Proves the abelian Zilber–Pink conjecture over \overline{\mathbb Q}: 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 \mathcal A_2 for Hodge-generic curves defined over \overline{\mathbb Q}, without boundary or reduction assumptions.
Manuscripts (4)
The abelian Zilber–Pink conjecture
September 24, 2026 · source · cite as \cite{OAI:The-Abelian-Zilber-Pink-Conjecture-September-24-2026}
We prove the abelian Zilber–Pink conjecture over \overline{\mathbb Q}. Every irreducible subvariety of an abelian variety has only finitely many maximal atypical subvarieties, where atypicality is measured inside its smallest containing torsion coset.
The E×CM component of Zilber–Pink for curves in A2
September 24, 2026 · source · cite as \cite{OAI:The-E-times-CM-Component-of-Zilber-Pink-for-Curves-in-A2-September-24-2026}
We prove the E\times\mathrm{CM} component of Zilber–Pink for Hodge-generic algebraic curves in \mathcal A_2 over \overline{\mathbb Q}. 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.
Quaternionic division points on curves in the Siegel threefold
September 24, 2026 · source · cite as \cite{OAI:Quaternionic-Division-Points-on-Curves-in-the-Siegel-Threefold-September-24-2026}
We prove the quaternionic-division component of Zilber–Pink for curves in \mathcal A_2 over \overline{\mathbb Q}. 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.
Elliptic squares and Zilber–Pink for curves in A2
September 24, 2026 · source · cite as \cite{OAI:Elliptic-Squares-and-Zilber-Pink-for-Curves-in-A2-September-24-2026}
We prove that every Hodge-generic algebraic curve in \mathcal A_2 over \overline{\mathbb Q} 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\times\mathrm{CM} and quaternionic-division finiteness theorems, we prove the curve case of Zilber–Pink in \mathcal A_2 over \overline{\mathbb Q}, without a boundary hypothesis.
Result 016 · Number theory. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.
Source: https://github.com/openai/math/blob/main/CONTENTS.md