This is probably well known but I am not an expert in the subject.

Given an abelian variety $A$ of dimension $g$ with CM by $O_K$ where $K$ is a CM field of degree $2g$, let $N_A$ be the norm of the conductor of $A$ as defined here http://en.wikipedia.org/wiki/Conductor_of_an_abelian_variety

My question is : can one compare N_A to d_K, the absolute discriminant of $K$?

More precisely, can one expect $log(N_A) < c log (d_K)$ where $c$ depends on $g$ only.

Many thanks in advance!

This is probably well known but I am not an expert in the subject.

Given an abelian variety $A$ of dimension $g$ with CM by $O_K$ where $K$ is a CM field of degree $2g$, let $N_A$ be the norm of the conductor of $A$ as defined here

http://en.wikipedia.org/wiki/Conductor_of_an_abelian_variety

My question is : can one compare $N_A$ to $d_K$, the absolute discriminant of $K$?

More precisely, can one expect $\log(N_A) < c \log (d_K)$ where $c$ depends on $g$ only.

Many thanks in advance!