Galois representations attached to abelian varieties of CM type

Lombardo, Davide

Bulletin de la Société Mathématique de France, Tome 145 (2017) no. 3, pp. 469-501

Let K be a number field, A/K be an absolutely simple abelian variety of CM type, and ℓ be a prime number. We give explicit bounds on the degree over K of the division fields K(A[ℓn]), and when A is an elliptic curve we also describe the full Galois group of K(Ators)/K. This makes explicit previous results of Serre [17] and Ribet [14], and strengthens a theorem of Banaszak, Gajda and Krasoń [2]. Our bounds are especially sharp when the CM type of A is nondegenerate.

Soient K un corps de nombres, A/K une variété abélienne géométriquement simple de type CM et ℓ un nombre premier. Nous donnons des bornes explicites sur le degré sur K des extensions K(A[ℓn]) engendrées par les points de ℓn-torsion de A, et quand A est une courbe elliptique nous décrivons le groupe de Galois de K(Ators)/K tout entier. Cela fournit une version explicite de résultats antérieurs de Serre [17] et Ribet [14], et renforce un théorème de Banaszak, Gajda and Krasoń [2]. Nos bornes sont particulièrement fines quand le type CM de A est non-dégénéré.

Classification : 14K22, 11F80, 11G10
Keywords: Complex multiplication, Galois representations, elliptic curves, Mumford-Tate group
Mots-clés : Multiplication complexe, représentations galoisiennes, courbes elliptiques, groupe de Mumford-Tate

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