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October 2016 On a certain family of asymmetric Riemann surfaces with the cyclic automorphism group
Ewa Kozłowska-Walania, Ewa Tyszkowska
Kodai Math. J. 39(3): 510-520 (October 2016). DOI: 10.2996/kmj/1478073768


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.


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Ewa Kozłowska-Walania. Ewa Tyszkowska. "On a certain family of asymmetric Riemann surfaces with the cyclic automorphism group." Kodai Math. J. 39 (3) 510 - 520, October 2016.


Published: October 2016
First available in Project Euclid: 2 November 2016

zbMATH: 1345.30058
MathSciNet: MR3567229
Digital Object Identifier: 10.2996/kmj/1478073768

Rights: Copyright © 2016 Tokyo Institute of Technology, Department of Mathematics


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Vol.39 • No. 3 • October 2016
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