Symmetry types of hyperelliptic Riemann surfaces

Symmetry types of hyperelliptic Riemann surfaces
[Types de symétrie des surfaces de Riemann hyperelliptiques]

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

Mémoires de la Société Mathématique de France, no. 86 (2001) , 128 p.

Résumé
Abstract

Une surface de Riemann compacte $X$ est dite symétrique si elle admet une involution antiholomorphe $τ : X \to X$ . On appelle structure réelle une telle involution. Deux structures réelles sont isomorphes si elles sont conjuguées par le groupe complet ${Aut}^{\pm} X$ des automorphismes holomorphes et anti-holomorphes de $X$ . Dans ce mémoire, nous classifions à isomorphisme près les structures réelles de toutes les surfaces de Riemann hyperelliptiques de genre $g \geq 2$ . Nous calculons aussi les invariants topologiques de chaque classe d’isomorphisme. Nous donnons la liste des groupes qui agissent comme le groupe des automorphismes holomorphes et anti-holomorphes d’une telle surface. De plus, nous décrivons la courbe algébrique complexe associée à une telle surface en terme d’équations polynomiales. Nous donnons enfin une formule explicite pour une structure réelle dans chaque classe d’isomorphisme.

A compact Riemann surface $X$ is symmetric if it admits an antianalytic involution $τ : X \to X$ . Such an involution is called a real structure. Two real structures are isomorphic if they are conjugate in the full group ${Aut}^{\pm} X$ of analytic and antianalytic automorphisms of $X$ . In this memoir we classify up to isomorphism the real structures of all symmetric hyperelliptic Riemann surfaces of genus $g \geq 2$ . The topological invariants of each isomorphism class are also computed. We give the list of groups which act as the full group of analytic and antianalytic automorphisms of such surfaces. Moreover, the complex algebraic curve associated to any such Riemann surface is described in terms of polynomial equations. We also find the explicit formula of a real structure in each isomorphism class.

MR Zbl

DOI : 10.24033/msmf.399

Classification : 14H, 30F, 20F, 20H
Keywords: Riemann surface, symmetry, automorphism group, real form, real algebraic curve
Mot clés : Surface de Riemann, symétrie, groupe d’automorphismes, forme réelle, courbe algébrique réelle

@book{MSMF_2001_2_86__1_0,
     author = {Bujalance, Emilio and Cirre, Francisco-Javier and Gamboa, J.-M. and Gromadzki, Grzegorz},
     title = {Symmetry types of hyperelliptic {Riemann} surfaces},
     series = {M\'emoires de la Soci\'et\'e Math\'ematique de France},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {86},
     year = {2001},
     doi = {10.24033/msmf.399},
     mrnumber = {1891804},
     zbl = {1078.14044},
     language = {en},
     url = {http://www.numdam.org/item/MSMF_2001_2_86__1_0/}
}

TY  - BOOK
AU  - Bujalance, Emilio
AU  - Cirre, Francisco-Javier
AU  - Gamboa, J.-M.
AU  - Gromadzki, Grzegorz
TI  - Symmetry types of hyperelliptic Riemann surfaces
T3  - Mémoires de la Société Mathématique de France
PY  - 2001
IS  - 86
PB  - Société mathématique de France
UR  - http://www.numdam.org/item/MSMF_2001_2_86__1_0/
DO  - 10.24033/msmf.399
LA  - en
ID  - MSMF_2001_2_86__1_0
ER  -

%0 Book
%A Bujalance, Emilio
%A Cirre, Francisco-Javier
%A Gamboa, J.-M.
%A Gromadzki, Grzegorz
%T Symmetry types of hyperelliptic Riemann surfaces
%S Mémoires de la Société Mathématique de France
%D 2001
%N 86
%I Société mathématique de France
%U http://www.numdam.org/item/MSMF_2001_2_86__1_0/
%R 10.24033/msmf.399
%G en
%F MSMF_2001_2_86__1_0

Bujalance, Emilio; Cirre, Francisco-Javier; Gamboa, J.-M.; Gromadzki, Grzegorz. Symmetry types of hyperelliptic Riemann surfaces. Mémoires de la Société Mathématique de France, Série 2, no. 86 (2001), 128 p. doi : 10.24033/msmf.399. http://numdam.org/item/MSMF_2001_2_86__1_0/

Bibliographie
Cité par

[1] N. L. Alling, Real Elliptic Curves, Mathematics Studies, 54, North-Holland, 1981. | MR | Zbl

[2] N. L. Alling and N. Greenleaf, Foundations of the Theory of Klein Surfaces, Lecture Notes in Math., 219, Springer, 1971. | MR | Zbl

[3] J. Bochnak, M. Coste and M. F. Roy, Géométrie Algébrique Réelle. Ergeb. der Math., 12, Springer-Verlag, Berlin, etc. 1987. | MR | Zbl

[4] R. Brandt and H. Stichtenoth, Die automorphismengruppen hyperelliptischer Kurven, Manuscripta Math, 55, 1986, pp. 83–92. | MR | EuDML | Zbl

