I'm using the same setup as Corollary 1.7 on p. 44 of de Shalit manuscript (Iwasawa theory of elliptic curves with complex multiplication).

I think there is a mistake in his Corollary 1.7 and I'm wondering if it is possible to fix it. So let $E/F$ be an elliptic curve defined over a number field $F$ such that

(1) $E$ has complex multiplication by $\mathcal{O}_K$,

(2) $F(E_{tor})$ is abelian over $K$.

In particular, it follows from (2) that $K(1)\subseteq F$, where $K(1)$ stands for the Hilbert class field of $K$. Let $\mathfrak{f}$ be the conductor of the Groessencharacter associated to $E/F$ (which depends just on the $F$-isogeny class of $E$).

For an integral ideal $\mathfrak{m}\subseteq\mathcal{O}_K$, we let

(a) $F[\mathfrak{m}]=F(E[\mathfrak{m}])$

(b) $F(\mathfrak{m})=F(x(P):P\in E[\mathfrak{m}])$

It is easy to see that $[L[\mathfrak{m}]:L(\mathfrak{m})]\leq 2$ and that we have an injection

$\theta_{\mathfrak{m}}:Gal(F[\mathfrak{m}]/F)\hookrightarrow (\mathcal{O}_K/\mathfrak{m})^{\times}$.

Let $\mathfrak{g}\subseteq\mathcal{O}_K$ be another ideal and let us assume that $(\mathfrak{m},\mathfrak{g}\mathfrak{f})=1$. Then de Shalit claims the following:

(c) $\theta$ is surjective

(d) $F[\mathfrak{m}]$ is linearly disjoint from $F[\mathfrak{g}]$.

In general, I don't see why $\theta_{\mathfrak{m}}$ is surjective, since in one of his computations, he forgets the contribution coming from the roots of unity of $K$. Moreover, if we take into account this omission, it seems to me that we only obtain that $F(\mathfrak{m})$ and $F(\mathfrak{g})$ are linearly disjoint over $F$.

Q1 Are (c) and (d) still true ?

There seems also to be something wrong with his proposition 1.6, since it seems to me that one could assume from the outset that $F$ contains the coordinates of the points $E[\mathfrak{m}]$ without changing the assumptions (1) and (2) above (but may his proposition is correct if $F=K(1)$).

Here is one related question to the previous paragraph which I don't know the answer:

Q2 Let us assume that $F=K(1)$. Then we know from CM theory that $F(\mathfrak{m})$ corresponds to the ray class field of $K$ of modulus $\mathfrak{m}$. Let us assume that $[F[\mathfrak{m}]:F(\mathfrak{m})]=2$. What is the conductor of the abelian extension $F[\mathfrak{m}]/K$ (in particular this conductor must be divisible by $\mathfrak{m}$) ?