I would like to know, what is known on algebraic cycles of dimension 2 modulo algebraic or rational equivalence on the square of a generic abelian surface.

First, let $A$ be a generic abelian surface (generic abelian variety of dimension 2) over $\mathbb{C}$. Then the group of cycles of dimension 1 (divisors) up to rational equivalence is known, it its the Picard group of $A$, see e.g. the answers to this question and Fulton, Intersection Theory, Chapter 19. Still I have a stupid question: Can one "write down" all the (positive?) divisors on $A$?

The real question concerns the square $A^2=A\times_{\mathbb{C}} A$ of a generic abelian surface $A$. This is an abelian variety of dimension 4. I am interested in algebraic cycles of dimension 2 on $A^2$. I think I know the group of algebraic cycles of dimension 2 on $A^2$ modulo homological equivalence and modulo torsion, it is $\mathbb{Z}^6$ (because the space of invariants of $\mathrm{Sp}_{4,\mathbb{Q}}$ in $\wedge^4(\mathbb{Q}^4\oplus\mathbb{Q}^4)$ is of dimension 6). What is known about the group of algebraic cycles of dimension 2 on $A^2$ modulo rational or algebraic equivalence? In particular, what is known about the Griffiths group? Again a stupid question: Can one "write down" all the cycles of dimension 2 on $A^2$ (in some sense)?