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, Volume 31 (2019) no. 1, pp. 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:
Accepted:
Published online:
DOI: 10.5802/jtnb.1075
Classification: 14K02, 11G10
Keywords: abelian varieties, complex multiplication, isogenies

Isabel Vogt 1

1 Department of Mathematics Massachusetts Institute of Technology 77 Massachusetts Avenue Cambridge, MA 02139, USA
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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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, Volume 31 (2019) no. 1, pp. 205-213. doi : 10.5802/jtnb.1075. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1075/

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

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

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

[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., Volume 154 (2018) no. 5, pp. 934-959 | DOI | MR | Zbl

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

[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), Volume 1716, Springer, 1997, pp. 167-234 | DOI | Zbl

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

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