Automorphism groups of algebraic curves with p-rank zero

TL;DR

This paper investigates automorphism groups of algebraic curves with zero p-rank in characteristic 2, showing that large automorphism groups typically have fixed points unless in specific exceptional cases.

Contribution

It establishes conditions under which automorphism groups of zero p-rank curves have fixed points, and classifies exceptions for genus g ≥ 2 in characteristic 2.

Findings

Automorphism groups with size > 24g^2 have fixed points on the curve.

Identifies exceptional cases with specific automorphism groups and genera.

Provides classification for automorphism groups in these exceptional cases.

Abstract

In positive characteristic, algebraic curves can have many more automorphisms than expected from the classical Hurwitz's bound. There even exist algebraic curves of arbitrary high genus g with more than 16g^4 automorphisms. It has been observed on many occasions that the most anomalous examples invariably have zero p-rank. In this paper, the K-automorphism group Aut(X) of a zero 2-rank algebraic curve X defined over an algebraically closed field K of characteristic 2 is investigated. The main result is that if the curve has genus g greater than or equal to 2, and |Aut(X)|>24g^2, then Aut(X) has a fixed point on X, apart from few exceptions. In the exceptional cases the possibilities for Aut(X) and g are determined.