Irreducible Canonical Representations in Positive Characteristic

TL;DR

This paper studies when the automorphism group action on the space of holomorphic differentials of a curve over a field of positive characteristic is irreducible, linking it to the curve's superspecial or ordinary nature and bounding automorphism groups.

Contribution

It characterizes conditions for irreducible canonical representations, bounds automorphism groups of superspecial and ordinary curves, and classifies superspecial curves with large genus having irreducible representations.

Findings

All automorphisms of an $ ext{F}_{q^2}$-maximal curve are defined over $ ext{F}_{q^2}$.

Classified superspecial curves with genus > 82 with irreducible representations.

Provided improved bounds on automorphism groups of ordinary curves.

Abstract

For $X$ a curve over a field of positive characteristic, we investigate when the canonical representation of $\text{Aut}(X)$ on $H^0(X, \Omega_X)$ 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 $g > 82$ 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.