  In this paper we consider the question of when Seshadri constants on abelian
surfaces are integers. Our first result concerns self-products $E\times E$ of
elliptic curves: If $E$ has complex multiplication in $\Z[i]$ or in
$\Z[\frac12(1+i\sqrt3)]$ or if $E$ has no complex multiplication at all, then
it is known that for every ample line bundle $L$ on $E\times E$, the Seshadri
constant $\eps(L)$ is an integer. We show that, contrary to what one might
expect, these are in fact the only elliptic curves for which this integrality
statement holds. Our second result answers the question how -- on any abelian
surface~-- integrality of Seshadri constants is related to elliptic curves.
