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Math-2475-A1 · revision r1 · on Math-2475

OpenAI claim summary for Math-2475

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 at unsolvedmath.com: 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 abelian result keeps algebraic-number constants and an abelian ambient variety. The Siegel-threefold papers cover Hodge-generic curves in $\mathcal A_2$, including their separate endomorphism branches. Arbitrary semiabelian or mixed-Shimura ambient spaces and higher-dimensional variations in the present statement are not covered.

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

Manuscript

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

Zilber-Pink Unlikely Intersection Conjecture

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 non-CM elliptic-square points on Hodge-generic curves in $\mathcal A_2$ are finite, and combining the three endomorphism branches gives the full curve-level Zilber--Pink assertion there; In manuscript 2 (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; In manuscript 3 (family 016; 2026-09-24), the asserted result is that subvarieties of abelian varieties over algebraic numbers have only finitely many maximal atypical intersections, measured inside their smallest containing torsion coset; In manuscript 4 (family 016; 2026-09-24), the asserted result is that a Hodge-generic curve in $\mathcal A_2$ over algebraic numbers meets the locus of surfaces isogenous to an elliptic product with a CM factor only finitely often.

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.
For a number field $K$, abelian variety $A/K$, and irreducible $X\subseteq A_{\bar K}$, let $S$ be the smallest special subvariety containing $X$; special subvarieties are torsion translates of abelian subvarieties. An irreducible component $Y$ of $X\cap Z$, $Z\subseteq S$ special, is atypical when \[ \dim Y>\dim X+\dim Z-\dim S. \] The claim is finiteness of maximal atypical subvarieties of $X$, equivalently existence of finitely many proper atypical $Y_j\subsetneq X$ containing every atypical subvariety. No dimension bound on $A$ or $X$ is imposed.
Manuscript 4.
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), a counterexample sequence is decomposed into distinct height scales. Uniform tangent positivity and weighted small sections give a height inequality despite growing homomorphism coefficients and point degrees; passing to suitable quotients rules out the remaining scale configurations. For manuscript 4 (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 manuscript 1 (family 016; 2026-09-24), manuscript 2 (family 016; 2026-09-24), manuscript 3 (family 016; 2026-09-24), manuscript 4 (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