Notation: For a finite abelian extension $F / K$, let $\mathfrak{f}_{F / K} \subset \mathcal{O}_K$ denotes its conductor such that $F$ is contained in the ray class field $K(\mathfrak{f})$. In particular, the set of primes dividing $\mathfrak{f}_{F / K}$ consists precisely of those prime ideals which ramify in $F / K$. Whenever $\chi: \mathbf{A}_L^{\times} \rightarrow \mathbf{C}^{\times}$is a Hecke character of some number field $L$, we let $\mathfrak{f}_\chi$ denote its conductor, i.e. the smallest ideal such that $\chi$ is trivial on $U_f\left(\mathfrak{f}_\chi\right)$.
Let $K$ be an imaginary quadratic field and $F / K$ a finite abelian extension. Let $E / F$ be an elliptic curve with complex multiplication by $\mathcal{O}_K$. Let $F\left(E_{\text {tors }}\right)$ denote the field extension of $F$ generated by all the torsion points of $E$. Let $\psi: \mathbf{A}_F^{\times} \rightarrow \mathbf{C}^{\times}$be the Hecke character associated to $E / F$. Let $N_{F / K}: \mathbf{A}_F^{\times} \rightarrow \mathbf{A}_K^{\times}$denote the idele norm. Assume that there exists a Hecke character $\varphi: \mathbf{A}_K^{\times} \rightarrow \mathbf{C}^{\times}$such that
$$\psi=\varphi \circ N_{F / K}$$
Show that $F\left(E_{\text {tors }}\right)$ is an abelian extension over $K$.
Hint: It is equivalent to show that the $G_K$-module $\operatorname{Ind}_{G_F}^{G_K}\left(E_{\mathrm{tors}}\right)$ is abelian. Note that the Artin map $$\left[-, K^{\mathrm{ab}} / K\right]: \mathbf{A}_K^{\times} \rightarrow \operatorname{Gal}\left(K^{\mathrm{ab}} / K\right)$$
factors through the finite ideles. We may use that the kernel of $\left[-, K^{\mathrm{ab}} / K\right]$ : $\mathbf{A}_{K, f}^{\times} \rightarrow \operatorname{Gal}\left(K^{\mathrm{ab}} / K\right)$ is the topological closure of $K^{\times}$in $\mathbf{A}_{K, f}^{\times}$.
I would really appreciate any help with this exercise with which I am stuck.
