Fields of definition of $ℚ$-curves
Journal de théorie des nombres de Bordeaux, Volume 13 (2001) no. 1, p. 275-285

Let $C$ be a $ℚ$-curve with no complex multiplication. In this note we characterize the number fields $K$ such that there is a curve ${C}^{\text{'}}$ isogenous to $C$ having all the isogenies between its Galois conjugates defined over $K$, and also the curves ${C}^{\text{'}}$ isogenous to $C$ defined over a number field $K$ such that the abelian variety Res${}_{K/ℚ}\left({C}^{\text{'}}/K\right)$ obtained by restriction of scalars is a product of abelian varieties of GL${}_{2}$-type.

Soit $C$ une $ℚ$-courbe sans multiplication complexe. Dans cet article, nous caractérisons les corps de nombres $K$ pour lesquels il existe une courbe ${C}^{\text{'}}$ isogène à $C$ dont toutes les isogénies entre les conjuguées par le groupe de Galois sont définies sur $K$. Nous caractérisons également les courbes ${C}^{\text{'}}$ isogènes à $C$ définies sur un corps de nombres $K$ telles que la variété abélienne Res${}_{K/ℚ}$ déduite de ${C}^{\text{'}}$ par restriction des scalaires est un produit de variétés abéliennes de type GL${}_{2}$.

@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},
publisher = {Universit\'e Bordeaux I},
volume = {13},
number = {1},
year = {2001},
pages = {275-285},
zbl = {1046.11044},
mrnumber = {1838087},
language = {en},
url = {http://www.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://www.numdam.org/item/JTNB_2001__13_1_275_0/

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