Zilber–Pink in abelian varieties and the Siegel threefold

Unlikely intersections in abelian varieties are claimed to be finite in number, in every dimension.

4 papers September 24, 2026 Explained by AI

An abelian variety is a geometric shape with a way to add its points. A dimension count predicts how big its intersections with special pieces should be. The abelian Zilber–Pink conjecture says the bigger-than-expected intersections fall into finitely many families. The first paper claims to prove it over the algebraic numbers in every dimension. Three more papers claim the curve case in A₂, a space whose points stand for abelian surfaces.

First step

2002

B. Zilber, who formulated the unlikely-intersection principle for semiabelian varieties

Significance

Major

Earlier results covered curves or added extra conditions; the paper claims every dimension and codimension, if the proof holds.

Trust

Claimed, not checked

Field

Number theory

4 papers

The explanation

Take the curve y² = x³ − x. Its points, with one extra point at infinity, can be added by a geometric rule. That makes it an elliptic curve, the simplest abelian variety (a shape whose points can be added). The points (0, 0), (1, 0) and (−1, 0) are torsion points. Adding any one of them to itself gives the zero point. Products of such curves give abelian varieties of any dimension.

Inside an abelian variety A, the special pieces are torsion cosets. A torsion coset is a smaller abelian variety shifted by a torsion pointSource 1. Now take a subvariety X, a curved shape inside A. Dimensions predict how big X meets a special piece T: dim X + dim T − dim A. Take A as a product of three elliptic curves, so dimension 3. Let X be a curve, dimension 1, and T a shifted elliptic curve, dimension 1. The prediction is 1 + 1 − 3 = −1, so they should not meet at all. A meeting point is atypical: bigger than predictedSource 2.

Such meetings can happen, because the pieces T vary without end. The abelian Zilber–Pink conjecture says they are rare in a precise sense. All the largest atypical pieces fall into a finite listSource 3. The case where every T is a single torsion point is the Manin–Mumford theorem, proved by Raynaud in 1983Source 4. Zilber stated the general principle in 2002, and Pink gave a related form in 2005Source 5.

Before this paper, the curve case was known. Habegger and Pila proved finiteness for curves in any abelian variety over number fields in 2016Source 6. Barroero, Kühne and Schmidt extended it to semiabelian varietiesSource 7. In higher dimensions, results needed extra conditions on how X sits under quotient mapsSource 8. Some covered powers of a single elliptic curve in codimension twoSource 9.

The first paper claims the conjecture over number fields in every dimensionSource 10. For every abelian variety and every subvariety X, the largest atypical pieces form a finite listSource 3. No condition on A is imposed, such as complex multiplication (extra symmetries of the curve)Source 11. The theorem gives no bound for how many pieces there areSource 12. In the example, a curve in the triple product has only finitely many such atypical meeting points.

The hard part is height. Height measures how complicated the coordinates of a point are. Earlier non-density results held below each fixed height boundSource 13. The paper uses a height argument in the tradition of Vojta and FaltingsSource 14. It adds geometric results of Ax and of Habegger and Pila. It then passes from non-density to finiteness with a reduction by Barroero and DillSource 15.

Three companion papers treat curves in A₂ (a space whose points stand for abelian surfaces with a fixed kind of polarization). A curve there is Hodge generic when it lies in no proper special subvarietySource 16. One paper claims finiteness for points whose surface is like E × E for a non-CM elliptic curve ESource 17. The other two cover products with a CM factor, and quaternion division pointsSource 18. Together they claim the curve case of Zilber–Pink in A₂Source 19.

How sure is this? No Lean formalization is listed for these papers in the openai/math release. So no computer proof checker has checked them. The A₂ result depends on all three linked papers. No specialist review has been published.

What it doesn't show. This does not bound how many atypical pieces there are, and it does not cover Zilber–Pink in general Shimura varieties.

Where it came from

This result comes from an unreleased OpenAI model, published in the openai/math release on 6 October 2026.

Hype check

Major only if the proofs hold. No Lean formalization is listed and no specialist review is published. The results are ineffective: they give no bound on how many atypical pieces occur.

Words used

Abelian variety: A geometric shape defined by polynomial equations whose points can be added like numbers.
Elliptic curve: An abelian variety of dimension one, such as the points of y² = x³ − x plus a point at infinity.
Torsion point: A point that returns to zero after being added to itself some number of times.
Torsion coset: A smaller abelian variety shifted by a torsion point; the special pieces in this problem.
Atypical intersection: An intersection piece of larger dimension than a dimension count predicts.
Height: A measure of how complicated the coordinates of an algebraic point are.
Hodge generic: For a curve in A₂, lying in no proper special subvariety.
Complex multiplication (CM): Extra symmetries of an elliptic curve beyond multiplication by whole numbers.

