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Math-2770 · mathematics · Number theory

Elliptic-square intersections of Hodge-generic curves in A2

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For a reduced irreducible closed Hodge-generic Qbar-curve in A2, are only finitely many of its abelian surfaces isogenous to E x E for an elliptic curve E?

Short English statement

For a reduced irreducible closed Hodge-generic Qbar-curve in A2, are only finitely many of its abelian surfaces isogenous to E x E for an elliptic curve E?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2770-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2770

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).

Elliptic-square 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 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.

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.

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.

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

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