Large p-group actions with a p-elementary abelian derived group

Rocher, Magali

doi:10.1016/j.jalgebra.2008.09.030

Cited by 14 publications

(13 citation statements)

References 14 publications

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“…Since |Q| is relatively small, Theorem 1.1 also shows that the order of the automorphism group is mostly prime-to-p. This contrasts with the situation in many recent papers about curves whose automorphism group is large [16][17][18].…”

Section: Introductioncontrasting

confidence: 71%

The automorphism groups of a family of maximal curves

2012

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The Hasse Weil bound restricts the number of points of a curve which are defined over a finite field; if the number of points meets this bound, the curve is called maximal. Giulietti and Korchmaros introduced a curve C3 which is maximal over F q 6 and determined its automorphism group. Garcia, Guneri, and Stichtenoth generalized this construction to a family of curves Cn, indexed by an odd integer n ≥ 3, such that Cn is maximal over F q 2n . In this paper, we determine the automorphism group Aut(Cn) when n > 3; in contrast with the case n = 3, it fixes the point at infinity on Cn. The proof requires a new structural result about automorphism groups of curves in characteristic p such that each Sylow p-subgroup has exactly one fixed point. keywords: Weil bound, maximal curve, automorphism, ramification.

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Section: Introductioncontrasting

confidence: 71%

The automorphism groups of a family of maximal curves

2012

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“…Later on, Matignon and Rocher [24] showed that the action of a p-subgroup of K-automorphisms S satisfying |S| > 4 (p 2 −1) 2 g(X ) 2 , corresponds to theétale cover of the affine line with Galois group S ∼ = (Z/pZ) n for n ≤ 3. These results have been refined by Rocher, see [31] and [32]. The essential tools used in the above mentioned papers are ramification theory and some structure theorems about finite p-groups.…”

Section: Introductionmentioning

confidence: 99%

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

Giulietti¹,

Korchmáros²

2017

Journal of Algebra

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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 γ defined over an algebraically closed field K of characteristic p ≥ 3. Nakajima [26] proved that if γ ≥ 2 then |S| ≤ p p−2 (g − 1). If equality holds, X is a Nakajima extremal curve. We prove that ifthen one of the following cases occurs.(i) γ = 0 and the extension K(X )|K(X ) S completely ramifies at a unique place, and does not ramify elsewhere.(ii) |S| = 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). There are exactly p − 1 such Galois extensions. 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.

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“…Matignon and Rocher [20,23,24] continued the work of Lehr and Matignon, especially in order to classify big actions in which |S| g 2 > 4 (p 2 − 1) 2 .…”

Section: Introductionmentioning

confidence: 99%

Algebraic curves with a large non-tame automorphism group fixing no point

Giulietti

Korchmáros

2010

Trans. Amer. Math. Soc.

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Abstract. Let K be an algebraically closed field of characteristic p > 0, and let X be a curve over K of genus g ≥ 2. Assume that the automorphism group Aut(X ) of X over K fixes no point of X . The following result is proven. If there is a point P on X whose stabilizer in Aut(X ) contains a p-subgroup of order greater than gp/(p − 1), then X is birationally equivalent over K to one of the irreducible plane curves (II), (III), (IV), (V) listed in the Introduction.

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Large p-group actions with a p-elementary abelian derived group

Cited by 14 publications

References 14 publications

The automorphism groups of a family of maximal curves

The automorphism groups of a family of maximal curves

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

Algebraic curves with a large non-tame automorphism group fixing no point

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