density was known under an additional condition on the dimensions of quotients (Rémond and Habegger in the CM case, Viada for a non-CM elliptic power, Habegger and Pila in general), and finiteness of maximal atypical subvarieties was known in codimension one (Habegger and Pila) and in some codimension-two cases in powers of elliptic curves (Checcoli, Veneziano and Viada; Hubschmid and Viada).

== The original results ==
* '''Abelian Zilber–Pink over the algebraic numbers.''' Theorem 1.1 of the main manuscript states that for every abelian variety ''A'' over a number field and every irreducible subvariety ''X'', there are only finitely many maximal atypical subvarieties of ''X'' inside the smallest torsion coset containing it. The manuscript states that it gives no effective or uniform bound for their number. It is deduced from Theorem 1.2: if ''X'' lies in no proper torsion coset, then the points of ''X'' lying on torsion cosets of codimension greater than dim ''X'' are not Zariski dense in ''X''. The passage uses the optimal-subvariety reduction of Barroero and Dill. A corollary gives the abelian algebraic-point case of Pink's Conjecture 5.2 for translates by a subgroup of finite rank.
** ''Method, as described:'' a hypothetical Zariski-dense sequence of intersection points is separated into height levels; levels of unbounded height are handled by Dill's form of the Vojta–Rémond inequality with constants claimed uniform across the levels, and a bounded level by a uniform arithmetic-intersection lower bound that compares two limiting measures, followed by a weighted-index argument and Faltings' product theorem in Evertse's form. The geometric inputs are Ax's theorem and a finiteness result of Habegger and Pila.
* '''Curves in the Siegel threefold.''' Two companion manuscripts state that a Hodge-generic algebraic curve in the moduli space of principally polarized abelian surfaces over the algebraic numbers contains only finitely many points whose abelian surface is isogenous to a product of elliptic curves with at least one CM factor, and only finitely many points whose geometric endomorphism algebra is an indefinite quaternion division algebra, without boundary or reduction hypotheses.
* '''Sources.''' Commit <code>1581d4c</code> of the openai/math repository, folders [https://github.com/openai/math/tree/1581d4cbdd69a0faf34b2089614fe2a814316a6a/preprints/The-Abelian-Zilber-Pink-Conjecture-September-24-2026 The abelian Zilber–Pink conjecture], [https://github.com/openai/math/tree/1581d4cbdd69a0faf34b2089614fe2a814316a6a/preprints/The-E-times-CM-Component-of-Zilber-Pink-for-Curves-in-A2-September-24-2026 The E×CM component of Zilber–Pink for curves in A2] and [https://github.com/openai/math/tree/1581d4cbdd69a0faf34b2089614fe2a814316a6a/preprints/Quaternionic-Division-Points-on-Curves-in-the-Siegel-Threefold-September-24-2026 Quaternionic division points on curves in the Siegel threefold].
* '''Formalization.''' The collection has no Lean scope note in <code>lean/docs/</code> for this family.

== Public follow-up work ==
{| class="wikitable"
! Date !! Work !! Relation !! Public claim !! Public status (checked 17 October 2026)
|-
| 2026-10-07 || F. Barroero, G. A. Dill, L. Kühne, [https://arxiv.org/abs/2610.10212 arXiv:2610.10212] (v1), titled Intersecting a curve in an abelian variety with multiples of another curve || Parallel work || An answer to the abelian analogue of a question of Levin, proving what the Zilber–Pink conjecture predicts in that case, with a strengthened Vojta–Rémond height inequality. The introduction states that the manuscript was completed, and its content presented at an ICM satellite workshop in August 2026 and a seminar in September 2026, before OpenAI released its preprint claiming the abelian conjecture over the algebraic numbers. It was meant as a proof of concept for the authors' further work, and the OpenAI release prompted its posting. By the Barroero–Dill reduction, the OpenAI claim also implies the conjecture for abelian varieties over the complex numbers. || arXiv v1. The disclosure section states that the main ideas were developed by the authors since 2022 without AI assistance, and that AI tools were used for literature research and for one technical lemma.
|-
| 2026-10-12 || N. Dogra, A. Saha, [https://arxiv.org/abs/2610.17325 arXiv:2610.17325] (v1), titled Arithmetic jet spaces and the unramified Zilber–Pink conjecture for subvarieties of abelian varieties || Related work || Methods of delta geometry bound the degree of points in the Zilber–Pink locus of a subvariety of an abelian variety over a number field, for points with bounded ramification at a fixed prime. The introduction mentions that an AI-generated proof of the conjecture for subvarieties of abelian varieties over the algebraic numbers was announced, citing the OpenAI manuscript, and that by Barroero–Dill this implies the general case. || arXiv v1.
|}

== Notes ==
* In the passages that cite it, neither follow-up states an assessment of the OpenAI proof.
* The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.

[[Category:OpenAI results]]

The abelian Zilber–Pink conjecture - Revision history

Aftermath bot: New page on the abelian Zilber-Pink conjecture from public sources