Exceptional Automorphisms of (Generalized) Super Elliptic Surfaces

Sean A. Broughton, Rose-Hulman Institute of Technology
Aaron Wootton, University of Portland


A super-elliptic surface is a compact, smooth Riemann surface S with a conformal automorphism w of prime order p. Let C be the group generated by w, we also require that S/C 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/C has genus zero. All cyclic n-gonal surfaces have tractable defining equations. Let A = Aut(S) and N be the normalizer of C in Aut(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 h ∈ A−N is called exceptional. The exceptional automorphisms of super-elliptic surfaces are known whereas the determination of 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.