On uniform lower bound of the Galois images associated to elliptic curves
Journal de Théorie des Nombres de Bordeaux, Volume 20 (2008) no. 1, pp. 23-43.

Let p be a prime and let K be a number field. Let ρ E,p :G K Aut(T p E)GL 2 ( p ) be the Galois representation given by the Galois action on the p-adic Tate module of an elliptic curve E over K. Serre showed that the image of ρ E,p is open if E has no complex multiplication. For an elliptic curve E over K whose j-invariant does not appear in an exceptional finite set (which is non-explicit however), we give an explicit uniform lower bound of the size of the image of ρ E,p .

Soit p un nombre premier et K un corps de nombres. Soit ρ E,p :G K Aut(T p E)GL 2 ( p ) la représentation Galoisienne donnée par l’action du groupe de Galois sur le module de Tate p-adique d’une courbe elliptique E définie sur K. Serre a prouvé que l’image de ρ E,p est ouverte si E n’a pas de multiplication complexe. Pour E une courbe elliptique définie sur K et dont l’invariant j n’appartient pas à un ensemble fini exceptionnel (qui est non explicite cependant), nous donnons une minoration uniforme et explicite de la taille de l’image de ρ E,p .

Received:
Published online:
DOI: 10.5802/jtnb.614
Keisuke Arai 1

1 Graduate School of Mathematical Sciences The University of Tokyo Tokyo 153-8914, Japan
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Keisuke Arai. On uniform lower bound of the Galois images associated to elliptic curves. Journal de Théorie des Nombres de Bordeaux, Volume 20 (2008) no. 1, pp. 23-43. doi : 10.5802/jtnb.614. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.614/

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