Abstract
A closed Riemann surface S is called a generalized Fermat curve of type (p, n), 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 (p, n). In this case, it is known that S is non-hyperelliptic and that H is its unique generalized Fermat group of type (p, n). 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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Hidalgo, R.A. Smooth quotients of generalized Fermat curves. Rev Mat Complut 36, 27–55 (2023). https://doi.org/10.1007/s13163-022-00422-5
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DOI: https://doi.org/10.1007/s13163-022-00422-5