On generalizations of Fermat curves over finite fields and their automorphisms

Arakelian, Nazar; Speziali, Pietro

doi:10.1080/00927872.2017.1287271

Cited by 5 publications

(7 citation statements)

References 8 publications

(14 reference statements)

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“…Remark 4.12. From the proof of 4.11 (2), it is easily seen that when p = 2, a tame (g + 1)curve is hyperelliptic if, and only if, it is birationally equivalent to a generalized Fermat curve, see [2]. Although results on such curves where stated (and proved) over an algebraic closure of a finite field, all the results regarding the automorphism groups of generalized Fermat curves are still valid over any algebraically closed field.…”

Section: Classificationmentioning

confidence: 99%

“…Although results on such curves where stated (and proved) over an algebraic closure of a finite field, all the results regarding the automorphism groups of generalized Fermat curves are still valid over any algebraically closed field. Thence, the full automorphism group of a tame, hyperelliptic (g + 1)-curve can be found in [2,Theorem 6.11]. Proposition 4.13.…”

Section: Classificationmentioning

confidence: 99%

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Algebraic curves with automorphism groups of large prime order

Arakelian¹,

Speziali²

2020

Preprint

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Let X be an algebraic curve of genus g defined over an algebraically closed field K of characteristic p ≥ 0, and q a prime dividing |Aut(X )|. We say that X is a q-curve. Homma proved that either q ≤ g + 1 or q = 2g + 1, and classified (2g + 1)-curves. In this note, we classify (g + 1)-curves, and fully characterize the automorphism groups of q-curves for q = 2g + 1, g + 1. We also give some partial results on q-curves for q = g, g − 1.

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Section: Classificationmentioning

confidence: 99%

Section: Classificationmentioning

confidence: 99%

Algebraic curves with automorphism groups of large prime order

Arakelian¹,

Speziali²

2020

Preprint

Self Cite

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“…This provides an example of a generalized Fermat curve, having recently been studied from the point of view of its automorphism group [3]. The number N q (G) of F q -rational points on the nonsingular model Y of G was first investigated in the context of Finite Geometry to study the number of chords of an affinely regular polygon in A 2 (F q ) passing through a given point (see [1], [7]).…”

Section: Introductionmentioning

confidence: 99%

“…This provides an example of a generalized Fermat curve, having recently been studied from the point of view of its automorphism group [3].…”

Section: Introductionmentioning

confidence: 99%

On some generalized Fermat curves and chords of an affinely regular polygon inscribed in a hyperbola

Borges

Coutinho

2020

Journal of Pure and Applied Algebra

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Let G be the projective plane curve defined over F q given bywhere ab / ∈ {0, 1}, and for each s ∈ {2, . . . , n − 1}, let D P1,P2 s be the base-point-free linear series cut out on G by the linear system of all curves of degree s passing through the singular points P 1 = (1 : 0 : 0) and P 2 = (0 : 1 : 0) of G. The present work determines an upper bound for the number N q (G) of F q -rational points on the nonsingular model of G in cases where D P1,P2 s is F q -Frobenius classical. As a consequence, when F q is a prime field, the bound obtained for N q (G) improves in several cases the known bounds for the number n P of chords of an affinely regular polygon inscribed in a hyperbola passing through a given point P distinct from its vertices. (2010): 11G20, 14G05, 51E15. * Mathematics Subject Classifications

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“…This provides an example of a generalized Fermat curve, having been studied from the point of view of its automorphism group in (ARAKELIAN; SPEZIALI, 2017).…”

Section: Introductionmentioning

confidence: 99%

Three topics in algebraic curves over finite fields

Coutinho¹

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Ao professor Herivelto, pelo imenso apoio, paciência e atenção ao longo dos últimos quatro anos; pelo exemplo de pessoa, professor e orientador; pelo entusiasmo que sempre colocou em cada explicação e questão a ser analisada; mas, especialmente, por ter me mostrado a importância de se fazer perguntas e de não se ter medo de errar.A cada um dos professores do ICMC com os quais tive a oportunidade conviver. Em especial, agradeço ao professor Sérgio Monari, pelo acolhimento em todos esses anos, e ao professor Daniel Levcovitz, por ter me aceitado como aluna tantas vezes e pela enorme ajuda.Aos professores Abramo Hefez, Daniel Levcovitz (novamente) e Fernando Torres, pela participação na banca que examinou esse trabalho, bem como por cada uma das sugestões e comentários que contribuíram para a melhoria desse texto.À equipe da cantina do ICMC, pela alegria com que sempre me recebeu.

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On generalizations of Fermat curves over finite fields and their automorphisms

Cited by 5 publications

References 8 publications

Algebraic curves with automorphism groups of large prime order

Algebraic curves with automorphism groups of large prime order

On some generalized Fermat curves and chords of an affinely regular polygon inscribed in a hyperbola

Three topics in algebraic curves over finite fields

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