Abstract: The Chow group of zero-cycles of a variety X points can roughly be described as follows: the generators are closed points of X, and the relations, also known as rational equivalences, are divisors of rational functions on curves in X. In general, it can be very difficult to tell whether a given zero-cycle is trivial in the Chow group, and the structure of the Chow group as a whole is very mysterious. Deep conjectures due to Bloch and Beilinson give some indication of what the structure should be and which zero-cycles should vanish, but very little has been proven in this direction.

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.