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Zilber-Pink for Hodge-generic curves in the Siegel modular threefold A_2

Let A2\mathcal A_2 be the moduli space of principally polarized abelian surfaces. Pink's Zilber-Pink conjecture for mixed Shimura varieties (2005, Conjecture 1.3) predicts that a Hodge-generic curve meets the union of special subvarieties of codimension at least two only finitely often; in A2\mathcal A_2 these are the special points and the special curves (products with a CM elliptic factor, squares of non-CM elliptic curves, quaternionic multiplication). Pila and Tsimerman proved Andre-Oort for A2\mathcal A_2, and Daw and Orr proved the curve loci under boundary or multiplicative-reduction hypotheses. For a Hodge-generic algebraic curve C⊂A2C\subset\mathcal A_2 over Qˉ\bar{\mathbb Q}, is C∩⋃dim⁡Z≤1ZC\cap\bigcup_{\dim Z\le1}Z finite?

Result
Proved(see note)
Status
Partial result
AI contribution
AI-discovered
Method
Argument
Field
Unlikely intersections in Shimura varieties
Posed by
Richard Pink (2005, Conjecture 1.3, for mixed Shimura varieties); the A_2 curve case studied by Christopher Daw and Martin Orr (2021-2025)
Year posed
2005
Years open
21y
Solved
2026-09-24
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Unreviewed
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
40 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Claims the full curve case of Zilber-Pink in A2\mathcal A_2 over Qˉ\bar{\mathbb Q}: a Hodge-generic algebraic curve contains only finitely many points whose abelian surface is isogenous to a product with a CM factor, to the square of a non-CM elliptic curve, or has quaternionic-division endomorphism algebra, besides finitely many special points. Previous unconditional results needed the curve to meet the Baily-Borel boundary or to have multiplicative degeneration. It does NOT treat higher-dimensional subvarieties of A2\mathcal A_2, Ag\mathcal A_g for g≥3g\ge3, or curves not defined over Qˉ\bar{\mathbb Q}.

What the AI did

The release README says the manuscripts were produced by an unreleased internal OpenAI model, the vast majority by one fixed procedure using on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier results produced by the models. Its named exceptions to that procedure (the Hodge conjecture for CM abelian varieties and the zeta zero-free region work) do not concern this family. All three manuscripts are credited to OpenAI with no human author named.

Verification

No independent mathematician has checked this yet. Theorem 1.3 of the principal manuscript was read against the posed problem: for every Hodge-generic curve in A2\mathcal A_2 over Qˉ\bar{\mathbb Q}, finitely many points on special subvarieties of dimension at most one, with no boundary, reduction, height or degree hypothesis. It assembles three branch theorems: elliptic squares (proved here, by establishing the Galois-orbit lower bound Daw and Orr state as their Conjecture 6.2 and applying their Theorem 1.3), and the E x CM and quaternionic-division branches from the two companions, plus Pila-Tsimerman for special points. The companions were read at the level of their main theorems only. No Lean formalization. Curves over C\mathbb C not defined over Qˉ\bar{\mathbb Q} are not covered.

Sources

Changelog1 change

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