Explicit Galois representations of automorphisms on holomorphic differentials in characteristic $p$

TL;DR

This paper explicitly determines how automorphism groups act on holomorphic differentials in cyclotomic function fields over characteristic p, including complex cases with wild ramification, and extends findings to rank one Drinfeld modules.

Contribution

It provides explicit bases and descriptions of automorphism actions on differentials, including cases with wild ramification and non-cyclic p-part groups, advancing understanding in positive characteristic function fields.

Findings

Explicit automorphism representations on differentials constructed

Includes cases with wild ramification and non-cyclic p-part groups

Results extended to rank one Drinfeld modules

Abstract

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