Let be a -curve with no complex multiplication. In this note we characterize the number fields such that there is a curve isogenous to having all the isogenies between its Galois conjugates defined over , and also the curves isogenous to defined over a number field such that the abelian variety Res obtained by restriction of scalars is a product of abelian varieties of GL-type.
Soit une -courbe sans multiplication complexe. Dans cet article, nous caractérisons les corps de nombres pour lesquels il existe une courbe isogène à dont toutes les isogénies entre les conjuguées par le groupe de Galois sont définies sur . Nous caractérisons également les courbes isogènes à définies sur un corps de nombres telles que la variété abélienne Res déduite de par restriction des scalaires est un produit de variétés abéliennes de type GL.
@article{JTNB_2001__13_1_275_0, author = {Quer, Jordi}, title = {Fields of definition of $\mathbb {Q}$-curves}, journal = {Journal de th\'eorie des nombres de Bordeaux}, pages = {275--285}, publisher = {Universit\'e Bordeaux I}, volume = {13}, number = {1}, year = {2001}, mrnumber = {1838087}, zbl = {1046.11044}, language = {en}, url = {http://archive.numdam.org/item/JTNB_2001__13_1_275_0/} }
Quer, Jordi. Fields of definition of $\mathbb {Q}$-curves. Journal de théorie des nombres de Bordeaux, Volume 13 (2001) no. 1, pp. 275-285. http://archive.numdam.org/item/JTNB_2001__13_1_275_0/
[1] Remarks on elliptic k-curves. Preprint, 1992.
,[2] Abelian varieties over Q with large endomorphism algebras and their simple components over Q. Ph.D. Thesis, Univ. of California at Berkeley, 1995.
,[3] Q-curves and Abelian varieties of GL2-type. Proc. London Math. Soc. (3) 81 (2000), 285-317. | MR | Zbl
,[4] Abelian varieties over Q and modular forms. Proceedings of KAIST Mathematics Workshop (1992), 53-79. | MR
,[5] Fields of definition of Abelian varieties with real multiplication. Contemp. Math. 174 (1994), 107-118. | MR | Zbl
,