On uniform lower bound of the Galois images associated to elliptic curves

Keisuke Arai

doi:10.5802/jtnb.614

Keisuke Arai ¹

¹ Graduate School of Mathematical Sciences The University of Tokyo Tokyo 153-8914, Japan

Journal de théorie des nombres de Bordeaux, Volume 20 (2008) no. 1, pp. 23-43.

Abstract
Résumé

Let $p$ be a prime and let $K$ be a number field. Let $ρ_{E, p} : G_{K} \to Aut (T_{p} E) ≅ {GL}_{2} (ℤ_{p})$ 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 $ρ_{E, p} : G_{K} \to Aut (T_{p} E) ≅ {GL}_{2} (ℤ_{p})$ 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 $ρ_{E, p}$ .

MR

DOI: 10.5802/jtnb.614

Author's affiliations:

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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