Abelian varieties isogenous to a power of an elliptic curve over a Galois extension
Journal de Théorie des Nombres de Bordeaux, Tome 31 (2019) no. 1, pp. 205-213.

Soient E/k une courbe elliptique et k ' /k une extension de Galois. On construit un foncteur exact de la catégorie des modules sans torsion sur l’anneau des endomorphismes EndE k ' munis d’une action semi-linéaire de Gal(k ' /k) vers la catégorie des variétés algébriques sur k qui sont k ' -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 ' /k, we construct an exact functor from torsion-free modules over the endomorphism ring EndE k ' with a semilinear Gal(k ' /k) action to abelian varieties over k that are k ' -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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DOI : https://doi.org/10.5802/jtnb.1075
Classification : 14K02,  11G10
Mots clés : abelian varieties, complex multiplication, isogenies
@article{JTNB_2019__31_1_205_0,
     author = {Isabel Vogt},
     title = {Abelian varieties isogenous to a power of an elliptic curve over a {Galois} extension},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {205--213},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {31},
     number = {1},
     year = {2019},
     doi = {10.5802/jtnb.1075},
     mrnumber = {3994726},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1075/}
}
Isabel Vogt. Abelian varieties isogenous to a power of an elliptic curve over a Galois extension. Journal de Théorie des Nombres de Bordeaux, Tome 31 (2019) no. 1, pp. 205-213. doi : 10.5802/jtnb.1075. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1075/

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