## Kodai Mathematical Journal

- Kodai Math. J.
- Volume 39, Number 3 (2016), 510-520.

### On a certain family of asymmetric Riemann surfaces with the cyclic automorphism group

Ewa Kozłowska-Walania and Ewa Tyszkowska

#### Abstract

A compact Riemann surface *X* of genus *g* ≥ 2 is called *asymmetric* or *pseudo-real* if it admits an anticonformal automorphism but no anticonformal involution. The order *d* = #(δ) of an anticonformal automorphism δ of such a surface is divisible by 4. In the particular case where *d* = 4, δ is a pseudo-symmetry and the surface is called *pseudo-symmetric*.

A Riemann surface *X* is said to be *p-hyperelliptic* if it admits a conformal involution ρ for which the orbit space *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 φ of prime order *n* such that *X*/φ has genus *q*. We are interested in possible values of *n* and *q* for which an asymmetric surface of given genus *g* ≥ 2 is (*q*,*n*)-gonal, and possible values of *p* for which the surface is *p*-hyperelliptic. Up till now, this problem was solved in the case where the surface is asymmetric and pseudo-symmetric. If an asymmetric Riemann surface *X* is not pseudo-symmetric then any anticonformal automorphism of *X* has order divisible by 2^{s}*n* for *s* ≥ 3 and *n* = 1 or *n* being an odd prime. In this paper we give the necessary and sufficient conditions on the existence of an asymmetric Riemann surface with the full automorphism group being *G* = *Z*_{2sn}, and we study (*q*,*n*)-gonal automorphisms and *p*-hyperelliptic involutions in *G*.

#### Article information

**Source**

Kodai Math. J., Volume 39, Number 3 (2016), 510-520.

**Dates**

First available in Project Euclid: 2 November 2016

**Permanent link to this document**

https://projecteuclid.org/euclid.kmj/1478073768

**Digital Object Identifier**

doi:10.2996/kmj/1478073768

**Mathematical Reviews number (MathSciNet)**

MR3567229

**Zentralblatt MATH identifier**

1345.30058

#### Citation

Kozłowska-Walania, Ewa; Tyszkowska, Ewa. On a certain family of asymmetric Riemann surfaces with the cyclic automorphism group. Kodai Math. J. 39 (2016), no. 3, 510--520. doi:10.2996/kmj/1478073768. https://projecteuclid.org/euclid.kmj/1478073768