On the Zilber-Pink conjecture over transcendental ground fields (PLEASE NOTE THE TIME!)

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-Oort.

After a general introduction to the conjecture, I will report on joint work with Fabrizio Barroero in which we proved that the Zilber-Pink conjecture for a complex abelian variety A can be deduced from the same statement 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 curves 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 with 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. Time permitting, I will introduce this setting and some results that we obtain in it (both conditional and unconditional).