Corps de définition et points rationnels
Journal de Théorie des Nombres de Bordeaux, Tome 15 (2003) no. 1, pp. 45-55.

Soit 𝔒 un objet algébrique (par exemple une courbe ou un revêtement) défini sur ¯ et de corps des modules un corps de nombres K. Il est bien connu que 𝔒 n’admet pas nécessairement de K-modèle. En utilisant deux résultats récents dus à P. Dèbes, J.-C. Douai et M. Emsalem nous donnerons un majorant pour le degré d’un corps de définition de 𝔒 sur K. Dans une deuxième partie, nous donnerons des conditions suffisantes sur l’ordre de Aut(𝔒) pour que 𝔒 admette un K-modèle.

Let 𝔒 be an algebraic object (e.g. a curve or a cover) defined over ¯ and of field of moduli an algebraic number field K. It is well known that 𝔒 does not necessarily admit a K-model. Using two recent results due to P. Dèbes, J.-C. Douai and M. Emsalem we shall give a bound from above for the degree of a field of definition of 𝔒 over K. In the second part, we shall give a sufficient condition on the order of Aut(𝔒) for 𝔒 to have a K-model.

@article{JTNB_2003__15_1_45_0,
     author = {Derome, Geoffroy},
     title = {Corps de d\'efinition et points rationnels},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {45--55},
     publisher = {Universit\'e Bordeaux I},
     volume = {15},
     number = {1},
     year = {2003},
     doi = {10.5802/jtnb.386},
     zbl = {1073.14520},
     mrnumber = {2019000},
     language = {fr},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.386/}
}
Geoffroy Derome. Corps de définition et points rationnels. Journal de Théorie des Nombres de Bordeaux, Tome 15 (2003) no. 1, pp. 45-55. doi : 10.5802/jtnb.386. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.386/

[1] W.L. Bailyjr., On the theory of theta functions, the moduli of abelian varieties, and the moduli of curves. Ann. of Math. 75 (1962), 342-381. | MR 162799 | Zbl 0147.39702

[2] G. Belyi, On the Galois extensions of the maximal cyclotomic field. Math. USSR Izv. 14 (1980), 247-256. | Zbl 0429.12004

[3] P. Dèbes, G. Derome, Finiteness results in descent theory. J. London Math. Soc., à paraître. | Zbl 01998683

[4] P. Dèbes, J.-C. Douai, Algebraic covers, field of moduli versus field of definition. Ann. Sci. École Norm. Sup. (4) 30 (1997), 303-338. | Numdam | MR 1443489 | Zbl 0906.12001

[5] P. Dèbes, M. Emsalem, On fields of moduli of curves. J. Algebra 211 (1999), 42-56. | MR 1656571 | Zbl 0934.14019

[6] E. Reyssat, Quelques aspects des surfaces de Riemann. Progress in Mathematics, 77. Birkhäuser Boston, Inc., Boston, MA, 1989. | MR 1034955 | Zbl 0689.30001

[7] R.E. Rodriguez, V. Gonzalez-Aguilera, Fermat's quartic curve and the tetrahedron. Extremal Riemann surfaces (San Francisco, CA, 1995), 43-62, Contemp. Math., 201, Amer. Math. Soc., Providence, RI, 1997. | MR 1429194 | Zbl 0911.14021

[8] G. Shimur.A, On the field of rationality for an abelian variety. Nagoya Math. J. 45 (1971), 167-178. | MR 306215 | Zbl 0243.14012

[9] J. Silverman, The arithmetic of elliptic curves. Graduate Texts in Mathematics, 106. Springer-Verlag, New York, 1986. | MR 817210 | Zbl 0585.14026

[10] A. Weil, The field of definition of a variety. Amer. J. Math. 78 (1956), 509-524. | MR 82726 | Zbl 0072.16001

[11] J. Wolfart, The "obvious" part of Belyi's theorem and Riemann surfaces with many automorphisms. Geometric Galois actions, 1, 97-112, London Math. Soc. Lecture Note Ser., 242, Cambridge Univ. Press, Cambridge, 1997. | MR 1483112 | Zbl 0915.14021