Irreducible canonical representations in positive characteristic

For X a curve over a field of positive characteristic, we investigate when the canonical representation of Aut(X) on H0(X , X) is irreducible. Any curve with an irreducible canonical representation must either be superspecial or ordinary. Having a small automorphism group is an obstruction to having irreducible canonical representation; with this motivation, the bulk of the paper is spent bounding the size of automorphism groups of superspecial and ordinary curves. After proving that all automorphisms of an Fq2 -maximal curve are defined over Fq2 , we find all superspecial curves with g > 82 having an irreducible representation. In the ordinary case, we provide a bound on the size of the automorphism group of an ordinary curve that improves on a result of Nakajima.