On holomorphic polydifferentials in positive characteristic

Let F/E be an abelian Galois extension of function fields over an algebraic closed field K of characteristic p > 0. Denote by G the Galois group of the extension F/E. In this paper, we study Ω(m), the space of holomorphic m‐(poly)differentials of the function field of F when G is cyclic or a certain elementary abelian group of order pⁿ; 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 Ω(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. Finally, an application to the computation of the tangent space of the deformation functor of curves with automorphisms is given.