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Elliptic squares and Zilber–Pink for curves in A2

OpenAI · September 24, 2026

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Abstract

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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Possibly relevant literature (found by the agent, not checked)

Fabrizio Barroero, Gabriel A. Dill, Distinguished categories and the Zilber–Pink conjecture (arXiv:2103.07422, to appear in Amer. J. Math.). Its base-change results include a case of complex curves in 𝒜₂ not defined over ℚ̄, which bears on extending Theorem 1.3 (Zilber–Pink for Hodge-generic curves in 𝒜₂ over ℚ̄) to curves over ℂ. [family-level]
Christopher Daw, Georgios Papas, Lang–Trotter phenomena and unlikely intersections (arXiv:2605.00759, May 2026). It uses André's G-function method to obtain Zilber–Pink cases for curves in 𝒜₃ with no boundary assumption, conditional on Lang–Trotter for pairs of elliptic curves; this is a point of comparison for the paper's boundary-free height bound in 𝒜₂. [family-level]
Christopher Daw, Unlikely intersections in Shimura varieties and beyond: a survey (arXiv:2506.02900, June 2025). A survey of the state of the art on unlikely intersections in Shimura varieties before this release, useful for placing the 𝒜₂ curve results. [family-level]

Elsewhere

VibeMathed: Problem record
Emergent Mind: family 016

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Cite

@misc{OAI:Elliptic-Squares-and-Zilber-Pink-for-Curves-in-A2-September-24-2026,
  author = {{OpenAI}},
  title = {{Elliptic squares and Zilber--Pink for curves in $\mathcal A_2$}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Elliptic-Squares-and-Zilber-Pink-for-Curves-in-A2-September-24-2026/paper.pdf}{OAI:Elliptic-Squares-and-Zilber-Pink-for-Curves-in-A2-September-24-2026}},
  year = {2026}
}