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Math-2430 · mathematics · Algebraic geometry

Finiteness of Optimal Points in Shimura 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.
Let $S$ be a Shimura variety and $V\subseteq S$ a subvariety. Call a subvariety $W\subseteq V$ optimal in $V$ if every subvariety $Y$ with $W\subsetneq Y\subseteq V$ has larger defect $\delta(Y)=\dim\langle Y\rangle-\dim Y$, where $\langle Y\rangle$ is the smallest special subvariety containing $Y$. Does $V$ contain only finitely many points that are optimal in $V$?

Formulation references: [S1], [S2].

Short English statement

Let $S$ be a Shimura variety and $V\subseteq S$ a subvariety. Call a subvariety $W\subseteq V$ optimal in $V$ if every subvariety $Y$ with $W\subsetneq Y\subseteq V$ has larger defect $\delta(Y)=\dim\langle Y\rangle-\dim Y$, where $\langle Y\rangle$ is the smallest special subvariety containing $Y$. Does $V$ contain only finitely many points that are optimal in $V$?

Sources

Catalogue entry

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

Contributions 1

Math-2430-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2430

For a Hodge-generic curve $C\subset\mathcal A_2$ over $\overline{\mathbb Q}$, the defect is $\delta(C)=2$. A point optimal in $C$ must have defect less than two, so its smallest special subvariety has dimension at most one. The source's finiteness theorem therefore proves this $\mathcal A_2$-curve case. It does not give the requested theorem for every subvariety of every Shimura variety.

Finiteness of Optimal Points in Shimura 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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