OpenAI · 016-04 · 2026-09-24 · Number theory

Elliptic squares and Zilber–Pink for curves in A2

preprints/Elliptic-Squares-and-Zilber-Pink-for-Curves-in-A2-September-24-2026

Family 016. Zilber–Pink in abelian varieties and the Siegel threefold.

The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.

Read against the standing problems

Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N

Proves the abelian Zilber–Pink conjecture over $Q$ : 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 $A_{2}$ for Hodge-generic curves defined over $Q$ , without boundary or reduction assumptions.

We prove that every Hodge-generic algebraic curve in $A_{2}$ over $Q$ 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 $E \times CM$ and quaternionic-division finiteness theorems, we prove the curve case of Zilber–Pink in $A_{2}$ over $Q$ , without a boundary hypothesis.

Preprint source and Lean scope

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No Lean scope note is in the repository for this family. The preprint source below is still the full manuscript, not only the abstract. No compilation is asserted.