  In this paper we study the space $\Omega(m)$, of holomorphic
$m$-(poly)differentials of a function field of a curve defined over an
algebraically closed field of characteristic $p>0$ when
  $G$ is cyclic or elementary abelian group of order $p^n$; we give bases for
each case when the base field is rational, introduce the Boseck invariants and
give an elementary approach to the $G$ module structure of $\Omega(m)$ in terms
of Boseck invariants. The last computation is achieved without any restriction
on the base field in the cyclic case, while in the elementary abelian case it
is assumed that the base field is rational. An application to the computation
of the tangent space of the deformation functor of curves with automorphisms is
given.
