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

Kodai Mathematical Journal

Ewa Tyszkowska¹  Ewa Kozłowska-Walania¹ 

[1] Institute of Mathematics Faculty of Mathematics Physics and Informatics University of Gdańsk

关键词: Riemann surface;    symmetry of Riemann surface;    asymmetric Riemann surface;    pseudo-symmetric Riemann surface;    Fuchsian groups;    NEC groups;   

DOI  :  10.2996/kmj/1478073768

学科分类：数学（综合）

来源: Tokyo Institute of Technology, Department of Mathematics

【 摘 要 】

References(18)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 2sn 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 = Z2sn, and we study (q,n)-gonal automorphisms and p-hyperelliptic involutions in G.

【 授权许可】

Unknown