Explicit Galois representations of automorphisms on holomorphic differentials in characteristic p

We determine the representation of the Galois group for the cyclotomic function fields in characteristic p > 0 induced by the natural action on the space of holomorphic differentials via construction of an explicit canonical basis of differentials. This includes those cases which present wild ramification and finite automorphism groups with non-cyclic p-part, which have remained elusive. We also obtain information on the gap sequences of ramified primes and extend these results to rank one Drinfel'd modules.