---
title: "Zilber–Pink in abelian varieties and the Siegel threefold"
authors:
  - "OpenAI (internal model)"
year: 2026
venue: "openai/math manuscript collection (GitHub), result family 016; unreviewed"
arxiv: null
doi: null
source: "https://github.com/openai/math/blob/main/preprints/The-Abelian-Zilber-Pink-Conjecture-September-24-2026/paper.pdf"
topics:
  - "openai-math"
  - "ai-generated-math"
  - "number-theory"
  - "unformalized"
seed_rank: 1876
seed_batch: "openai-math-2026-10-07"
reviewed: "2026-10-07"
pool: "maths-foundations"
relevance_score: 7
lineage: ai-mathematical-reasoning
cites:
  - title: "The abelian Zilber–Pink conjecture"
    url: "https://github.com/openai/math/blob/main/preprints/The-Abelian-Zilber-Pink-Conjecture-September-24-2026/paper.pdf"
    year: 2026
    arxiv: null
    doi: null
  - title: "The E×CM component of Zilber–Pink for curves in A2"
    url: "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"
    year: 2026
    arxiv: null
    doi: null
  - title: "Quaternionic division points on curves in the Siegel threefold"
    url: "https://github.com/openai/math/blob/main/preprints/Quaternionic-Division-Points-on-Curves-in-the-Siegel-Threefold-September-24-2026/paper.pdf"
    year: 2026
    arxiv: null
    doi: null
  - title: "Elliptic squares and Zilber–Pink for curves in A2"
    url: "https://github.com/openai/math/blob/main/preprints/Elliptic-Squares-and-Zilber-Pink-for-Curves-in-A2-September-24-2026/paper.pdf"
    year: 2026
    arxiv: null
    doi: null
---

# Zilber–Pink in abelian varieties and the Siegel threefold

## One-sentence takeaway

OpenAI's result family 016 (Number theory) claims: 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.

## Why it matters here

Pure mathematics with no direct engine use; it is in the library as part of the openai/math corpus, which records what an AI research model claims to have proved. More broadly, every card in this batch is evidence for the `ai-mathematical-reasoning` thread: per the repo README, an unreleased internal OpenAI model produced it, using about three hours of ChatGPT Pro thinking compute per result on average, across roughly 4,000 posed problems.

## Key ideas

- Family summary (repo): 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.
- *The abelian Zilber–Pink conjecture*: 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*: 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*: 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*: 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.

## Caveats

- Unreviewed: these are AI-generated manuscripts posted to GitHub with no peer review or arXiv deposit. The repo README says results are "at different stages of verification" and that "some of the unformalized results could have issues."
- This card restates the repo's family summary, manuscript abstracts and Lean scope notes. It is not an independent check of the proofs, and the scope of each theorem is exactly as stated there.
- Lean: not formalized. The repo has no Lean development for this family, so the claim rests on the prose manuscript(s) alone.

## Links

- Manuscript: [The abelian Zilber–Pink conjecture](https://github.com/openai/math/blob/main/preprints/The-Abelian-Zilber-Pink-Conjecture-September-24-2026/paper.pdf)
- Manuscript: [The E×CM component of Zilber–Pink for curves in A2](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)
- Manuscript: [Quaternionic division points on curves in the Siegel threefold](https://github.com/openai/math/blob/main/preprints/Quaternionic-Division-Points-on-Curves-in-the-Siegel-Threefold-September-24-2026/paper.pdf)
- Manuscript: [Elliptic squares and Zilber–Pink for curves in A2](https://github.com/openai/math/blob/main/preprints/Elliptic-Squares-and-Zilber-Pink-for-Curves-in-A2-September-24-2026/paper.pdf)
- Family entry: https://github.com/openai/math/blob/main/CONTENTS.md
- Overview PDF: https://github.com/openai/math/blob/main/overview.pdf
- Repo: https://github.com/openai/math