Zhang's conjecture and squares of abelian surfaces

TL;DR

This paper provides counterexamples involving squares of abelian surfaces that challenge Zhang's conjecture on the intersection of subvarieties and preperiodic points.

Contribution

It introduces specific counterexamples to Zhang's conjecture, advancing understanding of the conjecture's limitations in abelian surface contexts.

Findings

Counterexamples to Zhang's conjecture are constructed.

The results show the conjecture does not hold universally.

Insights into the intersection behavior of subvarieties and preperiodic points.

Abstract

We give in this paper some squares of abelian surfaces that are counterexamples to a conjecture formulated by Zhang about the intersection of subvarieties and preperiodic points.