p-Ranks and Automorphism Groups of Algebraic Curves

Nakajima, Shōichi

doi:10.2307/2000686

Cited by 37 publications

(59 citation statements)

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“…Stichtenoth's proof of Satz 3 in [26]). The second part follows from (1.2) and the same formula, since S. Nakajima has shown in [19], Theorem 2, that for ordinary X, the ramification groups of order ≥ 2 in π vanish. ✷ 2.…”

Section: Automorphisms and Uniformization In Positive Characteristicmentioning

confidence: 68%

Discontinuous groups in positive characteristic and automorphisms of Mumford curves

2001

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Section: Automorphisms and Uniformization In Positive Characteristicmentioning

confidence: 68%

Discontinuous groups in positive characteristic and automorphisms of Mumford curves

2001

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“…From [Nakajima 1987] we deduce that if G is a p-subgroup of Aut k (C) such that |G| > 2 p g/( p−1), the Hasse-Witt invariant of C is zero. The Deuring-Shafarevich formula (see [Bouw 2000], for instance) then implies that the genus of the quotient curve C/G is zero These results highlight the major role played by G 2 in the study of big actions.…”

Section: Introductionmentioning

confidence: 99%

Smooth curves having a large automorphismp-group in characteristicp >0

Matignon

Rocher

2008

ANT

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Let k be an algebraically closed field of characteristic p > 0 and C a connected nonsingular projective curve over k with genus g ≥ 2. This paper continues our study of big actions, that is, pairs (C, G) where G is a p-subgroup of the k-automorphism group of C such that |G|/g > 2 p/( p−1). If G 2 denotes the second ramification group of G at the unique ramification point of the cover C → C/G, we display necessary conditions on G 2 for (C, G) to be a big action, which allows us to pursue the classification of big actions.Our main source of examples comes from the construction of curves with many rational points using ray class field theory for global function fields, as initiated by J.-P. Serre and continued by Lauter and by Auer. In particular, we obtain explicit examples of big actions with G 2 abelian of large exponent.

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“…The following non-standard notation introduced by Nakajima in [19] can be useful. Let us denote the covering X → X /S by π.…”

Section: Background and Preliminary Resultsmentioning

confidence: 99%

“…For ordinary curves, Nakajima in his seminal work [19] proved that |Aut(X )| ≤ 84g(X )(g(X ) − 1). 1 However, Nakajima's bound seems to be far from optimal, and a bound of size at most g(X ) 8/5 is expected; see [16].…”

Section: Introductionmentioning

confidence: 99%

Large automorphism groups of ordinary curves in characteristic 2

Montanucci

Speziali

2019

Journal of Algebra

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Let X be a (projective, non-singular, geometrically irreducible) curve of even genus g(X ) ≥ 2 defined over an algebraically closed field K of odd characteristic p. If the prank γ(X ) equals g(X ), then X is ordinary. In this paper, we deal with large automorphism groups G of ordinary curves of even genus. We prove that |G| < 821.37g(X ) 7/4 . The proof of our result is based on the classification of automorphism groups of curves of even genus in positive characteristic, see [4]. According to this classification, for the exceptional cases Aut(X ) ∼ = Alt 7 and Aut(X ) ∼ = M 11 we show that the classical Hurwitz bound |Aut(X )| < 84(g(X ) − 1) holds, unless p = 3, g(X ) = 26 and Aut(X ) ∼ = M 11 ; an example for the latter case being given by the modular curve X(11) in characteristic 3.

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p-Ranks and Automorphism Groups of Algebraic Curves

Cited by 37 publications

References 0 publications

Discontinuous groups in positive characteristic and automorphisms of Mumford curves

Discontinuous groups in positive characteristic and automorphisms of Mumford curves

Smooth curves having a large automorphismp-group in characteristicp >0

Large automorphism groups of ordinary curves in characteristic 2

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