Galois representations attached to abelian varieties of CM type

Abstract

Let K be a number field, A/K be an absolutely simple abelian variety of CM type, and be a prime number. We give explicit bounds on the degree over K of the division fields K(A[^n]), and when A is an elliptic curve we also describe the full Galois group of K(A_tors)/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