  We study the (covering) gonality of abelian varieties and their orbits of
zero-cycles for rational equivalence. We show that any orbit for rational
equivalence of zero-cycles of degree $k$ has dimension at most $k-1$. Building
on the work of Pirola, we show that very general abelian varieties of dimension
$g$ have covering gonality $k\geq f(g)$ where $f(g)$ grows like ${\rm log}\,g$.
This answers a question asked by Bastianelli, De Poi, Ein, Lazarsfeld and B.
Ullery. We also obtain results on the Chow ring of very general abelian
varieties, eg. if $g\geq 2k-1$, for any divisor $D\in {\rm Pic}^0(A)$, $D^k$ is
not a torsion cycle.

Algebraic Geometry