  G. Pareschi and M. Popa give criterions for global generations and
surjectivity of multiplication maps of global sections of coherent sheaves on
abelian varieties in the theory of M-regularity. In this paper, we generalize
some of their criterions via the M-regularity of $\mathbb{Q}$-twisted sheaves.
As an application, we show that the M-regularity of a suitable
$\mathbb{Q}$-twisted sheaf implies property $(N_p)$ and jet-ampleness for ample
line bundles on abelian varieties.
