Large 3-groups of automorphisms of algebraic curves in characteristic 3

2013/12/18 by Massimo Giulietti, Giulietti, Massimo, Gábor Korchmáros +1
Mathematics · #14H37 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Finite Group Theory Research

paper · pdf · doi:10.48550/arxiv.1312.5108

openalex publication_date 2013/12/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let S be a p-subgroup of the \K-automorphism group \aut(\cX) of an algebraic curve \cX of genus ≫≥ 2 and p-rank γ defined over an algebraically closed field \mathbbK of characteristic p≥ 3.In this paper we prove that if |S|>2(≫-1) then one of the following cases occurs. \beginitemize \item[(i)] γ=0 and the extension \K(\cX)/\K(\cX)^S completely ramifies at a unique place, and does not ramify elsewhere. \item[(ii)] γ>0, p=3, \cX is a general curve, S attains the Nakajima's upper bound 3(γ-1) and \K(\cX) is an unramified Galois extension of the function field of a general curve of genus 2 with equation Y²=cX⁶+X⁴+X²+1 where c∈\K^*. \enditemize Case (i) was investigated by Stichtenoth, Lehr, Matignon, and Rocher.

Citations

• On a special class of p-groups
• Automorphisms groups for p-cyclic covers of the affine line

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