Transcendence degree one function fields over a finite field with many automorphisms

Korchmáros, Gábor; Montanucci, Maria; Speziali, Pietro

doi:10.1016/j.jpaa.2017.08.008

Cited by 5 publications

(9 citation statements)

References 21 publications

(27 reference statements)

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“…We are left with case (iii). In this case, (e Q 2 , d Q 2 ) ∈ {(2, 1), (4, 3), (5,4), (8,7), (11,10)}. Note that, apart from the case (8,7), in all the other cases we get an even genus if and only if (e Q 2 , d Q 2 ) = (72, 79).…”

Section: Note Thatmentioning

confidence: 92%

“…If Aut(X ) is solvable and p > 2, an even tighter bound (1) |Aut(X )| ≤ 34(g(X ) + 1) 3/2 holds; see [10]. The latter bound is sharp up to the constant term; see [11]. The object of this paper is two-fold.…”

Section: Introductionmentioning

confidence: 96%

“…If Aut(X ) is solvable and p > 2, an even tighter bound (1) |Aut(X )| ≤ 34(g(X ) + 1) 3/2 holds; see [12]. The latter bound is sharp up to the constant term; see [13]. More recently, the bound 1 has been proven to hold also for p = 2 and g(X ) even; see [15].…”

Section: Introductionmentioning

confidence: 98%

“…Two sporadic cases then arise: Aut(X ) ∼ = Alt 7 , M 11 , where M 11 is the Mathieu group of degree 11. In these cases, the classical Hurwitz bound for |Aut(X )| holds, unless p = 3, g(X ) = 26, Aut(X ) = M 11 , an example being given by the modular curve X (11), see Section 5.…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Note Thatmentioning

confidence: 92%

Section: Introductionmentioning

confidence: 96%

Section: Introductionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

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Large automorphism groups of ordinary curves in characteristic 2

Montanucci

Speziali

2019

Journal of Algebra

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“…Clearly from Henn's result f (g) ≤ 8g 3 . Also f (g) cannot be asymptotically of order less than g 3/2 as algebraic curves of positive p-rank with approximately g 3/2 automorphisms are known; see for example [10].…”

Section: Introductionmentioning

confidence: 99%

On algebraic curves with many automorphisms in characteristic p

Montanucci¹

2020

Preprint

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Let X be an irreducible, non-singular, algebraic curve defined over a field of odd characteristic p. Let g and γ be the genus and p-rank of X , respectively. The influence of g and γ on the automorphism group Aut(X ) of X is well-known in the literature. If g ≥ 2 then Aut(X ) is a finite group, and unless X is the so-called Hermitian curve, its order is upper bounded by a polynomial in g of degree four (Stichtenoth). In 1978 Henn proposed a refinement of Stichtenoth's bound of cube order in g up to few exceptions, all having p-rank zero. In this paper a further refinement of Henn's result is proposed. First, we prove that if an algebraic curve of genus g ≥ 2 has more than 336g 2 automorphisms then its automorphism group has exactly two short orbits, one tame and one non-tame. Then we show that if |Aut(X )| ≥ 900g 2 , the quotient curve X /Aut(X )(1) P where P is contained in the non-tame short orbit is rational, and the stabilizer of 2 points is either a p-group or a prime-to-p group, then the p-rank of X is equal to zero.

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On algebraic curves with many automorphisms in characteristic p

Montanucci

2022

Math. Z.

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Transcendence degree one function fields over a finite field with many automorphisms

Cited by 5 publications

References 21 publications

Large automorphism groups of ordinary curves in characteristic 2

Large automorphism groups of ordinary curves in characteristic 2

On algebraic curves with many automorphisms in characteristic p

On algebraic curves with many automorphisms in characteristic p

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