#### Journal Title

Contemporary Mathematics

#### Publication Date

2014

#### Abstract

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 *h* ∈ *A - 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.

#### Subjects

Riemann surfaces

#### Citation: Pilot Scholars Version (Modified MLA Style)

Broughton, S. Allen and Wootton, Aaron, "Exceptional automorphisms of (generalized) super elliptic surfaces" (2014). *Mathematics Faculty Publications and Presentations*. 11.

https://pilotscholars.up.edu/mth_facpubs/11

#### DOI

10.1090/conm/629

#### Peer-Reviewed

Yes

#### Document Type

Journal Article

## 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.