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 $\ell$ be a prime number. We give explicit bounds on the degree over $K$ of the division fields $K(A[{\ell }^n])$, and when $A$ is an elliptic curve we also describe the full Galois group of $K(A_{\text{tors}})/K$. This makes explicit previous results of Serre and Ribet, and strengthens a theorem of Banaszak, Gajda and Krasoń. Our bounds are especially sharp in case the CM type of $A$ is nondegenerate.

arXiv commentv2: theorem 1.2 and exposition improved, fixed some minor mistakes

References in corpus (1)

• The average size of the 2-Selmer group of Jacobians of hyperelliptic curves having a rational Weierstrass point

Cited by in corpus (1)

• How big is the image of the Galois representations attached to CM elliptic curves?