Schottky uniformizations and riemann matrices of maximal symmetric Riemann surfaces of genus 5


  • Rubén A. Hidalgo Universidad Técnica Federico Santa María.



Schottky groups, Riemann surfaces, Riemann matrices.


In this note we consider pairs (S, ? ), where S is a closed Riemann surface of genus five and ? : S ? S is some anticonformal involution with fixed points so that K(S, ? ) = {h ? Aut±(S) : h? = ?h} has the maximal order 96 and S/? is orientable. We observe that there are exactly two topologically different choices for ? . They give non-isomorphic groups K(S, ? ), each one acting topologically rigid on the respective surface S. These two cases give then two (connect) real algebraic sets of real dimension one in the moduli space of genus 5. In this note we describe these components by classical Schottky groups and with the help of these uniformizations we compute their Riemann matrices.

Author Biography

Rubén A. Hidalgo, Universidad Técnica Federico Santa María.

Departamento de Matemática.


[1] Bers, L.: Automorphic forms for Schottky groups. Adv. in Math. 16 (1975), 332-361.

[2] Burnside, W.: On a class of Automorphic Functions. Proc. London Math. Soc. Vol 23 (1892), 49-88.

[3] Conway, J. H. and Sloan, N. J. A.: it Sphere Packings, Lattices and Groups. Springer-Verlag, 1988.

[4] Costa, A. F. and Hidalgo, R. A.: Anticonformal automorphisms and Schottky coverings. Preprint.

[5] Chuckrow, V.: On Schottky groups with application to Kleinian groups. Ann. of Math. 88 (1968), 47-61.

[6] González-Díez, G.: Loci of curves which are prime Galois coverings of P1. Proc. London Math. Soc. 62 (1991), 469-489.

[7] Heltai, B.: Symmetric Riemann surfaces, torsion subgroups and Schottky coverings. Proc. Amer. Math. Soc. 100 (1987), 675-682.

[8] Hidalgo, R. A.: On Schottky groups with automorphisms. Ann. Acad. Scie. Fenn. Ser. AI Mathematica 19 (1994), 259-289.

[9] Hidalgo, R. A.: Schottky uniformizations of closed Riemann surfaces with Abelian groups of conformal automorphisms. Glasgow Math. J. 36 (1994), 17-32.

[10] Hidalgo, R. A.: Dihedral groups are of Schottky type. Revista Proyecciones. 18 (1999), 23-48.

[11] Hidalgo, R. A.: A4, A5 and S4 of Schottky type. Preprint. [12] Hidalgo, R. A.: Bounds for Conformal Automorphisms of Riemann Surfaces with Condition (A). Preprint.

[13] Keen, L.: On hyperelliptic Schottky groups. Ann. Acad. Sci. Fenn. Series A.I. Mathematica 5 (1980).

[14] Marden, A.:Schottky groups and circles. Contribution to Analysis, A collection of papers Dedicated to Lipman Bers. (1994), 273-278.

[15] Maskit, B.: Kleinian Groups. G.M.W. 287, Springer-Verlag, 1988.

[16] Maskit, B.: Special uniformizations of symmetric Riemann surfaces. Preprint

[17] Maskit, B.: A characterization of Schottky groups. J. d’Analyse Math. 19 (1967), 227-230.

[18] May, C. L.: Automorphisms of compact Klein surfaces with boundary. Pacific J. Math. 59 (1975), 199-210.

[19] May, C. L.: A bound for the number of automorphisms of a compact Klein surface with boundary. Proc. Amer. Math. Soc. 63 (1977), 273-280.

[20] Mumford, D.: Tata lectures on Theta II. Progress on Mathematics 43, Birkha¨user, Boston, 1984.

[21] Natanzon, S. M.: Differential equations for Prym theta-functions, a criterion for two-dimensional finite zone potential Schr¨odinger operators to be real. Funktsional Anal. i Prilozhen 26 (1992), 17-26.

[22] Quine, J. R. and Zhang, P.: Extremal symplectic lattices. Preprint.

[23] Schmutz, P.: Riemann surfaces with shortest geodesic of maximal lenght. Gemetric and Functional Analysis, 3 (1993), 564-631.

[24] Sibner, R. J.: Uniformization of Symmetric Riemann surfaces by Schottky groups. Trans. Amer. Math. Soc. 116 (1965), 79-85.



How to Cite

R. A. Hidalgo, “Schottky uniformizations and riemann matrices of maximal symmetric Riemann surfaces of genus 5”, Proyecciones (Antofagasta, On line), vol. 20, no. 1, pp. 93-126, Apr. 2017.