The Chow group of zero-cycles is a generalization to higher dimensions of the Picard group of a smooth projective curve. When X is a curve over an algebraically closed field k its Picard group can be fully understood by the Abel-Jacobi map, which gives an isomorphism between the degree zero elements of the Picard group and the k-points of the Jacobian variety of X. In higher dimensions however the situation is much more chaotic, as the Abel-Jacobi map in general has a kernel, which over large fields like the complex numbers can be enormous. On the other extreme, a famous conjecture of Beilinson predicts that if X is a smooth projective variety over the algebraic closure of the rational numbers, then this kernel is zero. For a variety X with positive geometric genus this conjecture is very hard to establish. In fact, there are hardly any examples in the literature. In this talk I will discuss joint work with Jonathan Love where we make substantial progress on this conjecture for an abelian surface A. First, we will describe a very large collection of relations in the kernel of the Abel-Jacobi arising from hyperelliptic curves mapping to A. Second, we will show that at least in the special case when A is isogenous to a product of two elliptic curves, such hyperelliptic curves are plentiful. Namely, we will describe a construction that produces for infinitely many values of g countably many hyperelliptic curves of genus g mapping birationally into A.