Automorphism groups of algebraic curves with p-rank zero

Massimo Giulietti, Gabor Korchmaros

Published 2008-01-22Version 1

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.