Journal Title

Contemporary Mathematics

Publication Date



A super-elliptic surface is a compact, smooth Riemann surface S with a conformal automorphism w of prime order p such that S/ has genus zero, extending the hyper-elliptic case p=2. More generally, a cyclic n-gonal surface S has an automorphism w of order n such that S/ has genus zero. All cyclic n-gonal surfaces have tractable defining equations. Let A = Aut(S) and N be the normalizer of C = in A. The structure of N, in principal, can be easily determined from the defining equation. If the genus of S is sufficiently large in comparison to n, and C satisfies a generalized super-elliptic condition, then A = N. For small genus A - N may be non-empty and, in this case, any automorphism hA - N is called exceptional. The exceptional automorphisms of all general cyclic n-gonal surfaces seems to be hard. We focus on generalized super-elliptic surfaces in which n is composite and the projection of S onto S/C is fully ramified. Generalized super-elliptic surfaces are easily identified by their defining equations. In this paper we discuss an approach to the determination of generalized super-elliptic surfaces with exceptional automorphisms.


Riemann surfaces

Publication Information

Contemporary Mathematics: Riemann and Klein Surfaces, Automorphisms and Moduli Spaces, Symmetries 2014, Volume 629, 29-42.

© 2014 American Mathematical Society

Archived version is the final published version.





Document Type

Journal Article

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Mathematics Commons