The Hurwitz bound on the size of the K-automorphism group Aut(X) of an algebraic curve X of genus g>1 defined over a field K of zero characteristic is |Aut(X )| <=84(g − 1). For a positive characteristic, algebraic curves can have many more automorphisms than expected from the Hurwitz 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 of algebraic curves with very large automorphism groups 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 2 and |Aut(X )| > 24g(g − 1), then Aut(X ) has a fixed point on X, apart from a few exceptions. In the exceptional cases, the possibilities for Aut(X) and g are determined.

Automorphism groups of algebraic curves with p-rank zero

GIULIETTI, Massimo;
2010

Abstract

The Hurwitz bound on the size of the K-automorphism group Aut(X) of an algebraic curve X of genus g>1 defined over a field K of zero characteristic is |Aut(X )| <=84(g − 1). For a positive characteristic, algebraic curves can have many more automorphisms than expected from the Hurwitz 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 of algebraic curves with very large automorphism groups 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 2 and |Aut(X )| > 24g(g − 1), then Aut(X ) has a fixed point on X, apart from a few exceptions. In the exceptional cases, the possibilities for Aut(X) and g are determined.
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11391/41565
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