Abstract: The Zilber-Pink conjecture is a very general statement that implies many well-known
results in diophantine geometry, e.g., Manin-Mumford, Mordell-Lang and André-
I will report on recent joint work with Gabriel Dill in which we proved that the Zilber-
Pink conjecture for a complex abelian variety A can be deduced from the same state-
ment for its trace, i.e., the largest abelian subvariety of A that can be defined over the
algebraic numbers. This gives some unconditional results, e.g., the conjecture for cur-
ves in complex abelian varieties (over the algebraic numbers this is due to Habegger
and Pila) and the conjecture for arbitrary subvarieties of powers of elliptic curves that
have transcendental j-invariant.
While working on this project we realised that many definitions, statements and proofs
were formal in nature and we came up with a categorical setting that contains most
known examples and in which (weakly) special subvarieties can be defined and a
Zilber-Pink statement can be formulated. We obtained some conditional as well as
some unconditional results.