# Exceptional Automorphisms of (Generalized) Super Elliptic Surfaces

#### Abstract

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.

*This paper has been withdrawn.*