CM Elliptic Curves and a result concerning ray class fields

Question

Let $K$ be an imaginary quadratic field and suppose that $E/K$ has complex multiplication by $\mathcal{O}_K$. Let $\psi$ be the Hecke character associated with $E$ and $\mathfrak{f}$ its conductor (i.e it is the integral ideal of $\mathcal{O}_K$ coinciding exactly with the primes of bad reduction of $E$). Let $\mathfrak{a} \triangleleft \mathcal{O}_K$ and $K(\mathfrak{a})$ denote the ray class field of $K$ modulo $\mathfrak{a}$.

Given that $\mathfrak{a}$ is prime to $6\mathfrak{f}$ and $E[\mathfrak{af}] \subseteq E(K(\mathfrak{af}))$ and that $\mathrm{Gal}(K(E[\mathfrak{a}])/K) \hookrightarrow (\mathcal{O}_K/\mathfrak{a})^\times$, show that in fact $\mathrm{Gal}(K(E[\mathfrak{a}]/K) \cong (\mathcal{O}_K/\mathfrak{a})^\times$.

I have been struggling to prove this for a long while now. It is effectively the content of Corollary 5.20 of Rubin's paper found here . I don't particularly understand the assertions regarding the kernel and isomorphism that follow the first paragraph in the proof. Does anyone have any ideas? Help with the other assertions of Corollary 5.20 (iii - v) would also be greatly appreciated.

Cheers!

P.S I have crossposted this from MathSE as I have yet to receive an answer there.

Answer 1

Given your assumption that $E/K$ has CM by $\mathcal{O}_K$, it follows that $K$ has class numer $1$. In particular, there are isomorphisms $\mathrm{Gal}(E[\mathfrak{a}])/K\cong\mathrm{Gal}(K(\mathfrak{a})/K)$ as well as $\mathrm{Gal}(K(\mathfrak{a})/K)\cong(\mathcal{O}_K/\mathfrak{a})^\times$, the first coming from the main theorem of Complex Multiplication stating that adding $\mathfrak{a}$-torsion points together with $j(\mathcal{O}_K)$ gives you the ray class field; and the second coming from Class Field Theory, which explicitely describes the difference between ray and Hilbert class fields. The inclusion in your statement needs therefore be an isomorphism.