Result 016, Number theory

Zilber–Pink in abelian varieties and the Siegel threefold

Proves the abelian Zilber–Pink conjecture over Q‾\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 A2\mathcal A_2 for Hodge-generic curves defined over Q‾\overline{\mathbb Q}, without boundary or reduction assumptions.

Proof

The bigger picture

Why it matters

Abelian varieties are geometric spaces with an algebraic addition law. These manuscripts claim that their unexpectedly large intersections with certain arithmetic structures fall into finitely many maximal pieces, helping researchers organize geometry that ordinary dimension-counting misses.

What changes?

Over the algebraic numbers, the manuscripts report that every irreducible subvariety of an abelian variety has finitely many maximal atypical subvarieties. These are components of unexpectedly large-dimensional intersections with torsion cosets, or algebraic subgroups shifted by finite-order points; expectations are measured inside the smallest containing torsion coset. They also report the full curve case in A2, the three-dimensional parameter space for principally polarized abelian surfaces, for Hodge-generic curves (not contained in a proper special subvariety), without boundary or reduction assumptions.

What does that help mathematicians do?

The abelian result would let researchers contain every atypical intersection component within a finite collection of maximal exceptional pieces. It does not say that all exceptional points are finite in number: those pieces can have positive dimension. For the specified curves in A2, the reported results do give pointwise finiteness, including surfaces related by an isogeny, a finite-kernel algebraic group map, to squares of elliptic curves without complex multiplication.

Are there practical applications?

The immediate value is foundational: the claims constrain how exceptional arithmetic structure can appear within algebraic families. In A2, they would rule out infinitely many points of the specified exceptional types along a Hodge-generic curve. The supplied statements establish finiteness, not numerical bounds or a procedure for listing those points or the maximal atypical pieces.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

4 manuscripts

The abelian Zilber–Pink conjecture

September 24, 2026 44 pages

We prove the abelian Zilber–Pink conjecture over Q‾\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.

PDF LaTeX source
Cite (BibTeX) 
@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

September 24, 2026 15 pages

We prove the E×CME\times\mathrm{CM} component of Zilber–Pink for Hodge-generic algebraic curves in A2\mathcal A_2 over Q‾\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.

PDF LaTeX source
Cite (BibTeX) 
@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

September 24, 2026 46 pages

We prove the quaternionic-division component of Zilber–Pink for curves in A2\mathcal A_2 over Q‾\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.

PDF LaTeX source
Cite (BibTeX) 
@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

September 24, 2026 65 pages

We prove that every Hodge-generic algebraic curve in A2\mathcal A_2 over Q‾\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×CME\times\mathrm{CM} and quaternionic-division finiteness theorems, we prove the curve case of Zilber–Pink in A2\mathcal A_2 over Q‾\overline{\mathbb Q}, without a boundary hypothesis.

PDF LaTeX source
Cite (BibTeX) 
@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}
}

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 8 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.