This question is mainly referring to the proof of Theorem 5.6, Chapter 2 of Silverman's "Advanced Topics in the AEC". Basically, let $K$ be an imaginary quadratic field, and $E$ be an elliptic curve with CM by $R_K$ (The ring of integer of $K$). Let $h : E \rightarrow \mathbb{P}^1$ be a Weber function for $E/H$ ($E$ is chosen to have a model over $H$, the Hilbert Class field of $K$). Let $\mathfrak{c}$ be an integral ideal of $R_K$, then the field $L = K(j(E), h(E[\mathfrak{c}]))$ is the ray class field modulo $\mathfrak{c}$ of $K$.

The proof in the book basically uses the characterisation of ray class field that $L$ is the ray class field if and only if it satisfies the following equivalent conditions:

\begin{align*} (\mathfrak{p}, L/K) =1 \leftrightarrow \mathfrak{p} \in P(\mathfrak{c}), \end{align*} where $(\cdot, L/K)$ is the Artin map and $P(\mathfrak{c})= \{(\alpha): \alpha \in K^*, \alpha \equiv 1 \; (\text{mod} \; \mathfrak{c}) \}$.

My question is, for us to apply the Artin map, one has first to assume that $L$ is abelian over $K$. But this is not given in the book, at least I did not see a direct proof. It is not even given that $L$ is galois over $K$. So is there an easy way to see this?

I think certainly this has something to do with the fact that $h$ is invariant under automorphism of $E$. But I don't have a clear explanation.


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