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

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.

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.

Received : 2018-04-29
Accepted : 2018-10-19
Published online : 2019-07-29
DOI : https://doi.org/10.5802/jtnb.1075
Classification:  14K02,  11G10
Keywords: 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},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {31},
     number = {1},
     year = {2019},
     pages = {205-213},
     doi = {10.5802/jtnb.1075},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/item/JTNB_2019__31_1_205_0}
}
Vogt, Isabel. Abelian varieties isogenous to a power of an elliptic curve over a Galois extension. Journal de Théorie des Nombres de Bordeaux, Volume 31 (2019) no. 1, pp. 205-213. doi : 10.5802/jtnb.1075. jtnb.centre-mersenne.org/item/JTNB_2019__31_1_205_0/

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

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

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

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

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

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

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