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Let $E$ be an elliptic curve with complex multiplication by an order $\mathcal O$ in an imaginary quadratic field $K$. Let $H=K(j(E))$ and $$L_N=K(j(E),E[N])=H(E[N]).$$

It is not hard to prove that $$\operatorname {Gal}(L_N/H)\hookrightarrow \left(\mathcal O/N\mathcal O\right)^\times.$$

However, we should have an isomorphism here, not only injection. It is possible to prove this without using the general theorems of class field theory?

Note that this is equivalent to my question Zeros of modular functions and automorphisms which asks how to prove that zeros of modular functions of level $N$ are preserved under those automorphisms of the modular function field which come from complex multiplication.

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  • $\begingroup$ The proof is the same as for $N=1$ and it is the fundamental theorem of CM. Everytime you write "It is not hard to prove" you mean "it is hard to prove" ? $\endgroup$
    – reuns
    Commented Sep 19, 2019 at 19:43
  • $\begingroup$ @reuns, that there is an injection is easy (see Silverman, Advanced Topics in the Arithmetic of Elliptic curves, p. 109). What do you mean by the fundamental theorem of CM? $\endgroup$
    – Shimrod
    Commented Sep 19, 2019 at 20:01

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