Automorphism groups of algebraic curves with p-rank zero

Details Automorphism groups of algebraic curves with p-rank zero Automorphism groups of algebraic curves with p-rank zero

2008-01-22

arxiv.org/abs/0801.3423v1

Mathematics

Algebraic Geometry

Scientific paper

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.

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