Zilber–Pink conjecture

Let $X$ be one of the following ambient varieties over $\mathbb C$, together with its collection $\mathcal S(X)$ of special subvarieties:

• a semiabelian variety, such as an algebraic torus $\mathbb G_m^n$ or an abelian variety. Its special subvarieties are the torsion cosets $t+H$, where $H\subseteq X$ is a connected algebraic subgroup and $t\in X$ is a torsion point;
• a mixed Shimura variety, such as the moduli space $\mathcal A_g$ of principally polarized abelian varieties, the product $Y(1)^n$ of modular curves, or the universal abelian variety over $\mathcal A_g$. Its special subvarieties are the irreducible components of images of mixed Shimura subvarieties under Hecke correspondences, as defined by Pink [4].

Let $V\subseteq X$ be an irreducible closed subvariety. An irreducible closed subvariety $W\subseteq V$ is atypical for $V$ if there is a special subvariety $T\in\mathcal S(X)$ such that $W$ is an irreducible component of $V\cap T$ and

$$\dim W\ >\ \dim V+\dim T-\dim X.$$

In words, $V$ and $T$ meet in a larger dimension than a generic intersection would.

Conjecture (Zilber–Pink). For every such $X$ and every irreducible closed subvariety $V\subseteq X$, the set of maximal atypical subvarieties of $V$ is finite.

A closely related form, due to Pink [4], states the following. If $V$ is not contained in any proper special subvariety of $X$, then the union of the intersections $V\cap T$ over all special $T$ with $\dim T<\dim X-\dim V$ is not Zariski dense in $V$. A solution must prove the conjecture for all semiabelian varieties and all mixed Shimura varieties.

Background

Zilber formulated the conjecture for algebraic tori in 2002, motivated by the model theory of complex exponentiation and Schanuel's conjecture [1]. Bombieri, Masser and Zannier independently studied unlikely intersections of curves with algebraic subgroups of $\mathbb G_m^n$ [2, 3]. Pink stated the conjecture for mixed Shimura varieties [4].

The conjecture unifies several earlier problems. Atypical points of $V$ are points lying in special subvarieties of too small dimension. When $T$ ranges over special points, the conjecture specializes to the Manin–Mumford conjecture for semiabelian varieties and to the André–Oort conjecture for Shimura varieties. Pink showed that it also implies the Mordell–Lang conjecture [4]. Many problems of the form "only finitely many parameters make two special conditions hold at once" are special cases [5, 6].