  Given a closed subscheme $Z$ of a polarized abelian variety $(A,\ell)$ we
define its vanishing threshold with respect to $\ell$ and relate it to the
Seshadri constant of the ideal defining $Z.$ As a particular case, we introduce
the notion of jets-separation thresholds, which naturally arise as the
vanishing threshold of the $p$-infinitesimal neighborhood of a point.
Afterwards, by means of Fourier-Mukai methods we relate the jets-separation
thresholds with the surjectivity of certain higher Gauss-Wahl maps. As a
consequence we obtain a criterion for the surjectivity of those maps in terms
of the Seshadri constant of the polarization $\ell.$
