Open Access

Irreducible canonical representations in positive characteristic

Research in Number Theory20151:3

https://doi.org/10.1007/s40993-015-0004-8

Received: 26 September 2014

Accepted: 19 November 2014

Published: 29 May 2015

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.