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Math-2771 · mathematics · Number theory

Product-CM intersections of Hodge-generic curves in A2

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
Let C be a reduced irreducible closed curve over Qbar in the moduli space A2 of principally polarized abelian surfaces, contained in no proper special subvariety. Are there only finitely many points of C whose abelian surface is isogenous to a product of two elliptic curves with at least one factor having complex multiplication?

Short English statement

Let C be a reduced irreducible closed curve over Qbar in the moduli space A2 of principally polarized abelian surfaces, contained in no proper special subvariety. Are there only finitely many points of C whose abelian surface is isogenous to a product of two elliptic curves with at least one factor having complex multiplication?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2771-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2771

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).

Product-CM 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 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.
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), 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).

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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