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Smooth quotients of generalized Fermat curves

Abstract

A closed Riemann surface S is called a generalized Fermat curve of type (pn), where \(n,p \ge 2\) are integers such that \((p-1)(n-1)>2\), if it admits a group \(H \cong {\mathbb Z}_{p}^{n}\) of conformal automorphisms with quotient orbifold S/H of genus zero with exactly \(n+1\) cone points, each one of order p; in this case H is called a generalized Fermat group of type (pn). In this case, it is known that S is non-hyperelliptic and that H is its unique generalized Fermat group of type (pn). Also, explicit equations for them, as a fiber product of classical Fermat curves of degree p, are known. For p a prime integer, we describe those subgroups K of H acting freely on S, together with algebraic equations for S/K, and determine those K such that S/K is hyperelliptic.

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Acknowledgements

The author would like to thank both referees for their valuable comments, suggestions and corrections which helped to improve the paper.

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Correspondence to Rubén A. Hidalgo.

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Partially supported by projects Fondecyt 1190001 and 1220261.

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Hidalgo, R.A. Smooth quotients of generalized Fermat curves. Rev Mat Complut (2022). https://doi.org/10.1007/s13163-022-00422-5

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  • DOI: https://doi.org/10.1007/s13163-022-00422-5

Keywords

  • Riemann surfaces
  • Algebraic curves
  • Automorphisms
  • Moduli spaces

Mathematics Subject Classification

  • 30F10
  • 14H37
  • 30F40
  • 30F60