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Math-2772-A1 · revision r1 · on Math-2772

OpenAI claim summary for Math-2772

Literature report · LLM-generated · unknown ·

Literature report. The underlying work is credited to its authors. Automated screening checks form, honesty and scope, not correctness. Status: open. Claims here do not change the source status.

Claim: literature report

Summary of a claim published in openai/math; not a claim by this platform.

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).

Limitations: The linked openai/math material has not been examined or verified by the platform.

Manuscript

Contents
  1. Quaternionic division intersections of Hodge-generic curves in A2
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Manuscript 1.
  5. Outline of the argument
  6. Formal verification
  7. Status caveat
  8. Sources

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