# Lang's proof concerning ray class fields of imaginary quadratic number fields

Crosspost from Math.SE as I did not receive an answer there:

In Lang's book Elliptic Functions, he shows how to generate the ray class field with conductor $$N$$ of an imaginary quadratic number field $$k$$ using the $$j$$-invariant of an elliptic curve $$A/\mathbb{C}$$ with $$\mathrm{End}(A)\cong\frak{o}_\text{k}$$ and the values of the Weber function at $$N$$-torsion points of $$A$$. Namely, Theorem 2 of Chapter 10 reads as follows:

Let $$A$$ be an elliptic curve whose ring of endomorphisms is the ring of algebraic integers $$\frak{o}_\text{k}$$ in an imaginary quadratic number field $$k$$, and $$A$$ is defined over $$k(j_A)$$. Let $$h$$ be the Weber function on $$A$$, giving the quotient of $$A$$ by its group of automorphisms. Then $$k(j_A, h(A_N))$$ is the ray class field of $$k$$ with conductor $$N$$.

However, his proof starts out by saying "[l]et $$K$$ be the smallest Galois extension of $$k$$ containing $$j_A=j(\frak{a})$$ and all coordinates $$h(A_N)$$" and then he goes on to prove that $$K$$ is the ray class field of $$k$$ with conductor $$N$$. After that, he concludes that "[t]his proves Theorem 2", without ever mentioning the fact that we have not yet proved $$K=k(j_A, h(A_N))$$, i.e. we do not yet know that $$k(j_A, h(A_N))$$ is Galois. Is there an obvious reason I am missing here?

Notice that he does a similar thing when he proves that the Hilbert class field of $$k$$ is $$k(j(\frak{a}))$$ with $$\frak{a}$$ some fractional ideal of $$k$$: He starts by defining $$K$$ as the smallest Galois extension of $$k$$ containing all $$j(\frak{a}_\text{i})$$, where the $$\frak{a}_\text{i}$$ are a set of representatives for the ideal class group, proves that $$K$$ is the Hilbert class field of $$k$$ and then decides to be done. However, here I can conclude the argument: It was shown in the course of the proof that all $$j(\frak{a}_\text{i})$$ are conjugate, hence $$j(\frak{a})$$ has at least degree $$h_k$$ over $$k$$ and since this is also the degree of the Hilbert class field of $$k$$ over $$k$$, the Hilbert class field must already be $$k(j(\frak{a}))$$. Maybe something similar is possible for the ray class field?

## 1 Answer

I have found out what's going on here and it is so trivial that I wonder why this did not occur to me earlier: The conclusion of the proof is that $$K$$ is the ray class field of $$k$$ modulo $$N$$ - but this means that $$K$$ is abelian over $$k$$, so the intermediate field $$k(j_A, h(A_N))$$ must be Galois over $$k$$ because all intermediate extensions of abelian extensions are Galois! Duh ...