Automorphism groups of hyperelliptic curves of $2$-rank zero

TL;DR

This paper classifies the automorphism groups of hyperelliptic curves of small genus and zero 2-rank in characteristic 2, providing explicit structures and conjectures based on computational experiments.

Contribution

It determines the automorphism group structures for small genus hyperelliptic curves of 2-rank zero in characteristic 2, including conjectures inspired by computational data.

Findings

Explicit automorphism group structures for small genus curves.

Semidirect product structures of automorphism groups clarified.

Formulation of two conjectures analogous to the Oort conjecture.

Abstract

In this paper, we determine the reduced automorphism groups of hyperelliptic curves of a small genus in characteristic $2$, when they are of $2$-rank $0$. Such a curve is an Artin-Schreier curve defined in the form $y^2-y=f(x)$ for a polynomial $f(x)$. After we clarify semidirect-product structures of the automorphism groups for an arbitrary genus, we derive the detailed group structures for the reduced automorphism groups of the curves of a small genus, through computations using the computational algebra system Magma. With these experiments, we formulate two conjectures, which are analogues for our curves of the Oort conjecture on automorphism groups of generic principally polarized supersingular abelian varieties.