Cuerpo de moduli y curvas de Fermat generalizadas

Field of Moduli and Generalized Fermat Curves

Una curva de Fermat generalizada de tipo (p,n) es una superficie de Riemann cerrada S la cual admite un grupo H \cong Z_pⁿ de automorfismos conformales de manera que S/H sea de género cero y tenga exactamente n+1 puntos cónicos, cada uno de orden p. Si (p-1)(n-1) ≥ 3, entonces se sabe que S no es hiperelíptica y genéricamente no es casiplatónica. Denotemos porAut_H(S) el normalizador de H en Aut(S). Si p es primo y tenemos que (i) n=4 o bien (ii) n es par y Aut_H(S)/H no es un grupo cíclico no trivial o bien (iii) n es impar y Aut_H(S)/H no es un grupo cíclico, entonces verificamos que S se puede definir sobre su cuerpo de moduli. Más aún, si n ε{3,4}, entonces determinamos tal cuerpo de moduli.

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+1cone 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 Aut_H(S) the normalizer of H in Aut(S). If p is a prime, and either (i) n=4 or (ii) n is even and Aut_H(S)/H is not a non-trivial cyclic group or (iii) nis odd and Aut_H(S)/H is not a cyclic group, then we prove that S can be defined over its field of moduli. Moreover, if n ε {3,4}, then we also compute the field of moduli of S.

Field of Moduli and Generalized Fermat Curves Cuerpo de moduli y curvas de Fermat generalizadas RUBEN A. HIDALGO¹, SEBASTIÁN REYES-CAROCCA², MARÍA ELISA VALDÉS³

¹Universidad Técnica Federico Santa María, Valparaíso, Chile. Email: ruben.hidalgo@usm.cl
²Universidad Autónoma de Madrid, Madrid, España. Email: sebastian.reyes@uam.es
³Universidad de Concepción, Concepción, Chile. Email: mariaevaldes@udec.cl

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Abstract

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 Aut_H(S) the normalizer of H in Aut(S). If p is a prime, and either (i) n=4 or (ii) n is even and Aut_H(S)/H is not a non-trivial cyclic group or (iii) n is odd and Aut_H(S)/H is not a cyclic group, then we prove that S can be defined over its field of moduli. Moreover, if n ε {3,4}, then we also compute the field of moduli of S.

Key words: Algebraic curves, Riemann surfaces, Field of moduli, Field of definition.

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2000 Mathematics Subject Classification: 14H37, 14H10, 14H45, 30F10.

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Resumen

Una curva de Fermat generalizada de tipo (p,n) es una superficie de Riemann cerrada S la cual admite un grupo H \cong Z_pⁿ de automorfismos conformales de manera que S/H sea de género cero y tenga exactamente n+1 puntos cónicos, cada uno de orden p. Si (p-1)(n-1) ≥ 3, entonces se sabe que S no es hiperelíptica y genéricamente no es casiplatónica. Denotemos por Aut_H(S) el normalizador de H en Aut(S). Si p es primo y tenemos que (i) n=4 o bien (ii) n es par y Aut_H(S)/H no es un grupo cíclico no trivial o bien (iii) n es impar y Aut_H(S)/H no es un grupo cíclico, entonces verificamos que S se puede definir sobre su cuerpo de moduli. Más aún, si n ε {3,4}, entonces determinamos tal cuerpo de moduli.

Palabras clave: Curvas algebraicas, superficies de Riemann, cuerpo de moduli, cuerpo de definición.

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Texto completo disponible en PDF

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References

[1] M. Artebani and S. Quispe, `Fields of Moduli and Fields of Definition of Odd Signature Curves', Archiv der Mathematik 99, (2012), 333-343.

[2] A. F. Beardon, The Geometry of Discrete Groups, Vol. 91 of Graduate Texts in Mathematics, Springer-Verlag, New York, USA, 1983.

[3] A. Carocca, V. González-Aguilera, R. A. Hidalgo, and R. E. Rodríguez, `Generalized Humbert Curves', Israel Journal of Math. 164, (2008), 165-192.

[4] P. Dèbes and M. Emsalem, `On Fields of Moduli of Curves', J. of Algebra 211, (1999), 42-56.