The original papers (4)

The abelian Zilber–Pink conjecture (PDF, opens in a new tab)
September 24, 2026
The E×CM component of Zilber–Pink for curves in A2 (PDF, opens in a new tab)
September 24, 2026
Quaternionic division points on curves in the Siegel threefold (PDF, opens in a new tab)
September 24, 2026
Elliptic squares and Zilber–Pink for curves in A2 (PDF, opens in a new tab)
September 24, 2026

All quoted lines (19)

A special subvariety of $A_{\bar K}$ is a torsion coset $B+\tau$ , where $B$ is an abelian subvariety and $\tau$ is a torsion point.
The abelian Zilber–Pink conjecture Back to text
An irreducible component $Y$ of $X\cap T$ , for a special $T\subseteq S$ , is \emphatypical for $X$ in $S$ if $\dim Y>\dim X+\dim T-\dim S.$
The abelian Zilber–Pink conjecture Back to text
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 $S=\langle X\rangle_{\mathrm{sp}}$ .
The abelian Zilber–Pink conjecture Back to text
The torsion-point case belongs to the Manin–Mumford theorem of Raynaud: a subvariety with dense torsion points is a torsion coset.
The abelian Zilber–Pink conjecture Back to text
Zilber formulated the unlikely-intersection principle for semiabelian varieties.
The abelian Zilber–Pink conjecture Back to text
Habegger and Pila proved finiteness for curves in arbitrary abelian varieties over number fields.
The abelian Zilber–Pink conjecture Back to text
Barroero, K\"uhne and Schmidt subsequently established the corresponding curve theorem for general semiabelian varieties over number fields.
The abelian Zilber–Pink conjecture Back to text
Higher-dimensional non-density was known under additional geometric or height restrictions.
The abelian Zilber–Pink conjecture Back to text
For codimension-two subvarieties of powers of an elliptic curve contained in no proper torsion coset, maximal-atypical finiteness was proved in the CM case by Checcoli, Veneziano and Viada and without the CM restriction by Hubschmid and Viada.
The abelian Zilber–Pink conjecture Back to text
We prove the conjecture over number fields in arbitrary dimension.
The abelian Zilber–Pink conjecture Back to text
No bound on the degrees or heights of the intersection points is assumed, and no simplicity, endomorphism, or complex multiplication hypothesis is imposed on $A$ .
The abelian Zilber–Pink conjecture Back to text
The theorem concerns torsion cosets in abelian varieties; it gives no effective or uniform bound for the number of maximal atypical subvarieties.
The abelian Zilber–Pink conjecture Back to text
Without that condition, their Theorem 1.3(i) gives non-density at every fixed height bound when $X$ is contained in no proper torsion coset.
The abelian Zilber–Pink conjecture Back to text
The height argument follows Vojta's curve method, Faltings' extension to abelian varieties, and R\'emond's generalized inequality.
The abelian Zilber–Pink conjecture Back to text
The passage from this assertion to Theorem … uses the optimal-subvariety reduction of Barroero and Dill.
The abelian Zilber–Pink conjecture Back to text
Let $\A_2$ be the coarse moduli variety of principally polarized abelian surfaces. A curve in $\A_2$ is Hodge generic if it is contained in no proper Shimura-special subvariety.
Elliptic squares and Zilber–Pink for curves in A2 Back to text
If $C$ is Hodge generic, then $\Sigma_{E^2}(C)$ is finite.
Elliptic squares and Zilber–Pink for curves in A2 Back to text
The two remaining special-curve loci are treated by the following companion theorems.
Elliptic squares and Zilber–Pink for curves in A2 Back to text
… resolves positively the Hodge-generic-curve case of the Zilber–Pink conjecture in $\A_2$ over $\Qbar$ .
Elliptic squares and Zilber–Pink for curves in A2 Back to text

Other sources

006Goldfeld’s conjecture: densities and mean analytic rankNumber theory
002The full BSD formula from low Selmer corankNumber theory
030Modularity of elliptic curves over imaginary quadratic fieldsNumber theory
001Milne’s rationality conjecture and algebraic specializationNumber theory
005Irrationality of Catalan’s constantNumber theory

Browse Number theory Elliptic curves and abelian varieties Open since the 2000s Claimed, not yet checked