The Zilber--Pink conjecture for Shimura varieties

About 5 years old · traced to 2021

Let $S$ be a Shimura variety. A special subvariety of $S$ is an irreducible component of a Shimura subvariety of $S$ . An irreducible subvariety $V\subset S$ is Hodge generic if it is not contained in any special subvariety other than a component of $S$ itself.

Zilber--Pink conjecture. If $V$ is an irreducible Hodge generic subvariety of $S$ , then the intersection of $V$ with the special subvarieties of $S$ having codimension greater than $\dim V$ is not Zariski dense in $V$ .

This is a general unlikely-intersections conjecture for Shimura varieties. The paper proves particular cases for curves in the moduli space of principally polarised abelian varieties, while the general statement remains open.

References

Primary source

Christopher Daw and Martin Orr, “Lattices with skew-Hermitian forms over division algebras and unlikely intersections”, arXiv:2111.13056 (2023).

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The manuscript claims finiteness of points on every reduced irreducible closed Hodge-generic curve in A_2 over the algebraic closure of Q whose abelian surface is isogenous to a product of elliptic curves with at least one CM factor, without a boundary or reduction hypothesis. This is the E × CM component of the curve case, rather than the general Shimura-variety conjecture.

Repository: https://github.com/openai/math

Manuscript: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-E-times-CM-Component-of-Zilber-Pink-for-Curves-in-A2-September-24-2026/paper.pdf

OpenAI-016-02-The-E-CM-component-of-Zilber-Pink-for-curves-in-A2.pdf413,183 bytesOpen

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The manuscript claims finiteness of points on every reduced irreducible closed Hodge-generic curve in A_2 over the algebraic closure of Q whose full geometric rational endomorphism algebra is an indefinite quaternion division algebra over Q, without a boundary or reduction hypothesis. This records only the quaternionic-division component of the curve case.

Repository: https://github.com/openai/math

Manuscript: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Quaternionic-Division-Points-on-Curves-in-the-Siegel-Threefold-September-24-2026/paper.pdf

OpenAI-016-03-Quaternionic-division-points-on-curves-in-the-Siegel-threefold.pdf571,090 bytesOpen

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Claimed by OpenAI.

The manuscript claims finiteness of non-CM elliptic-square points on every reduced irreducible closed Hodge-generic curve in A_2 over the algebraic closure of Q; combining its two companion branch theorems, it also claims finiteness of intersections with all special subvarieties of dimension at most one, without a boundary hypothesis. This records the Hodge-generic-curve case in A_2, rather than the general Shimura-variety conjecture.

Repository: https://github.com/openai/math

Manuscript: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Elliptic-Squares-and-Zilber-Pink-for-Curves-in-A2-September-24-2026/paper.pdf

OpenAI-016-04-Elliptic-squares-and-Zilber-Pink-for-curves-in-A2.pdf719,203 bytesOpen