Browse › Number theory › 016 Zilber–Pink in abelian varieties and the Siegel threefold
The abelian Zilber–Pink conjecture
Open PDFCite Report a problem Add a reference or comment Interested in working on this Start a discussion
Edit NOTES.md · Existing issues · Source on GitHub
Abstract
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.
If the PDF does not display here, open it directly: Open PDF.
Notes
No notes yet. Add a reference, a comment or your name with the buttons above, or edit NOTES.md on GitHub.
Automatically collected notes (unreviewed)
work in progress
Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Follow-up works
- Netan Dogra, Arnab Saha, Arithmetic jet spaces and the unramified Zilber–Pink conjecture for subvarieties of abelian varieties (arXiv:2610.17325, 12 Oct 2026). Its introduction cites this preprint as an announced AI-generated proof of Zilber–Pink for subvarieties of abelian varieties over ℚ̄, noting that by Barroero–Dill this implies the general abelian case; the paper itself bounds degrees of Zilber–Pink points with bounded ramification at a fixed prime using delta geometry.
- Fabrizio Barroero, Gabriel A. Dill, Lars Kühne, Intersecting a curve in an abelian variety with multiples of another curve (arXiv:2610.10212, 7 Oct 2026). This independently completed work on an abelian analogue of Levin's question notes that OpenAI released a preprint "claiming a proof of the Zilber-Pink conjecture" for abelian varieties over ℚ̄, which prompted the authors to post their manuscript; its results do not depend on it.
Related prior literature mentioned by others
- Fabrizio Barroero, Gabriel A. Dill, On the Zilber–Pink conjecture for complex abelian varieties (arXiv:1909.01271; Ann. Sci. ÉNS 55 (2022)). Barroero–Dill–Kühne (arXiv:2610.10212) and Dogra–Saha (arXiv:2610.17325) invoke this reduction to explain that the ℚ̄ statement claimed here implies the abelian conjecture over ℂ; it is also in the paper's own bibliography.
Possibly relevant literature (found by the agent, not checked)
- Fabrizio Barroero, Gabriel A. Dill, Distinguished categories and the Zilber–Pink conjecture (arXiv:2103.07422, to appear in Amer. J. Math.). An axiomatic framework for unlikely intersections relating Zilber–Pink statements under base change; it may matter for extending Theorem 1.1 beyond abelian varieties over ℚ̄.
Digests and expositions
- Orply / Perplexity, Abelian Zilber–Pink Gives Finitely Many Maximal Atypical Subvarieties (Orply, 2026-10-08). This AI-assisted exposition describes the manuscript's special-closure formulation, maximal atypical subvarieties and non-density strategy. [AI-generated per source]
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 016
Independent sites; their content is not reviewed here.
Cite
@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}
}Family 016: Zilber–Pink in abelian varieties and the Siegel threefold
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.
Other papers in this family:
- The E×CM component of Zilber–Pink for curves in A2 · September 24, 2026
- Quaternionic division points on curves in the Siegel threefold · September 24, 2026
- Elliptic squares and Zilber–Pink for curves in A2 · September 24, 2026