[5] C. J. Earle, On the Moduli of Closed Riemann Surfaces with Symmetries, `Advances in the Theory of Riemann Surfaces', (1971), Princeton Univ. Press, Princeton, p. 119-130.

[6] H. M. Farkas and I. Kra., Riemann Surfaces, Vol. 71 of Graduate Text in Math., second edn, Springer-Verlag, 1991.

[7] Y. Fuertes, G. González, R. A. Hidalgo, and M. Leyton, `Automorphism Group of Generalized Fermat Curves of Type (k,3)', Journal of Pure and Applied Algebra 217, (2013), 1791-1806.

[8] G. González, R. A. Hidalgo, and M. Leyton, `Generalizated Fermat Curves', Journal of Algebra 321, (2009), 1643-1660.

[9] A. Grothendieck, Esquisse d'un programme, `Geometric Galois actions 1. around Grothendieck's Esquisse d'un Programme', (1997), Vol. 242 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, p. 5-48.

[10] H. Hammer and F. Herrlich, `A Remark on the Moduli Field of a Curve', Arch. Math. 81, (2003), 5-10.

[11] R. A. Hidalgo, `Non-Hyperelliptic Riemann Surfaces with Real Field of Moduli but not Definable over the Reals', Archiv der Mathematik 93, (2009a), 219-222.

[12] R. A. Hidalgo, `Non-Hyperelliptic Riemann Surfaces with Real Field of Moduli but not Definable over the Reals', Archiv der Mathematik 93, (2009b), 219-222.

[13] R. A. Hidalgo, `Erratum to: Non-hyperelliptic Riemann Surfaces with Real Field of Moduli but not Definable over the Reals', Archiv der Math. 98, (2012), 449-451.

[14] R. A. Hidalgo and M. Leyton, `On Uniqueness of Automorphisms Groups of Riemann Surfaces', Revista Matematica Iberoamericana 23, 3 (2007), 793-810.

[15] R. A. Hidalgo and S. Quispe, Fields of Moduli of Curves, Preprint, 2013.

[16] R. A. Hidalgo and S. Reyes-Carocca, `Fields of Moduli of Classical Humbert Curves', Quarterly Journal of Math. 63, 4 (2012), 919-930.

[17] B. Huggins, `Fields of Moduli of Hyperelliptic Curves', Math. Res. Lett. 14, 2 (2007), 249-262.

[18] S. Koizumi, `Fields of Moduli for Polarized Abelian Varieties and for Curves', Nagoya Math. J. 48, (1972), 37-55.

[19] A. Kontogeorgis, `Field of Moduli versus Field of Definition for Cyclic Covers of the Projective Line', J. de Theorie des Nombres de Bordeaux 21, (2009), 679-692.

[20] B. Maskit, Kleinian Groups, GMW, Springer-Verlag, 1987.

[21] J. Quer and G. Cardona, `Fields of Moduli and Field of Definition for Curves of Genus 2. Computational Aspects of Algebraic Curves', Lecture Notes Series Comput. 13, (2005), 71-83.

[22] S. Reyes-Carocca, `Field of Moduli of Generalized Fermat Curves', Quarterly Journal Math. 63, 2 (2012), 467-475.

[23] G. Shimura., `On the Field of Rationality for an Abelian Variety', Nagoya Math. J. 45, (1972), 167-178.

[24] A. Weil, `The field of definition of a variety', Amer. J. Math. 78, (1956), 509-524.

[25] J. Wolfart, ABC for polynomials, dessins d'enfants and uniformization-a survey, `Elementare und Analytische Zahlentheorie', (2006), Vol. 20, Schr. Wiss. Ges. Johann Wolfgang Goethe Univ. Frankfurt am Main, Franz Steiner Verlag Stuttgart, Stuttgart, p. 313-345.

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(Recibido en julio de 2013. Aceptado en septiembre de 2013)

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Este artículo se puede citar en LaTeX utilizando la siguiente referencia bibliográfica de BibTeX:

@ARTICLE{RCMv47n2a07,
    AUTHOR  = {Hidalgo, Ruben A. and Reyes-Carocca, Sebastián and Valdés, María Elisa},
    TITLE   = {{Field of Moduli and Generalized Fermat Curves}},
    JOURNAL = {Revista Colombiana de Matemáticas},
    YEAR    = {2013},
    volume  = {47},
    number  = {2},
    pages   = {205--221}
}

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