Field of moduli versus field of definition for cyclic covers of the projective line
Journal de théorie des nombres de Bordeaux, Volume 21 (2009) no. 3, pp. 679-693.

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 that can not be defined over is also given.

Nous donnons un critère, dépendant du groupe des automorphismes, pour que certains revêtements cycliques de la droite projective soient définis sur leur corps de modules. Nous donnons aussi un exemple de revêtement cyclique de la droite projective complexe de corps de module qui ne peut pas être défini sur .

DOI: 10.5802/jtnb.694
Kontogeorgis, Aristides 1

1 Department of Mathematics, University of the Ægean, 83200 Karlovassi, Samos, Greece
@article{JTNB_2009__21_3_679_0,
     author = {Kontogeorgis, Aristides},
     title = {Field of moduli versus field of definition for cyclic covers of the projective line},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {679--693},
     publisher = {Universit\'e Bordeaux 1},
     volume = {21},
     number = {3},
     year = {2009},
     doi = {10.5802/jtnb.694},
     zbl = {1201.14020},
     mrnumber = {2605539},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/jtnb.694/}
}
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
SP  - 679
EP  - 693
VL  - 21
IS  - 3
PB  - Université Bordeaux 1
UR  - http://archive.numdam.org/articles/10.5802/jtnb.694/
DO  - 10.5802/jtnb.694
LA  - en
ID  - JTNB_2009__21_3_679_0
ER  - 
%0 Journal Article
%A Kontogeorgis, Aristides
%T Field of moduli versus field of definition for cyclic covers of the projective line
%J Journal de théorie des nombres de Bordeaux
%D 2009
%P 679-693
%V 21
%N 3
%I Université Bordeaux 1
%U http://archive.numdam.org/articles/10.5802/jtnb.694/
%R 10.5802/jtnb.694
%G en
%F JTNB_2009__21_3_679_0
Kontogeorgis, Aristides. Field of moduli versus field of definition for cyclic covers of the projective line. Journal de théorie des nombres de Bordeaux, Volume 21 (2009) no. 3, pp. 679-693. doi : 10.5802/jtnb.694. http://archive.numdam.org/articles/10.5802/jtnb.694/

[1] Jannis A. Antoniadis and Aristides Kontogeorgis, On cyclic covers of the projective line. Manuscripta Math. 121 (2006), no. 1, 105–130. MR2258533 | MR | Zbl

[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)

[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) | Numdam | MR | Zbl

[4] Pierre Dèbes and Michel Emsalem, On fields of moduli of curves. J. Algebra 211 (1999), no. 1, 42–56. MR1656571 (99k:14044) | MR | Zbl

[5] Helmut Hasse, Theorie der relativ-zyklischen algebraischen Funktionenkörper, insbesondere bei endlichem Konstantenkörper. J. Reine Angew. Math. 172 (1934), 37–54. | Zbl

[6] Bonnie Huggins, Fields of moduli of hyperelliptic curves. Math. Res. Lett. 14 (2007), no. 00, 10001–10014. | MR | Zbl

[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) | MR | Zbl

[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) | MR | Zbl

[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. | MR | Zbl

[10] Heinrich-Wolfgang Leopoldt, Über die Automorphismengruppe des Fermatkörpers. J. Number Theory 56 (1996), no. 2, 256–282. | MR | Zbl

[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 | MR | Zbl

[12] Henning Stichtenoth, Algebraic function fields and codes. Springer-Verlag, Berlin, 1993. 94k:14016 | MR | Zbl

[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 | MR | Zbl

[14] André Weil, The field of definition of a variety. Amer. J. Math. 78 (1956), 509–524. MR 0082726 (18,601a) | MR | Zbl

Cited by Sources: