Abstract: The Chow group of zero-cycles of an algebraic variety under rational equivalence is a mysterious object whose structure depends in a crucial way on the arithmetic properties of the base field. In general, it can be very difficult to tell whether a given zero-cycle is trivial in the Chow group, but deep conjectures due to Bloch and Beilinson give some indication of what the structure should be and which zero-cycles should vanish.

In this talk we will focus on certain abelian surfaces A, and discuss a collection of methods that can take one of the zero-cycles that is predicted to vanish in Chow and verify that it is indeed a rational equivalence. The key idea behind these methods is a relation between hyperelliptic curves in A and rational curves in the Kummer surface of A. This is joint work with Evangelia Gazaki.