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Math-2426 · mathematics · Algebraic geometry

Finiteness of Atypical Hodge Subvarieties

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Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
For an irreducible smooth quasi-projective variety $S$ with a variation $V$ of mixed rational Hodge structures, is the atypical locus $S_{\mathrm{atyp}}(V)$ a finite union of maximal atypical subvarieties, equivalently a proper algebraic subset and hence not Zariski dense; and, in the generalized André–Oort case, does Zariski density of CM points force $V$ to arise by pullback from a connected mixed Shimura variety?

Formulation references: [S1], [S2].

Short English statement

For an irreducible smooth quasi-projective variety $S$ with a variation $V$ of mixed rational Hodge structures, is the atypical locus $S_{\mathrm{atyp}}(V)$ a finite union of maximal atypical subvarieties, equivalently a proper algebraic subset and hence not Zariski dense; and, in the generalized André–Oort case, does Zariski density of CM points force $V$ to arise by pullback from a connected mixed Shimura variety?

Sources

Catalogue entry

Statement licence: CC-BY-SA-4.0 · Source file

Contributions 1

Math-2426-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2426

The selected paper proves finiteness of intersections with special subvarieties of dimension at most one for Hodge-generic curves in $\mathcal A_2$ over $\overline{\mathbb Q}$. This supplies a curve-level pure weight-one Hodge case. The question allows arbitrary mixed variations on smooth quasi-projective bases and also asks a general consequence of dense CM points. Those wider clauses are not asserted by this manuscript.

Finiteness of Atypical Hodge Subvarieties

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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