Field of moduli versus field of definition for cyclic covers of the projective line

Aristides Kontogeorgis[1]

  • [1] Department of Mathematics, University of the Ægean, 83200 Karlovassi, Samos, Greece

Journal de Théorie des Nombres de Bordeaux (2009)

  • Volume: 21, Issue: 3, page 679-693
  • ISSN: 1246-7405

Abstract

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We give a criterion, based on the automorphism group, for certain cyclic covers of the projective line to be defined over their field of moduli. An example of a cyclic cover of the complex projective line with field of moduli ℝ that can not be defined over ℝ is also given.

How to cite

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Kontogeorgis, Aristides. "Field of moduli versus field of definition for cyclic covers of the projective line." Journal de Théorie des Nombres de Bordeaux 21.3 (2009): 679-693. <http://eudml.org/doc/10904>.

@article{Kontogeorgis2009,
abstract = {$\!$ We give a criterion, based on the automorphism group, for certain cyclic covers of the projective line to be defined over their field of moduli. An example of a cyclic cover of the complex projective line with field of moduli $\mathbb\{R\}$ that can not be defined over $\mathbb\{R\}$ is also given.},
affiliation = {Department of Mathematics, University of the Ægean, 83200 Karlovassi, Samos, Greece},
author = {Kontogeorgis, Aristides},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {field of moduli; field of definition; automorphism group},
language = {eng},
number = {3},
pages = {679-693},
publisher = {Université Bordeaux 1},
title = {Field of moduli versus field of definition for cyclic covers of the projective line},
url = {http://eudml.org/doc/10904},
volume = {21},
year = {2009},
}

TY - JOUR
AU - Kontogeorgis, Aristides
TI - Field of moduli versus field of definition for cyclic covers of the projective line
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2009
PB - Université Bordeaux 1
VL - 21
IS - 3
SP - 679
EP - 693
AB - $\!$ We give a criterion, based on the automorphism group, for certain cyclic covers of the projective line to be defined over their field of moduli. An example of a cyclic cover of the complex projective line with field of moduli $\mathbb{R}$ that can not be defined over $\mathbb{R}$ is also given.
LA - eng
KW - field of moduli; field of definition; automorphism group
UR - http://eudml.org/doc/10904
ER -

References

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  1. Jannis A. Antoniadis and Aristides Kontogeorgis, On cyclic covers of the projective line. Manuscripta Math. 121 (2006), no. 1, 105–130. MR2258533 Zbl1103.14015MR2258533
  2. Gunter Cornelissen, Fumiharu Kato and Aristides Kontogeorgis, Three examples of the relation between rigid-analytic and algebraic deformation parameters. ArXiv:0809.4579 (to appear in Israel Journal of Mathematics) Zbl1213.14046
  3. Pierre Dèbes and Jean-Claude Douai, Algebraic covers: field of moduli versus field of definition. Ann. Sci. École Norm. Sup. (4) 30 (1997), no. 3, 303–338. MR1443489 (98k:11081) Zbl0906.12001MR1443489
  4. Pierre Dèbes and Michel Emsalem, On fields of moduli of curves. J. Algebra 211 (1999), no. 1, 42–56. MR1656571 (99k:14044) Zbl0934.14019MR1656571
  5. Helmut Hasse, Theorie der relativ-zyklischen algebraischen Funktionenkörper, insbesondere bei endlichem Konstantenkörper. J. Reine Angew. Math. 172 (1934), 37–54. Zbl0010.00501
  6. Bonnie Huggins, Fields of moduli of hyperelliptic curves. Math. Res. Lett. 14 (2007), no. 00, 10001–10014. Zbl1126.14036MR2318623
  7. David Goss, Basic structures of function field arithmetic. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 35, Springer-Verlag, Berlin, 1996. MR1423131 (97i:11062) Zbl0892.11021MR1423131
  8. Aristides Kontogeorgis, The group of automorphisms of cyclic extensions of rational function fields, J. Algebra 216 (1999), no. 2, 665–706. MR1692965 (2000f:12005) Zbl0938.11056MR1692965
  9. Aristides Kontogeorgis, The group of automorphisms of the function fields of the curve x n + y m + 1 = 0 . J. Number Theory 72 (1998), no. 1, 110–136. Zbl0914.11060MR1643304
  10. Heinrich-Wolfgang Leopoldt, Über die Automorphismengruppe des Fermatkörpers. J. Number Theory 56 (1996), no. 2, 256–282. Zbl0854.11062MR1373551
  11. Henning Stichtenoth, Über die Automorphismengruppe eines algebraischen Funktionenkörpers von Primzahlcharakteristik. I. Eine Abschätzung der Ordnung der Automorphismengruppe. Arch. Math. (Basel) 24 (1973), 527–544. 49 #2749 Zbl0282.14006MR337980
  12. Henning Stichtenoth, Algebraic function fields and codes. Springer-Verlag, Berlin, 1993. 94k:14016 Zbl0816.14011MR1251961
  13. Robert C. Valentini and Manohar L. Madan, A Hauptsatz of L. E. Dickson and Artin-Schreier extensions. J. Reine Angew. Math. 318 (1980), 156–177. 82e:12030 Zbl0426.12016MR579390
  14. André Weil, The field of definition of a variety. Amer. J. Math. 78 (1956), 509–524. MR 0082726 (18,601a) Zbl0072.16001MR82726

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