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A TOWER OF RIEMANN SURFACES WHICH CANNOT BE DEFINED OVER THEIR FIELD OF MODULI

Part of: Riemann surfaces

Published online by Cambridge University Press:  10 June 2016

MICHELA ARTEBANI ,
MARIELA CARVACHO ,
RUBEN A. HIDALGO  and
SAÚL QUISPE
MICHELA ARTEBANI
Affiliation:
Departamento de Matemática, Universidad de Concepción., Casilla 160-C, Concepción, Chile e-mail: martebani@udec.cl
MARIELA CARVACHO
Affiliation:
Departamento de Matemática, Universidad Técnica Federico Santa María., Casilla 110-V, Valparaíso, Chile e-mail: mariela.carvacho@usm.cl
RUBEN A. HIDALGO
Affiliation:
Departamento de Matemática y Estadística, Universidad de La Frontera., Casilla 54-D, Temuco, Chile e-mails: ruben.hidalgo@ufrontera.cl, saul.quispe@ufrontera.cl
SAÚL QUISPE
Affiliation:
Departamento de Matemática y Estadística, Universidad de La Frontera., Casilla 54-D, Temuco, Chile e-mails: ruben.hidalgo@ufrontera.cl, saul.quispe@ufrontera.cl
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Abstract

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Explicit examples of both hyperelliptic and non-hyperelliptic curves which cannot be defined over their field of moduli are known in the literature. In this paper, we construct a tower of explicit examples of such kind of curves. In that tower there are both hyperelliptic curves and non-hyperelliptic curves.

MSC classification

Primary: 30F10: Compact Riemann surfaces and uniformization
Secondary: 30F20: Classification theory of Riemann surfaces 30F40: Kleinian groups

Information

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 59 , Issue 2 , May 2017 , pp. 379 - 393
Copyright
Copyright © Glasgow Mathematical Journal Trust 2016 

References

REFERENCES

1. Ackermann, P., Arithmetische Fuchssche Gruppen der Signatur (2;-), Dissertation (2005), Universität Dortmund.Google Scholar
2. Bujalance, E. and Turbek, P., Asymmetric and pseudo-symmetric hyperelliptic surfaces, Manuscr. Math. 108 (2002), 1–11.Google Scholar
3. Carocca, A., Gonzalez, V., Hidalgo, R. A. and Rodríguez, R., Generalized Humbert curves, Israel J. Math. 64 (1) (2008), 165–192.Google Scholar
4. Earle, C. J., On the moduli of closed Riemann surfaces with symmetries, in Advances in the theory of Riemann surfaces (Ahlfors, L. V. et al., Editors) (Princeton Univ. Press, Princeton, 1971), 119–130.Google Scholar
5. Earle, C. J., Diffeomorphisms and automorphisms of compact hyperbolic 2-orbifolds, in Geometry of Riemann surfaces (Gardiner, F., González-Diez, G. and Kourouniotis, C. Editors), London Math. Soc. Lecture Note Ser., vol. 368 (Cambridge Univ. Press, Cambridge, 2010), 139–155.Google Scholar
6. Gonzalez-Diez, G., Hidalgo, R. A. and Leyton, M., Generalized fermat curves, J. Algebra 321 (2009), 1643–1660.CrossRefGoogle Scholar
7. Hidalgo, R. A., Non-hyperelliptic Riemann surfaces with real field of moduli but not definable over the reals, Arch. Math. 93 (2009), 219–222.Google Scholar
8. Hidalgo, R. A., Erratum: Non-hyperelliptic Riemann surfaces with real field of moduli but not definable over the reals, Arch. Math. 98 (2012), 449–45.Google Scholar
9. Hidalgo, R. A., Homology closed Riemann surfaces, Q. J. Math. 63 (2012), 931–952 Google Scholar
10. Huggins, B., Fields of moduli and fields of definition of curves, PhD Thesis (University of California, 2005).Google Scholar
11. Huggins, B., Fields of moduli of hyperelliptic curves, Math. Res. Lett. 14 (2) (2007), 249–262.Google Scholar
12. Koizumi, S., Fields of moduli for polarized Abelian varieties and for curves, Nagoya Math. J. 48 (1972), 37–55.Google Scholar
13. Kontogeorgis, A., Field of Moduli versus field of definition for cyclic covers of the projective line, J. Theorie des Nombres de Bordeaux 21 (2009), 679–693.CrossRefGoogle Scholar
14. Kuribayashi, A. and Kimura, H., Automorphism groups of compact Riemann surfaces of genus five, J. Algebra 134 (1990), 80–103.Google Scholar
15. Kuribayashi, I. and Kuribayashi, A., Automorphism groups of compact Riemann surfaces of genera three and four, J. Pure Appl. Algebra 65 (3) (1990), 277–292.Google Scholar
16. Magaard, K., Shaska, T., Shpectorov, S. and Völklein, H., The locus of curves with prescribed automorphism group, Communications in arithmetic fundamental groups (Kyoto, 1999/2001). Surikaisekikenkyusho Kokyuroku No. 1267 (2002), 112–141.Google Scholar
17. Matsusaka, T., Polarized varieties, the field of moduli and generalized Kummer varieties of polarized abelian varieties, Amer. J. Math. 80 (1958), 45–82.CrossRefGoogle Scholar
18. Shimura, G., On the field of rationality for an abelian variety, Nagoya Math. J. 45 (1971), 167–178.Google Scholar
19. Silhol, R., Moduli problems in real algebraic geometry, in Real Algebraic Geometry (Coste, M. et al., Editors) (Springer-Verlag, Berlin, 1972), 110–119.Google Scholar
20. Weil, A., The field of definition of a variety, Amer. J. Math. 78 (1956), 509–524.Google Scholar