Zilber–Pink conjecture

The intersection of two algebraic varieties is called atypical if its dimension is larger than expected. More precisely, given three varieties ${\displaystyle X,Y\subseteq U}$, a component ${\displaystyle Z}$ of the intersection ${\displaystyle X\cap Y}$ is said to be atypical in ${\displaystyle U}$ if ${\displaystyle \dim Z>\dim X+\dim Y-\dim U}$. Since the expected dimension of ${\displaystyle X\cap Y}$ is ${\displaystyle \dim X+\dim Y-\dim U}$, atypical intersections are "atypically large" and are not expected to occur. When ${\displaystyle \dim X+\dim Y-\dim U<0}$, the varieties ${\displaystyle X}$ and ${\displaystyle Y}$ are not expected to intersect at all, so when they do, the intersection is said to be unlikely. For example, if in a 3-dimensional space two lines intersect, then it is an unlikely intersection, for two randomly chosen lines would almost never intersect.

Special varieties of a Shimura variety are certain arithmetically defined subvarieties. They are higher dimensional versions of special points. For example, in semiabelian varieties special points are torsion points and special varieties are translates of irreducible algebraic subgroups by torsion points. In the modular setting special points are the singular moduli and special varieties are irreducible components of varieties defined by modular equations.

Given a mixed Shimura variety ${\displaystyle X}$ and a subvariety ${\displaystyle V\subseteq X}$, an atypical subvariety of ${\displaystyle V}$ is an atypical component of an intersection ${\displaystyle V\cap T}$ where ${\displaystyle T\subseteq X}$ is a special subvariety.

Let ${\displaystyle X}$ be a mixed Shimura variety or a semiabelian variety defined over ${\displaystyle \mathbb {C} }$, and let ${\displaystyle V\subseteq X}$ be a subvariety. Then ${\displaystyle V}$ contains only finitely many maximal atypical subvarieties.[4]

The abelian and modular versions of the Zilber–Pink conjecture are special cases of the conjecture for Shimura varieties, while in general the semiabelian case is not. However, special subvarieties of semiabelian and Shimura varieties share many formal properties which makes the same formulation valid in both settings.