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
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     title = {Abelian varieties isogenous to a power of an elliptic curve over a {Galois} extension},
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Vogt, Isabel. 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. http://archive.numdam.org/articles/10.5802/jtnb.1075/

[1] Benson, David J. Representations and cohomology. I. Basic representation theory of finite groups and associative algebras, Cambridge Studies in Advanced Mathematics, 30, Cambridge University Press, 1998 | Zbl 0908.20001

[2] Bourdon, Abbey; Pollack, Paul Torsion subgroups of CM elliptic curves over odd degree number fields, Int. Math. Res. Not., Volume 2017 (2017) no. 16, pp. 4923-4961 | MR 3687120 | Zbl 1405.11072

[3] Clark, Pete L.; Cook, Brian; Stankewicz, James Torsion points on elliptic curves with complex multiplication, Int. J. Number Theory, Volume 9 (2013) no. 2, pp. 447-479 | Article | Zbl 1272.11075

[4] Jordan, Bruce W.; Keeton, Allan G.; Poonen, Bjorn; Rains, Eric M.; Shepherd-Barron, Nicholas; Tate, John T. Abelian varieties isogenous to a power of an elliptic curve, Compos. Math., Volume 154 (2018) no. 5, pp. 934-959 | Article | MR 3798590 | Zbl 1400.14116

[5] Kwon, Soonhak Degree of isogenies of elliptic curves with complex multiplication, J. Korean Math. Soc., Volume 36 (1999) no. 5, pp. 945-958 | MR 1724020 | Zbl 0943.11031

[6] Rubin, Karl Elliptic curves with complex multiplication and the conjecture of Birch and Swinnerton–Dyer, Arithmetic theory of elliptic curves (Cetraro, 1997) (Lecture Notes in Mathematics), Volume 1716, Springer, 1997, pp. 167-234 | Article | Zbl 0991.11028

[7] Vogt, Isabel A local-global principle for isogenies of composite degree (2018) (https://arxiv.org/abs/1801.05355)

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