$F_p$-espaces vectoriels de formes diff\'erentielles logarithmiques sur la droite projective

Let k be an algebraically closed field of characteristic p >0. Let $m \in \N$, (m,p)=1. We study $\fp$-vector spaces of logarithmic differential forms on the projective line such that each non zero form has a unique zero at $\infty$ of given order m-1. We discuss the existence of such vectors spaces according to the value of m. We give applications to the lifting to characteristic 0 of $(\Z /p\Z)^n$ actions as k-automorphisms of $k[[t]]$.