Zilber–Pink in abelian varieties and the Siegel threefold

What OpenAI claims

Proves the abelian Zilber–Pink conjecture over ℚ̅: every irreducible subvariety has finitely many maximal atypical subvarieties relative to its smallest containing torsion coset. It also proves the full curve case in the Siegel threefold 𝒜₂ for Hodge-generic curves defined over ℚ̅, without boundary or reduction assumptions.

Papers (4)

1. The abelian Zilber–Pink conjecture

   24 September 2026 · Preprint, not peer reviewed · PDF

   We prove the abelian Zilber–Pink conjecture over ℚ̅. Every irreducible subvariety of an abelian variety has only finitely many maximal atypical subvarieties, where atypicality is measured inside its smallest containing torsion coset.

2. The E×CM component of Zilber–Pink for curves in A2

   24 September 2026 · Preprint, not peer reviewed · PDF

   We prove the E×CM component of Zilber–Pink for Hodge-generic algebraic curves in 𝒜₂ over ℚ̅. Each such curve contains only finitely many points whose abelian surface is isogenous to a product of elliptic curves with at least one factor having complex multiplication.

3. Quaternionic division points on curves in the Siegel threefold

   24 September 2026 · Preprint, not peer reviewed · PDF

   We prove the quaternionic-division component of Zilber–Pink for curves in 𝒜₂ over ℚ̅. A Hodge-generic algebraic curve contains only finitely many points whose full geometric rational endomorphism algebra is an indefinite quaternion division algebra over ℚ. No boundary or reduction hypothesis is required.

4. Elliptic squares and Zilber–Pink for curves in A2

   24 September 2026 · Preprint, not peer reviewed · PDF

   We prove that every Hodge-generic algebraic curve in 𝒜₂ over ℚ̅ contains only finitely many points whose abelian surface is isogenous to the square of an elliptic curve without complex multiplication (CM). Combining this result with the companion E×CM and quaternionic-division finiteness theorems, we prove the curve case of Zilber–Pink in 𝒜₂ over ℚ̅, without a boundary hypothesis.

Status

Preprint. The papers have not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.