# Abelian group extensions

Let $K$ be an imaginary quadratic field and $E$ be an elliptic curve with CM by $\mathcal{O}_K$. Is there a way to see that $K(j(E), h(E[\mathfrak{p}]))/K$ is an Abelian extension for some $\mathfrak{p}$ where $h$ is the Weber function and $j(E)$ is the $j$-invariant, without invoking the fact that it is the ray class field?

I am looking for a proof which is more elementary and probably along the lines of showing that $Gal(K(j(E), h(E[\mathfrak{p}]))/K)$ injects into an Abelian group, sort of like how we prove $K(j(E), E_{tors})/ K(j(E))$ is abelian.

Any proof I have found in literature involves showing that it is the ray class field of $K$ modulo $\mathfrak{p}$ using a characterization involving the splitting of primes.

• Relevant, but in some sense in the opposite direction to your question: mathoverflow.net/questions/100276/… – Asvin Feb 8 '18 at 2:08
• I believe Takagi proved CM with only his existence theorem and in particular, without explicitly knowing the splitting of primes in ray class groups. Perhaps you could look up how he did it. Weber's "Algebra" might also contain something. – Asvin Feb 8 '18 at 2:26