Abelian varieties isogenous to a power of an elliptic curve over a Galois extension

Soient $E/k$ une courbe elliptique et $k^{\text{'}}/k$ une extension de Galois. On construit un foncteur exact de la catégorie des modules sans torsion sur l’anneau des endomorphismes $End{E}_{k^{\text{'}}}$ munis d’une action semi-linéaire de $Gal(k^{\text{'}}/k)$ vers la catégorie des variétés algébriques sur $k$ qui sont $k^{\text{'}}$-isogènes à une puissance de $E$. Comme application, on donne une preuve simple du fait que toute courbe elliptique sur $k$ qui est géométriquement à multiplication complexe, est isogène sur $k$ à une courbe elliptique à multiplication complexe par un ordre maximal.

Given an elliptic curve $E/k$ and a Galois extension $k^{\text{'}}/k$, we construct an exact functor from torsion-free modules over the endomorphism ring $End{E}_{k^{\text{'}}}$ with a semilinear $Gal(k^{\text{'}}/k)$ action to abelian varieties over $k$ that are $k^{\text{'}}$-isogenous to a power of $E$. As an application, we give a simple proof that every elliptic curve with complex multiplication geometrically is isogenous over the ground field to one with complex multiplication by a maximal order.

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