Computing the Field of Moduli of the KFT family
DOI:
https://doi.org/10.4067/S0716-09172014000100005Keywords:
Closed Riemann surfaces, modulus field, automorphisms, superficies cerradas de Riemann, campo de módulo, automorfismos.Abstract
The computation of the field of moduli of a given closed Riemann surface is in general a very difficult task. In this note we consider the family of closed Riemann surfaces of genus three admitting the symmetric group in four letters as a group of conformai automorphisms and we provide the computations of the corresponding field of moduli.References
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[2] S. A. Broughton. Classifying finite group actions on surfaces of low genus, J. Pure Applied Algebra 69, pp. 233-270, (1990).
[3] I. Dolgachev and V. Kanev. Polar covariants of plane cubics and quartics. Advances in Math. 98, pp. 216-301, (1992).
[4] C. J. Earle. On the moduli of closed Riemann surfaces with symmetries. Advances in the Theory of Riemann Surfaces (1971) 119-130. Ed. L.V. Ahlfors et al. (Princeton Univ. Press, Princeton).
[5] H. Farkas and I. Kra. Riemann Surfaces. Second edition. Graduate Texts in Mathematics 71. Springer-Verlag, New York, (1992).
[6] Y. Fuertes and M. Streit. Genus 3 normal coverings of the Riemann sphere branched over 4 points. Rev. Mat. Iberoamericana 22 No. 2, pp.
413-454, (2006).
[7] M. Guizatullin. Bialgebra and geometry of plane quartics, Preprint Max-Planck-Institute für Mathematik 46, (2000).
[8] H. Hammer and F. Herrlich. A Remark on the Moduli Field of a Curve. Arch. Math. 81, pp. 5-10, (2003).
[9] R. A. Hidalgo. Non-hyperelliptic Riemann surfaces with real field of moduli but not definable over the reals. Archiv der Mathematik 93, pp.
219-222, (2009).
[10] R. A. Hidalgo. Schottky uniformizations of genus 3 and 4 reflecting S4. Journal of the London Math. Soc. 72, No. 1, pp. 185-204, (2005).
[11] A. Hurwitz. Uber algebraische Gebilde mit eindeutigen Transformationen in sich. Math. Ann. 41, pp. 403-442, (1893).
[12] L. Keen. Canonical polygons for finitely generated Fuchsian groups. Acta Mathematica 115 No.1, pp. 1-16, (1966).
[13] S. Koizumi. The fields of moduli for polarized abelian varieties and for curves. Nagoya Math. J. 48, pp. 37-55, (1972).
[14] J. Ries. Splittable jacobian varieties. Contemporary Mathematics, 136, pp. 305-326, (1992).
[15] R. E. Rodríguez and V. González-Aguilera. Fermat’s Quartic Curve, Klein’s Curve and the Tetrahedron. In Contemporary Mathematics, 201, pp. 43-62, (1997).
[16] G. Shimura. On the field of rationality for an abelian variety. Nagoya Math. J. 45, 167-178, (1972).
[17] J. Silverman. The Arithmetic of Elliptic Curves. GTM, SpringerVerlag, (1986).
[18] D. Singerman. Finitely Maximal Fuchsian Groups. J. London Math. Soc. 6, No. 2, pp. 29-38, (1972).
[19] A. Weil. The field of definition of a variety. Amer. J. Math., 78, pp. 509-524, (1956).
[20] A. Wiman. Uber die hyperelliptischen curven und diejenigen vom geschlechte p=3 welche eindeutige transformationen in sich zulassen. Bilhang till Kongl Svenska VeteskjapsHandliger Stockholm, 21, pp. 1-23, (1985).
[21] J. Wolfart. ABC for polynomials, dessins d’enfants and uniformization a survey. Elementare und analytische Zahlentheorie, 313—345, Schr. Wiss. Ges. Johann Wolfgang Goethe Univ. Frankfurt am Main, 20, Franz Steiner Verlag Stuttgart, Stuttgart, (2006).
Published
2017-03-23
How to Cite
[1]
R. A. Hidalgo, “Computing the Field of Moduli of the KFT family”, Proyecciones (Antofagasta, On line), vol. 33, no. 1, pp. 61-75, Mar. 2017.
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