Zilber–Pink in abelian varieties and the Siegel threefold

4 papers

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.

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▶Introduction: the problem and why it mattersin preparation

▶The presented solutionin preparation

Papers 4

The abelian Zilber–Pink conjecture

OpenAI · 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.

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@misc{OAI:The-Abelian-Zilber-Pink-Conjecture-September-24-2026,
  author = {{OpenAI}},
  title = {{The abelian Zilber--Pink conjecture}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/The-Abelian-Zilber-Pink-Conjecture-September-24-2026/paper.pdf}{OAI:The-Abelian-Zilber-Pink-Conjecture-September-24-2026}},
  year = {2026}
}

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

OpenAI · 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.

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@misc{OAI:The-E-times-CM-Component-of-Zilber-Pink-for-Curves-in-A2-September-24-2026,
  author = {{OpenAI}},
  title = {{The $E\times\mathrm{CM}$ component of Zilber--Pink for curves in $\mathcal A_2$}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/The-E-times-CM-Component-of-Zilber-Pink-for-Curves-in-A2-September-24-2026/paper.pdf}{OAI:The-E-times-CM-Component-of-Zilber-Pink-for-Curves-in-A2-September-24-2026}},
  year = {2026}
}

Quaternionic division points on curves in the Siegel threefold

OpenAI · 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.

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@misc{OAI:Quaternionic-Division-Points-on-Curves-in-the-Siegel-Threefold-September-24-2026,
  author = {{OpenAI}},
  title = {{Quaternionic division points on curves in the Siegel threefold}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Quaternionic-Division-Points-on-Curves-in-the-Siegel-Threefold-September-24-2026/paper.pdf}{OAI:Quaternionic-Division-Points-on-Curves-in-the-Siegel-Threefold-September-24-2026}},
  year = {2026}
}

Elliptic squares and Zilber–Pink for curves in A2

OpenAI · 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.

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Cite

@misc{OAI:Elliptic-Squares-and-Zilber-Pink-for-Curves-in-A2-September-24-2026,
  author = {{OpenAI}},
  title = {{Elliptic squares and Zilber--Pink for curves in $\mathcal A_2$}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Elliptic-Squares-and-Zilber-Pink-for-Curves-in-A2-September-24-2026/paper.pdf}{OAI:Elliptic-Squares-and-Zilber-Pink-for-Curves-in-A2-September-24-2026}},
  year = {2026}
}

Formal verification

No Lean formalization is listed for this result yet. The repository notes that unformalized results could have issues.