[5] S. A. Broughton, E. Bujalance, A. F. Costa, J. M. Gamboa and G. Gromadzki, Symmetries of Riemann surfaces in which PSL $(2, q)$ acts as a Hurwitz automorphism group. J. Pure App. Alg. 106, 2, 1996, pp. 113–126. | Zbl

[6] S. A. Broughton, E. Bujalance, A. F. Costa, J. M. Gamboa and G. Gromadzki, Symmetries of Accola-Maclachlan and Kulkarni surfaces. Proc. Amer. Math. Soc., 127, 1999, pp. 637–646. | MR | Zbl

[7] E. Bujalance, Normal $N E C$ signatures. Illinois J. Math., 26, 1982, pp. 519–530. | MR | Zbl

[8] E. Bujalance, M. D. E. Conder, J. M. Gamboa, G. Gromadzki and M. Izquierdo, Double coverings of Klein surfaces by a given Riemann surface. To appear in J. Pure Appl. Alg. | MR | Zbl

[9] E. Bujalance and A. F. Costa, A combinatorial approach to symmetries of $M$ and $(M - 1)$ -Riemann surfaces, Lect. Notes Series of London Math. Soc., 173, 1992, pp. 16–25. | MR | Zbl

[10] E. Bujalance and A. F. Costa, Estudio de las simetrías de la superficie de Macbeath. Libro homenaje al Prof. Etayo, Ed. Complutense, 1994, pp. 375–386.

[11] E. Bujalance and A. F. Costa, On symmetries of $p$ -hyperelliptic Riemann surfaces. Math. Ann., 308, 1997, pp. 31–45. | MR | Zbl

[12] E. Bujalance, A. F. Costa and J. M. Gamboa, Real parts of complex algebraic curves. Lecture Notes in Math., 1420. Springer-Verlag, Berlin, 1990, pp. 81–110. | MR | Zbl

[13] E. Bujalance, A. F. Costa, S. M. Natanzon and D. Singerman, Involutions on compact Klein surfaces. Math. Z., 211, 1992, pp. 461–478. | MR | EuDML | Zbl

[14] E. Bujalance, J. J. Etayo, J. M. Gamboa and G. Gromadzki, Automorphism Groups of Compact Bordered Klein Surfaces. Lecture Notes in Math., 1439, Springer-Verlag, Berlin, 1990. | MR | Zbl

[15] E. Bujalance, J. M. Gamboa and G. Gromadzki, The full automorphism group of hyperelliptic Riemann surfaces. Manuscripta Math, 79, 1993, pp. 267–282. | MR | EuDML | Zbl

[16] E. Bujalance, G. Gromadzki and M. Izquierdo, On real forms of a complex algebraic curve. J. Austral. Math. Soc., 70, 2001, no 1, pp 134–142. | MR | Zbl

[17] E. Bujalance, G. Gromadzki and D. Singerman, On the number of real curves associated to a complex algebraic curve. Proc. Amer. Math. Soc., 120, 1994, pp. 507–513. | MR | Zbl

[18] E. Bujalance and D. Singerman, The symmetry type of a Riemann Surface. Proc. London Math. Soc., (3), 51, 1985, pp. 501–519. | MR | Zbl

[19] J. A. Bujalance, Hyperelliptic compact non-orientable Klein surfaces without boundary. Kodai Math. J., 12, 1989, pp. 1–8. | MR | Zbl

[20] P. Buser and R. Silhol, Geodesics, periods and equations of real hyperelliptic curves. Duke Math. J., 108, 2001, no 2, pp. 211–250. | MR | Zbl

[21] F. J. Cirre, Complex automorphism groups of real algebraic curves of genus $2$ . J. Pure Appl. Alg., 157, 2001, no 2–3, pp. 157–181. | MR | Zbl

[22] F. J. Cirre, On the birational classification of hyperelliptic real algebraic curves in terms of their equations. Submitted. | MR

[23] F. J. Cirre, On a family of hyperelliptic Riemann surfaces. Submitted.

[24] C. J. Earle, On the moduli of closed Riemann surfaces with symmetries. Advances in the theory of Riemann surfaces. L. V. Ahlfors et al. (eds), Ann. of Math. Studies, 66, pp. 119–130, Princeton University Press and University of Tokio Press, 1971.

[25] The GAP Group, GAP – Groups, Algorithms and Programming, Version 4b5, 1998. School of Mathematical and Computational Sciences, University of St Andrews, Scotland and Lehrstuhl D für Mathematik, Rheinisch Westfälische Technische Hochschule, Aachen, Germany.

