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Exceptional automorphisms of (generalized) super elliptic surfaces
Contemporary Mathematics (2014)
  • Sean A. Broughton
  • Aaron Wootton
A super-elliptic surface is a compact, smooth Riemann surface S
with a conformal automorphism w of prime order p such that S/<w> 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/<w> has genus zero.
All cyclic n−gonal surfaces have tractable defining equations. Let A = Aut(S)
and N be the normalizer of C = <w> 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 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.
  • Riemann surface,
  • Automorphisms,
  • n-gonal,
  • super elliptic
Publication Date
Citation Information
Sean A. Broughton and Aaron Wootton. "Exceptional automorphisms of (generalized) super elliptic surfaces" Contemporary Mathematics Vol. 629 (2014) p. 29 - 42 ISSN: 0271-4132
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