The Zilber-Pink conjecture predicts finiteness of maximal atypical intersections with special subvarieties in arithmetic algebraic varieties. The abelian formulation uses
torsion cosets, while the Shimura formulation uses
Shimura-special subvarieties and their Hecke
translates. For an irreducible subvariety 
of an abelian variety,
put 
equal to its smallest containing torsion coset.
An irreducible component 
of 
, with 
special, is atypical when
 |
Maximality means maximality under inclusion among the atypical subvarieties of 
in 
.
A subvariety of a Shimura variety is Hodge generic when it is contained in no proper special subvariety.
Curves in 
." Ann. Scuola Norm. Sup. Pisa Cl. Sci. 22,
1705-1745, 2021.