Irreducible canonical representations in positive characteristic

Abstract

For X a curve over a field of positive characteristic, we investigate when the canonical representation of Aut(X) on H 0 (X,Ω 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 $\mathbb {F}_{q^{2}}$ ‐maximal curve are defined over $\mathbb {F}_{q^{2}}$ , we find all superspecial curves with g>8 2 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.