Large 2-groups of automorphisms of algebraic curves over a field of characteristic 2

JOURNAL OF ALGEBRA

卷:427

Article

Giulietti, Massimo¹  Korchmaros, Gabor² 

[1] Univ Perugia, Dipartimento Matemat & Informat, I-06123 Perugia, Italy

[2] Univ Basilicata, Dipartimento Matemat, I-85100 Potenza, Italy

关键词: Algebraic curves;    Positive characteristic;    Automorphism groups;   

DOI  :  10.1016/j.jalgebra.2014.12.019

来源: Elsevier

【 摘 要 】

Let S be a 2-subgroup of the K-automorphisrn group Aut(chi) of an algebraic curve chi of genus g(chi) defined over an algebraically closed field K of characteristic 2. It is known that S may be quite large compared to the classical Hurwitz bound 84(g(chi)-1). However, if S fixes no point, then the size of S is smaller than or equal to 4(g(chi) - 1). In this paper, we investigate algebraic curves chi with a 2-subgroup S of Aut(chi) having the following properties: (I) vertical bar S vertical bar >= 8 and vertical bar S vertical bar > 2(g(chi) - 1), (II) S fixes no point on chi. Theorem 1.2 shows that chi is a general curve and that either vertical bar S vertical bar = 4(g(chi) - 1), or vertical bar S vertical bar = 2g(chi) + 2, or, for every involution U is an element of Z(S), the quotient curve chi/< u > inherits the above properties, that is, it has genus >= 2, and its automorphism group S/< u > still has properties (I) and (II). In the first two cases, S is completely determined. We also give examples illustrating our results. In particular, for every g = 2(h) +1 >= 9, we exhibit a (general bielliptic) curve chi of genus g whose K-automorphism group has a dihedral 2-subgroup S of order 4(g - 1) that fixes no point in chi. (c) 2014 Elsevier Inc. All rights reserved.

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