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Math-79-A3 · revision r1 · on Math-79

OpenAI claim summary for Math-79

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 selected results apply to Hodge-generic curves in the pure Shimura variety $\mathcal A_2$ over $\overline{\mathbb Q}$, including varying endomorphism algebras, CM fields and isogeny degrees. A pure Shimura variety is a special case of a mixed Shimura variety. The question ranges over arbitrary mixed Shimura ambient spaces and subvarieties, so these curve results supply a partial claim rather than the full requested finiteness statement.

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

Manuscript

Contents
  1. Zilber-Pink Conjecture on Unlikely Intersections
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Manuscript 1.
  5. Manuscript 2.
  6. Manuscript 3.
  7. Outline of the argument
  8. Formal verification
  9. Status caveat
  10. Sources

Zilber-Pink Conjecture on Unlikely Intersections

OpenAI's claim

The selected manuscripts address Hodge-generic curves in the Siegel threefold $\mathcal A_2$ over $\overline{\mathbb Q}$. Manuscript 1 claims the full finiteness assertion for intersections with special subvarieties of dimension at most one. Manuscripts 2 and 3 give the quaternion-division and elliptic-product-with-CM branches, respectively. These are curve cases of the Shimura intersection question; manuscript numbers follow Sources.

Mathematical statement of the claimed result

Manuscript 1.
Let $\mathcal A_2$ be the moduli space of principally polarized abelian surfaces over $\overline{\mathbb Q}$, and let $C$ be any nonempty reduced irreducible closed Hodge-generic algebraic curve therein. The complete asserted conclusion is \[ \#\left(C(\overline{\mathbb Q})\cap \bigcup_{\substack{Z\subset\mathcal A_2\ \mathrm{special}\\\dim Z\le1}} Z(\overline{\mathbb Q})\right)<\infty. \] In particular the points whose surface is isogenous to an elliptic square are finite. Two additional assertions give finiteness for surfaces isogenous to $E_1\times E_2$ with a CM factor, and for surfaces having a full rational endomorphism algebra that is quaternionic, a division algebra and indefinite. There are no bounds on CM fields, isogeny degrees, endomorphism orders or polarizations, and no height, boundary or integrality restrictions.
Manuscript 2.
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.
Manuscript 3.
Take a closed curve $C$ over $\overline{\mathbb Q}$, with its reduced structure and assumed irreducible, in $\mathcal A_2$, the moduli of principally polarized abelian surfaces. Assume Hodge genericity: $C$ lies in no proper special subvariety. The claimed finite set is \[ \{s\in C(\overline{\mathbb Q}):A_s\text{ is isogenous to }E_1\times E_2, \text{ with at least one }E_i\text{ having CM}\}. \] Both the CM fields and the isogeny degrees may vary; the theorem does not restrict to a fixed product locus.

Outline of the argument

For manuscript 1 (family 016; 2026-09-24), two auxiliary markings control bad-center proximity in a five-dimensional orthogonal variation. Common-parameter interpolation gives arithmetic height bounds, and orbit counting handles elliptic squares; classification of special curves then joins this branch to the CM-product and quaternionic branches.

For manuscript 2 (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.

For manuscript 3 (family 016; 2026-09-24), canonical product isogenies and degeneration comparisons bound heights along the curve. Period estimates produce a Galois-orbit lower bound, which conflicts with the definable counting bound unless only finitely many such points occur.

Formal verification

No Lean formalization is listed at this snapshot for manuscripts 1, 2 or 3 (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