Abstract

HIDALGO, RUBEN A.; REYES-CAROCCA, SEBASTIÁN  and  VALDES, MARÍA ELISA. Field of Moduli and Generalized Fermat Curves. Rev.colomb.mat. [online]. 2013, vol.47, n.2, pp.205-221. ISSN 0034-7426.

A generalized Fermat curve of type (p,n) is a closed Riemann surface S admitting a group H \cong Z_pⁿ of conformal automorphisms with S/H being the Riemann sphere with exactly n+1 cone points, each one of order p. If (p-1)(n-1) ≥ 3, then S is known to be non-hyperelliptic and generically not quasiplatonic. Let us denote by \operatornameAut_H(S) the normalizer of H in \operatornameAut(S). If p is a prime, and either (i) n=4 or (ii) n is even and \operatornameAut_H(S)/H is not a non-trivial cyclic group or (iii) n is odd and \operatornameAut_H(S)/H is not a cyclic group, then we prove that S can be defined over its field of moduli. Moreover, if n ∈ {3,4}, then we also compute the field of moduli of S.

Keywords : Algebraic curves; Riemann surfaces; Field of moduli; Field of definition.

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