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Proyecciones (Antofagasta)

Print version ISSN 0716-0917

Proyecciones (Antofagasta) vol.33 no.1 Antofagasta Mar. 2014

http://dx.doi.org/10.4067/S0716-09172014000100005 

 

Computing the Field of Moduli of the KFT family*

 

Rubén A. Hidalgo
U. Técnica Federico Santa Maria, Chile


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.

Subjclass [2010] : 30F10, 14H37, I4H45, 14Q05.

*Partially supported by project Fondecyt 1110001 and UTFSM 12.13.01


 

REFERENCES

[1] G.V.Belyi.On Galoisextensionsofamaximalcyclotomic field. Izv. Akad. Nauk SSSR Ser. Mat. 43, pp. 267-276, 479, (1979).

[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 quar-tics. Advances in Math. 98, pp. 216-301, (1992).

[4] C.J.Earle.On the moduliofclosed 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 fur 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. XJber algebraische Gebilde mit eindeutigen Transforma-tionen 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. Rodriguez and V. Gonzalez-Aguilera. Fermat's Quartic Curve, Klein's Curve and the Tetrahedron. 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,Springer-Verlag, (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. JJber die hyperelliptischen curven und diejenigen vom geschlechte p=3 welche eindeutige transformationen in sich zulassen. Bilhang till Kongl Svenska Veteskjaps Handliger 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).

 

Rubén Hidalgo
Departamento de Matemaitica, Universidad Tecnica Federico Santa María, Casilla 110-V, Valparaiso
e-mail : ruben.hidalgo@usm.cl

Received : May 2012. Accepted : October 2013.

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