Large p-groups of automorphisms of algebraic curves in characteristic p

JOURNAL OF ALGEBRA

卷:481

Article

Giulietti, Massimo¹  Korehmaros, Gabor² 

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

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

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

DOI  :  10.1016/j.jalgebra.2017.02.024

来源: Elsevier

【 摘 要 】

Let S be a p-subgroup of the K-automorphism group Aut(X) of an algebraic curve X of genus g >= 2 and p-rank gamma defined over an algebraically closed field K of characteristic p >= 3. Nakajima [27] proved that if gamma >= 2 then vertical bar S vertical bar <= p/p-2(g - 1). If equality holds, X is a Nakajima extremal curve. We prove that if vertical bar S vertical bar > p(2)/p(2) - p - 1(g - 1) then one of the following cases occurs. (i)gamma = 0 and the extension K(X)vertical bar K(X)(S) completely ramifies at a unique place, and does not ramify elsewhere. (ii)vertical bar S vertical bar =p, and X is an ordinary curve of genus g = p - 1. (iii) X is an ordinary, Nakajima extremal curve, and K(X) is an unramified Galois extension of a function field of a curve given in (ii). (iii) X is an ordinary, Nakajima extremal curve, and K(X) is an unramified Galois extension of a function field of a curve given in (ii). There are exactly p - 1 subgroups M of S such that K(X)vertical bar K(X)(M) is such a Galois extension. Moreover, if some of them is an abelian extension then S has maximal nilpotency class. The full K-automorphism group of any Nakajima extremal curve is determined, and several infinite families of Nakajima extremal curves are constructed by using their pro-p fundamental groups. (C) 2017 Elsevier Inc. All rights reserved.

【 授权许可】

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