[26] P. Gianni, M. Seppälä, R. Silhol and B. Trager, Riemann surfaces, plane algebraic curves and their periods matrices. J. Symbolic Computation, 12, 1998, pp. 789–803. | MR | Zbl

[27] L. Greenberg, Maximal Fuchsian groups. Bull. Amer. Math. Soc., 69, (1963), pp. 569–573. | MR | Zbl

[28] G. Gromadzki, On a Harnack-Natanzon theorem for the family of real forms of Riemann surfaces. J. Pure Appl. Alg., (3) 121, 1997, pp. 253–269. | MR | Zbl

[29] G. Gromadzki, On ovals on Riemann surfaces. Rev. Mat. Ibero-Americana, 16, 2000, no 3, pp. 515–527. | MR | EuDML | Zbl

[30] G. Gromadzki and M. Izquierdo, Real forms of a Riemann surface of even genus. Proc. Amer. Math. Soc., 126, (6), 1998, pp. 3475–3479. | MR | Zbl

[31] B. H. Gross and J. Harris, Real algebraic curves. Ann. Sci. École Norm. Sup., 14, 1981, pp. 157–182. | MR | EuDML | Zbl | Numdam

[32] A. Harnack, Über die Vieltheiligkeit der ebenen algebraischen Kurven. Math. Ann., 10, 1876, pp. 189–198. | MR | EuDML

[33] A. H. M. Hoare, Subgroups of NEC groups and finite permutation groups. Quart. J. Math. Oxford, (2), 41, 1990, pp. 45–59. | MR | Zbl

[34] M. Izquierdo and D. Singerman, Pairs of symmetries of Riemann surfaces. Ann. Acad. Sci. Fenn. Math., (1), 23, 1998, pp. 3–24. | MR | EuDML | Zbl

[35] F. Klein, Über Realitätsverhältnisse bei der einem beliebigen Geschlechte zugehörigen Normalkurve der $φ$ . Math. Ann., 42, 1893, pp. 1–29. | MR | EuDML

[36] A. M. Macbeath, The classification of non-euclidean plane crystallographic groups. Canad. J. Math., 19, 1967, pp. 1192–1205. | MR | Zbl

[37] C. Maclachlan, Smooth coverings of hyperellyptic surfaces, Quart. J. Math. Oxford, (2), 22, 1971, pp. 117–123. | MR | Zbl

[38] C. L. May, Large automorphism groups of compact Klein surfaces with boundary I. Glasgow Math. J., 18, 1977, pp. 1–10. | MR | Zbl

[39] A. Melekoğlu, Symmetries of Riemann Surfaces and Regular Maps. Doctoral thesis, Faculty of Mathematical Studies, University of Southampton, (1998). | MR

[40] R. Miranda, Algebraic Curves and Riemann Surfaces, Graduate Studies in Math., Vol. 5, A. M. S., 1995. | MR | Zbl

[41] S. M. Natanzon, Lobachevskian geometry and automorphisms of complex $M$ -curves. Geometric Methods in Problems of Analysis and Algebra, Yaroslav, Gos. Univ., Yaroslavl’ 1978, pp. 130–151; English transl. Selecta Math. Sovietica 1, 1981, pp. 81–99. | MR | Zbl

[42] S. M. Natanzon, The number and topological types of real hyperelliptic curves isomorphic over $ℂ$ . Constructive Algebraic Geometry (Z. A. Skopets, editor), Sb. Nauchn. Trudov Yaroslav. Gos. Ped. Inst. Vyp. 200, 1982, pp. 82–93, (Russian).

[43] S. M. Natanzon, Finite groups of homeomorphisms of a surface and real forms of complex algebraic curves. Trans. Moscow Math. Soc., 51, 1989, pp. 1–51. | MR | Zbl

[44] S. M. Natanzon, On the order of a finite group of homeomorphisms of a surface into itself and the number of real forms of a complex algebraic curve. Dokl. Akad. Nauk SSSR, 242, 1978, pp. 765–768. English transl. in Soviet Math. Dokl. (5), 19, 1978, pp. 1195–1199. | MR | Zbl

[45] S. M. Natanzon, On the total number of ovals of real forms of complex algebraic curves. Uspekhi Mat. Nauk, (1), 35, 1980, pp. 207–208. (Russian Math. Surveys (1), 35, 1980, pp. 223–224. | MR | Zbl

[46] R. Preston, Projective structures and fundamental domains on compact Klein surfaces. Ph. D. Thesis, Univ. of Texas, 1975.

[47] D. Singerman, Finitely maximal Fuchsian groups. J. London Math. Soc., (2), 6, 1972, pp. 29-38. | MR | Zbl

[48] D. Singerman, Symmetries of Riemann surfaces with large automorphism group. Math. Ann., 210, 1974, pp. 17–32. | MR | EuDML | Zbl

[49] D. Singerman, Symmetries and pseudo-symmetries of hyperelliptic surfaces. Glasgow Math. J., 21, 1980, pp. 39–49. | MR | Zbl

[50] D. Singerman, Mirrors on Riemann surfaces. Contemporary Mathematics, 184, 1995, pp. 411–417. | MR | Zbl

[51] P. Turbek, The full automorphism group of the Kulkarni surface. Rev. Mat. Univ. Compl. Madrid, (2), 10, 1997, pp. 265–276. | MR | EuDML | Zbl

[52] G. Weichold, Über symmetrische Riemannsche Flächen und die Periodizitätsmodulen der zugerhörigen Abelschen Normalintegrale erstes Gattung, Dissertation, Leipzig, 1883.

[53] H. C. Wilkie, On non-euclidean crystallographic groups. Math. Z., 91, 1966, pp. 87–102. | MR | EuDML | Zbl

Cité par Sources :