Symmetry types of hyperelliptic Riemann surfaces

Bujalance, E.; Cirre, Francisco-Javier; Gamboa, J. M.; Gromadzki, Grzegorz

doi:10.24033/msmf.399

Cited by 23 publications

(38 citation statements)

References 35 publications

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“…However x 2 g(gz) = gx 3 gx 2 (gz) and so there exists one more point z 4 ∈ F such that g(x 3 gz) = z 4 . Thus g(z 4 ) = x 2 gz and σ g = (1,5,2,8,4,7,3,6). So for C 8 = (4; 4, 0, 0),G has the presentation 8.5.…”

Section: Theoremmentioning

confidence: 99%

“…Hyperelliptic Riemann surfaces and their automorphisms have received a good deal of attention in the literature. In [1] and [10] the authors determined the full groups of conformal automorphisms of such surfaces which made possible to classify symmetry types of such actions in [3]. The p-hyperelliptic (p ≥ 1) surfaces at large have been studied in [4][5][6][7][8][9][13][14][15] and [24], where the most attention has been paid to a study of groups of automorphisms of such surfaces and their symmetries.…”

Section: Introductionmentioning

confidence: 99%

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On gonality automorphisms of p-hyperelliptic Riemann surfaces

Tyszkowska

2010

Rev. R. Acad. Cien. Serie A. Mat.

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A compact Riemann surface X of genus g > 1 is said to be a p-hyperelliptic if X admits a conformal involution ρ for which X/ρ has genus p. This notion is the particular case of so called cyclic (q, n)-gonal surface which is defined as the one admitting a conformal automorphism δ of order n such that X/δ has genus q. It is known that for g > 4p + 1, ρ is unique and so central in the automorphism group of X. We give necessary and sufficient conditions on p and g for the existence of a Riemann surface of genus g admitting commuting p-hyperelliptic involution ρ and (q, n)-gonal automorphism δ for some prime n and we study its group of automorphisms and the number of fixed points of δ. Furthermore, we deal with automorphism groups of Riemann surfaces admitting central automorphism with at most 8 fixed points. The condition on the small number of fixed points of such an automorphism is justified by the study of p-hyperelliptic surfaces. Sobre automorfismos de gonalidad de superficies de Riemann p-hiperelípticasResumen. Una superficie de Riemann compacta X de género g > 1 se dice p-hiperelíptica si X admite una involución conforme ρ, tal que X/ρ tiene género p. Las superficies p-hiperelípticas son un caso particular de las superficies (q, n)-gonales cíclicas que se definen como aquellas superficies que admiten un automorfismo conforme δ de orden q y de modo que X/δ tiene género q. En este trabajo nos restringiremos al caso en que q es un número primo mayor que 2. Es un hecho conocido que si g > 4p+1, la involución ρ esúnica y central en el grupo de automorfismos de X. Obtenemos condiciones necesarias y suficientes sobre p y g para la existencia de superficies de Riemann de género g que admiten una involución p-hiperelíptica y un automorfismo (q, n)-gonal que conmutan. Se determina la presentación de un cociente de los grupos de automorfismos de las superficies de Riemann que admiten un automorfismo (q, n)-gonal que sea central y con 8 puntos fijos como máximo. Esta restricción sobre el número de puntos fijos se justifica por el estudio anterior de las superfices que son a la vez p-hiperelípticas y (q, n)-gonales cíclicas.

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Section: Theoremmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On gonality automorphisms of p-hyperelliptic Riemann surfaces

Tyszkowska

2010

Rev. R. Acad. Cien. Serie A. Mat.

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“…(1) The curve S : w 2 = z(z 2g + 1), which corresponds to λ = e πi/2g , has four real forms, but only τ 1 , which has species −g, and τ 2 , which has species 0, have allowable species [7,Theorem 3.4.7(e)]. In addition, their centralizer in Aut S is precisely the group generated by u and v. Hence Aut(S, τ i ) = D 2g for i = 1, 2; however, the equality Aut S = D 2g is no longer true since this curve satisfies | Aut S| = 8g.…”

Section: Remarks 33mentioning

confidence: 99%

“…If |λ| = 1, then S λ also has four real forms, and again τ 1 and τ 1 ρ, both with species −g, are the unique ones which commute with u and v [7, Theorem 3.4.7(d)]. Finally, if arg(λ) = π/(2g), then S λ admits four real forms, but only τ 2 and τ 2 ρ have allowable species, see [7,Theorem 3.3.2], and commute with u and v. 2…”

Section: Case G Oddmentioning

confidence: 99%

Riemann surfaces with real forms which have maximal cyclic symmetry

2005

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We determine all the Klein surfaces which have a cyclic automorphism group of the maximum possible order, and find their topological types. We also compute their full automorphism groups and show that, except for a finite number of exceptions, they coincide with the full automorphism groups of their Riemann double covers. Explicit algebraic equations of the surfaces and the formulae of their real forms and automorphisms are also given.  2004 Elsevier Inc. All rights reserved.

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“…Bujalance et al (5) have calculated the groups of automorphisms of hyperelliptic Riemann surfaces and more recently Bujalance, Cirre and Gromadzki list the automorphisms groups of cyclic trigonal Riemann surfaces (see (6)). Recently Sanjeewa (18) has obtained the automorphisms groups of cyclic ngonal algebraic curves over fields of any characteristics.…”

Section: Introductionmentioning

confidence: 99%