On the Zilber-Pink conjecture for complex abelian varieties

TL;DR

This paper demonstrates that the Zilber-Pink conjecture for complex abelian varieties over any characteristic zero field can be deduced from its validity over algebraic numbers, reducing the problem to a more specific case.

Contribution

It establishes that the conjecture over arbitrary fields of characteristic zero follows from its validity over algebraic numbers, simplifying the scope of the problem.

Findings

The conjecture for subvarieties of dimension at most m over any characteristic zero field is implied by the case over algebraic numbers.

The proof reduces the general case to the case over algebraic numbers, focusing on the largest abelian subvariety.

The result connects the validity of the Zilber-Pink conjecture across different fields of definition.

Abstract

In this article, we prove that the Zilber-Pink conjecture for abelian varieties over an arbitrary field of characteristic $0$ is implied by the same statement for abelian varieties over the algebraic numbers. More precisely, the conjecture holds for subvarieties of dimension at most $m$ in the abelian variety $A$ if it holds for subvarieties of dimension at most $m$ in the largest abelian subvariety of $A$ that is isomorphic to an abelian variety defined over $\bar{ \mathbb{Q}}$.