# Irreducible canonical representations in positive characteristic

Research paper by Benjamin Gunby, Alexander Smith, Allen Yuan

Indexed on: 29 May '15Published on: 29 May '15Published in: Research in Number Theory

#### Abstract

For X a curve over a field of positive characteristic, we investigate when the canonical representation of Aut(X) on H0(X,ΩX) 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>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.