  This paper is mainly concerned with applying the theory of M-regularity
developed in the previous math.AG/0110003 to the study of linear series given
by multiples of ample line bundles on abelian varieties. We define a new
invariant of a line bundle, called M-regularity index, which is seen to govern
the higher order properties and (partly conjecturally) the defining equations
of such embeddings. We prove a general result on the behavior of higher
syzygies in embeddings given by multiples of ample bundles whose base locus has
no fixed components, extending a conjecture of Lazarsfeld known to be true by
work of the first author. This approach also unifies essentially all the
previously known results in this direction and is mainly based on vanishing
theorems rather than representations of theta groups.
