Quaternionic division intersections of Hodge-generic curves in A2
Status OpenLLM status Open unverified
Top Open Problems unknownunknown Not reported cataloguedOpenOpenStatement
Definitions and mathematical statement
Statement recorded in the supplied source.
For a reduced irreducible closed Hodge-generic Qbar-curve C in A2, are there only finitely many points whose abelian surface has rational endomorphism algebra an indefinite quaternion division algebra over Q?Short English statement
For a reduced irreducible closed Hodge-generic Qbar-curve C in A2, are there only finitely many points whose abelian surface has rational endomorphism algebra an indefinite quaternion division algebra over Q?Sources
- S1 Original question source or mathematical context. https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Number_theory.
- S2 Original question source or mathematical context. https://ems.press/content/serial-article-files/52570.
- S3 Original question source or mathematical context. https://arxiv.org/html/2504.00865v1.
- S4 Original question source or mathematical context. https://arxiv.org/html/2209.07967.
- S5 OpenAI, manuscript citation page. Source of the formulation motivating this local problem. https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Quaternionic-Division-Points-on-Curves-in-the-Siegel-Threefold-September-24-2026/README.md.
Statement licence: CC-BY-4.0 · Source file
Contributions 1
OpenAI claim summary for Math-2772
The result supplies the conclusion in this problem with its displayed hypotheses and quantifiers. The broader statements and applications in the manuscript are kept separate from the particular question above. The supporting result is manuscript 1 (family 016; 2026-09-24).
Quaternionic division intersections of Hodge-generic curves in A2
OpenAI's claim
OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 016; 2026-09-24), the asserted result is that a Hodge-generic algebraic curve in $\mathcal A_2$ contains only finitely many points whose full rational endomorphism algebra is quaternion division, with no boundary condition.Mathematical statement of the claimed result
Manuscript 1.
Let $C$ be a closed algebraic curve, reduced, irreducible and Hodge generic, in the Siegel threefold $\mathcal A_2$ over $\overline{\mathbb Q}$, the asserted finiteness is \[ \#\{s\in C(\overline{\mathbb Q}): \operatorname{End}_{\overline{\mathbb Q}}(A_s)\otimes\mathbb Q \text{ is an indefinite quaternion division algebra over }\mathbb Q\}<\infty. \] The full endomorphism algebra, not merely a contained quaternion order, is specified; the quaternion algebra and order may vary.Outline of the argument
For manuscript 1 (family 016; 2026-09-24), a rank-five variation detects the endomorphism locus. Two-argument interpolation and an ordinary-point zero estimate produce a height estimate unaffected by how many places exhibit supersingular closeness; Galois-orbit counting then gives finiteness.Formal verification
No Lean formalization is listed at this snapshot for manuscript 1 (family 016; 2026-09-24).This site has not executed the Lean code. Described statement coverage is not an independent proof check.
Status caveat
The manuscripts were produced by an internal OpenAI model. OpenAI cautions that results without formalization can contain mistakes. This site has not independently checked the proofs; the label records the scope of OpenAI's claim.Sources
- OpenAI, Quaternionic division points on curves in the Siegel threefold, 2026-09-24; citation key
OAI:Quaternionic-Division-Points-on-Curves-in-the-Siegel-Threefold-September-24-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:main. - OpenAI catalogue entry.