Publicacions Matemàtiques

On $p$-hyperellipticity of doubly symmetric Riemann surfaces

Ewa Kozłowska-Walania

Full-text: Open access


Studying commuting symmetries of $p$-hyperelliptic Riemann surfaces, Bujalance and Costa found in "On symmetries of $p$-hyperelliptic Riemann surfaces" (E. Bujalalance, A.F. Costa, Math. Ann. 308(1) (1997), 31–45) upper bounds for the degree of hyperellipticity of the product of commuting $(M-q)$- and $(M-q')$-symmetries, depending on their separabilities. Here, we find necessary and sufficient conditions for an integer $p$ to be the degree of hyperellipticity of the product of two such symmetries, taking into account their separabilities. We also give some results concerning the existence and uniqueness of symmetries from which we obtain a series of important results of Natanzon concerning $M$- and $(M-1)$-symmetries.

Article information

Publ. Mat., Volume 51, Number 2 (2007), 291-307.

First available in Project Euclid: 31 July 2007

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 30F
Secondary: 14H

Riemann surface symmetry of Riemann surface oval of a symmetry of a Riemann surface


Kozłowska-Walania, Ewa. On $p$-hyperellipticity of doubly symmetric Riemann surfaces. Publ. Mat. 51 (2007), no. 2, 291--307.

Export citation


  • E. Bujalance, A. F. Costa and D. Singerman, Application of Hoare's theorem to symmetries of Riemann surfaces, Ann. Acad. Sci. Fenn. Ser. A I Math. 18(2) (1993), 307\Ndash322.
  • E. Bujalance and A. F. Costa, A combinatorial approach to the symmetries of $M$ and $M-1$ Riemann surfaces, in: “Discrete groups and geometry” (Birmingham, 1991), London Math. Soc. Lecture Note Ser. 173, Cambridge Univ. Press, Cambridge, 1992, pp. 16\Ndash25.
  • E. Bujalance and A. F. Costa, On symmetries of $p$-hyperelliptic Riemann surfaces, Math. Ann. 308(1) (1997), 31\Ndash45.
  • E. Bujalance, J. J. Etayo, J. M. Gamboa and G. Gromadzki, “Automorphism groups of compact bordered Klein surfaces. A combinatorial approach”, Lecture Notes in Mathematics 1439, Springer-Verlag, Berlin, 1990.
  • H. M. Farkas and I. Kra, “Riemann surfaces”, Graduate Texts in Mathematics 71, Springer-Verlag, New York-Berlin, 1980.
  • S. M. Natanzon, Finite groups of homeomorphisms of surfaces, and real forms of complex algebraic curves, (Russian), Trudy Moskov. Mat. Obshch. 51 (1988), 3\Ndash53, 258; translation in: Trans. Moscow Math. Soc. (1989), 1\Ndash51.
  • D. Singerman, On the structure of non-Euclidean crystallographic groups, Proc. Cambridge Philos. Soc. 76 (1974), 233\Ndash240.
  • E. Tyszkowska, On $pq$-hyperelliptic Riemann surfaces, Colloq. Math. 103(1) (2005), 115\Ndash120.