Zilber–Pink in abelian varieties and the Siegel threefold

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Proves the abelian Zilber–Pink conjecture over \(\overline{\mathbb Q}\): every irreducible subvariety has finitely many maximal atypical subvarieties relative to its smallest containing torsion coset. It also proves the full curve case in the Siegel threefold \(\mathcal A_2\) for Hodge-generic curves defined over \(\overline{\mathbb Q}\), without boundary or reduction assumptions.

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The paper states

Theorem 1.1

In The abelian Zilber–Pink conjecture, 24 September 2026

Possible match to OpenAI’s description, selected from the papers’ main statements because it shares the most wording. No review confirming the selection is recorded.

Eight other possible matches are listed below the card, among them Theorem 1.2 (The two companion branches) in Elliptic squares and Zilber–Pink for curves in A2 and Theorem 1.1 in The E×CM component of Zilber–Pink for curves in A2.

For every number field K, every abelian variety A/K, and every irreducible subvariety X\subseteq A_{\bar K}, there are only finitely many maximal atypical subvarieties of X in S=\langle X\rangle_{\mathrm{sp}}. Equivalently, there are finitely many proper atypical subvarieties Y_1,\ldots,Y_m\subsetneq X such that every atypical subvariety is contained in one of the Y_j.

Eight more possible matches

Theorem 1.2 (The two companion branches) in Elliptic squares and Zilber–Pink for curves in A2, 24 September 2026.
Theorem 1.1 in The E×CM component of Zilber–Pink for curves in A2, 24 September 2026.
Theorem 1.2 in The abelian Zilber–Pink conjecture, 24 September 2026.
Theorem 1.1 (The elliptic-square case) in Elliptic squares and Zilber–Pink for curves in A2, 24 September 2026.
Theorem 1.3 (Zilber–Pink for curves in \A_2) in Elliptic squares and Zilber–Pink for curves in A2, 24 September 2026.
Theorem 1.1 in Quaternionic division points on curves in the Siegel threefold, 24 September 2026.
Corollary 1.3 (Finite-rank translated intersections) in The abelian Zilber–Pink conjecture, 24 September 2026.
Theorem 1.2 in The E×CM component of Zilber–Pink for curves in A2, 24 September 2026.

Reported proof verification: The release has no Lean statement for this family. The papers are available to read and assess.
Papers and explanations: Four current papers, each readable here as a page beside its PDF.

Papers

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The abelian Zilber–Pink conjecture

24 September 2026, current edition.

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We prove the abelian Zilber–Pink conjecture over \(\overline{\mathbb Q}\). Every irreducible subvariety of an abelian variety has only finitely many maximal atypical subvarieties, where atypicality is measured inside its smallest containing torsion coset.

The E×CM component of Zilber–Pink for curves in A2

24 September 2026, current edition.

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We prove the \(E\times\mathrm{CM}\) component of Zilber–Pink for Hodge-generic algebraic curves in \(\mathcal A_2\) over \(\overline{\mathbb Q}\). Each such curve contains only finitely many points whose abelian surface is isogenous to a product of elliptic curves with at least one factor having complex multiplication.

Quaternionic division points on curves in the Siegel threefold

24 September 2026, current edition.

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We prove the quaternionic-division component of Zilber–Pink for curves in \(\mathcal A_2\) over \(\overline{\mathbb Q}\). A Hodge-generic algebraic curve contains only finitely many points whose full geometric rational endomorphism algebra is an indefinite quaternion division algebra over ℚ. No boundary or reduction hypothesis is required.

Elliptic squares and Zilber–Pink for curves in A2

24 September 2026, current edition.

Read it here PDF

We prove that every Hodge-generic algebraic curve in \(\mathcal A_2\) over \(\overline{\mathbb Q}\) contains only finitely many points whose abelian surface is isogenous to the square of an elliptic curve without complex multiplication (CM). Combining this result with the companion \(E\times\mathrm{CM}\) and quaternionic-division finiteness theorems, we prove the curve case of Zilber–Pink in \(\mathcal A_2\) over \(\overline{\mathbb Q}\), without a boundary hypothesis.

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Independent comparisons and proof-verification reports

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Within this family

Elliptic squares and Zilber–Pink for curves in A2 cites The E×CM component of Zilber–Pink for curves in A2 and Quaternionic division points on curves in the Siegel threefold.

Papers elsewhere in the release that this family’s papers cite

The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane \(\Re s\gt 7/8\)Current edition; has an accompanying Lean statement Family The quasi-Riemann hypothesisCited by The E×CM component of Zilber–Pink for curves in A2.

Papers elsewhere in the release that cite this family’s papers

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References outside the release

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David J. Smith, Points and their multiples on curves in powers of simple abelian varieties, 7 October 2026. Source
Smith lists OpenAI’s abelian Zilber-Pink announcement alongside independent forthcoming work of Barroero, Dill and Kuehne as the context for their curve-multiples theorem.

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