Math Conjectures

Zilber–Pink Conjecture for Abelian Varieties

OPENMajorConjectureProposed 1999–2005 · Standard version

Canonical statement

Let AA be a complex abelian variety. For s≥0s\ge0, let A[s]A^{[s]} be the union of all torsion cosets τ+B\tau+B, where τ∈A\tau\in A is a torsion point and B⊆AB\subseteq A is an abelian subvariety with codim⁡AB≥s\operatorname{codim}_A B\ge s. If X⊆AX\subseteq A is an irreducible closed subvariety not contained in any proper torsion coset, then
X∩A[dim⁡X+1] X\cap A^{[\dim X+1]}
is not Zariski dense in XX.
View source LaTeX
Let \(A\) be a complex abelian variety. For \(s\ge0\), let \(A^{[s]}\) be the union of all torsion cosets \(\tau+B\), where \(\tau\in A\) is a torsion point and \(B\subseteq A\) is an abelian subvariety with \(\operatorname{codim}_A B\ge s\). If \(X\subseteq A\) is an irreducible closed subvariety not contained in any proper torsion coset, then
\[
  X\cap A^{[\dim X+1]}
\] is not Zariski dense in \(X\).
AG-UNLIKELY-008:[Zilber2002Exponential][Pink2005Conjecture][HabeggerPila2016Unlikely]
Algebraic geometryNumber theory

Notes

Let AA be a complex abelian variety, and call a translate of an abelian subvariety of AA by a torsion point a torsion coset. The Zilber–Pink conjecture predicts that if an irreducible closed subvariety X⊆AX\subseteq A is not contained in any proper torsion coset, then the intersection of XX with the union of all torsion cosets of codimension at least dim⁡X+1\dim X+1 is not Zariski dense in XX. Such intersections are “unlikely”: a subvariety whose codimension exceeds dim⁡X\dim X should typically miss XX entirely. The conjecture emerged in stages between 1999 and 2005, through Zilber's work around exponential-sums equations and Schanuel-type problems [Zilber2002Exponential] and Pink's formulation for mixed Shimura varieties, of which the statement here isolates the abelian-variety core [Pink2005Conjecture].

The conjecture unifies a family of diophantine finiteness statements: important boundary cases are covered by Manin–Mumford and André–Oort, the latter part of Pink's common generalization [Pink2005Conjecture]. The o-minimality and point-counting strategy of Habegger and Pila has proved many low-dimensional or bounded-degree instances of the atypical-intersection statement [HabeggerPila2016Unlikely].

The general assertion remains open; a resolution must control atypical intersections uniformly, beyond the reach of current height and point-counting bounds.

References (3)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.