Zilber–Pink for Hodge-generic curves in A₂
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Does Zilber–Pink hold for Hodge-generic curves in the Siegel threefold A₂ over algebraic number fields without a boundary hypothesis? The studied exceptional branches include non-CM elliptic squares, E×CM cases and quaternionic-division points.
Why it matters and what counts as progress
Why it matters: Removing the boundary restriction would extend finiteness beyond previously treated degeneration and fiber cases.
What counts as progress: Finiteness for every required exceptional branch under the same curve hypotheses, with the combined implication justified rather than inferred from only one branch.
How the problem developed
- 24 Sept 2026 · Reported theorem [1]
OpenAI Reported finiteness of points on a Hodge-generic curve in A₂ whose surface is isogenous to the square of a non-CM elliptic curve.
- 24 Sept 2026 · Reported theorem [2]
OpenAI Reported finiteness for the E×CM locus: products of elliptic curves with at least one CM factor on Hodge-generic curves in A₂ over the algebraic closure of a number field.
- 24 Sept 2026 · Reported theorem [3]
OpenAI Reported finiteness of points on Hodge-generic curves in A₂ whose endomorphism algebra is an indefinite quaternion division algebra.
AI contribution
Elliptic squares and Zilber–Pink for curves in A2
Three OpenAI manuscripts jointly claim Zilber–Pink for Hodge-generic curves in the Siegel threefold A₂ over algebraic number fields without a boundary hypothesis. The combined result uses separate finiteness theorems for non-CM elliptic squares, E×CM cases and quaternionic-division points.
Sources
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Publication dates and editorial changes.