Borne uniforme pour les homoth\'eties dans l'image de Galois associ\'ee aux courbes elliptiques

TL;DR

This paper establishes a uniform bound depending on number field invariants, ensuring that for large primes, the Galois representation on elliptic curve torsion points contains a large subgroup of homotheties.

Contribution

It provides a bound C(K) depending only on number field invariants, guaranteeing a large subgroup of homotheties in Galois images for large primes.

Findings

Bound C(K) depends only on degree, class number, and discriminant of K.

For primes larger than C(K), the Galois image contains a subgroup of homotheties of index less than 12.

Results apply uniformly across elliptic curves over fixed number fields.

Abstract

Let K be a fixed number field and G its absolute Galois group. We give a bound C(K), depending only on the degree, the class number and the discriminant of K, such that for any elliptic curve E defined over K and any prime number p strictly larger than C(K), the image of the representation of G attached to the p-torsion points of E contains a subgroup of homotheties of index smaller than 12.