Abelian varieties isogenous to a power of an elliptic curve over a Galois extension
Isabel Vogt1
1 Department of Mathematics Massachusetts Institute of Technology 77 Massachusetts Avenue Cambridge, MA 02139, USA
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 : 10.5802/jtnb.1075
Classification : 14K02, 11G10
Mots-clés : abelian varieties, complex multiplication, isogenies

Isabel Vogt 1

1 Department of Mathematics Massachusetts Institute of Technology 77 Massachusetts Avenue Cambridge, MA 02139, USA
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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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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