Automorphism groups of algebraic curves with p-rank zero
Identifieur interne : 000306 ( Istex/Checkpoint ); précédent : 000305; suivant : 000307Automorphism groups of algebraic curves with p-rank zero
Auteurs : Massimo Giulietti [Italie] ; Gábor Korchmáros [Italie]Source :
- Journal of the London Mathematical Society [ 0024-6107 ] ; 2010-04.
Abstract
The Hurwitz bound on the size of the 𝕂-automorphism group Aut(𝒳) of an algebraic curve 𝒳 of genus g ≥ 2 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 16g4 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.
Url:
DOI: 10.1112/jlms/jdp066
Affiliations:
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