On the Zilber-Pink conjecture for complex abelian varieties and distinguished categories [1]

Data:

16/01/2023 - 11:00

Aula Seminari

Fabrizio Barroero

Seminari di Algebra e Geometria Algebrica

Università di Roma Tre

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-

Oort.

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.