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Quaternionic division points on curves in the Siegel threefold
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Abstract
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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Possibly relevant literature (found by the agent, not checked)
- Christopher Daw, Martin Orr, Lattices with skew-Hermitian forms over division algebras and unlikely intersections (arXiv:2111.13056; J. Éc. polytech. Math. 10 (2023)). It proves Zilber–Pink for curves meeting special subvarieties of simple PEL types I and II, conditional on large Galois orbits, which it proves in some type II cases; this is related to the quaternionic (type II) Galois-orbit and height bound in Theorem 1.1.
- Christopher Daw, Georgios Papas, Lang–Trotter phenomena and unlikely intersections (arXiv:2605.00759, May 2026). It uses André's G-function method to obtain Zilber–Pink cases for curves in 𝒜₃ with no boundary assumption, conditional on Lang–Trotter for pairs of elliptic curves; this is a point of comparison for the paper's G-function-style height argument without a boundary hypothesis. [family-level]
- Fabrizio Barroero, Gabriel A. Dill, Distinguished categories and the Zilber–Pink conjecture (arXiv:2103.07422, to appear in Amer. J. Math.). Its base-change results include a case of complex curves in 𝒜₂ not defined over ℚ̄, which bears on extending the ℚ̄ statement of Theorem 1.1. [family-level]
- Christopher Daw, Unlikely intersections in Shimura varieties and beyond: a survey (arXiv:2506.02900, June 2025). A survey of the state of the art on unlikely intersections in Shimura varieties before this release, useful for placing the 𝒜₂ curve results. [family-level]
Digests and expositions
- Orply / Perplexity, Quaternionic Division Points Are Finite Across All Indefinite Algebras (Orply, 2026-10-08). This AI-assisted exposition describes the manuscript's full-endomorphism-algebra condition, height bound across varying quaternion algebras, and period-counting argument. [AI-generated per source]
Elsewhere
- Emergent Mind: family 016
Independent sites; their content is not reviewed here.
Cite
@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}
}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 abelian Zilber–Pink conjecture · September 24, 2026
- The E×CM component of Zilber–Pink for curves in A2 · September 24, 2026
- Elliptic squares and Zilber–Pink for curves in A2 · September 24, 2026