Let $X_s$ be the compact modular curve of level $\Gamma_0(N)\cap\Gamma_1(p^s)$, with $N\in\mathbb{N}$ and $p$ prime, $(N,p)=1$. Then noncuspidal points on $X_s$ correspond to triples $(E,\frak{n},\pi)$ with $E$ elliptic curve over $\mathbb{C}$, $\frak{n}$ cyclic subgroup of $E$ of order $N$ and $\pi$ point of $E$ of order $p^s$.
Let now $z\in X_s$ correspond to $(E_z,\frak{n}_z,\pi_z)$ with $E_z$ with complex multiplication, and let $K$ be the imaginary quadratic field that contains the order by which $E_z$ has CM. I have heard that Shimura reciprocity law says that:
(a) $z\in X_s(K_{ab})$, where $K_{ab}$ is the maximal abelian extension of $K$.
(b) $z^\sigma=(E_z,\frak{n}_z,\pi_z)^\sigma=$ $(E_z^\sigma,\frak{n}_z^\sigma,\pi_z^\sigma)$ for $\sigma\in Gal(K_{ab}/K)$.
Is this true? Does anyone have a reference for a proof and/or an explanation of this fact?
Searching in the internet, the only real reference is the book "Introduction to the arithmetic theory of automorphic functions" of Shimura, Chapter 6, Theorem 6.31. I have read that chapter, but I really don't get how one can get point (b) from Shimura's theorem.
Any other references that I found online on Shimura reciprocity law don't talk about the Galois action on points of $X_s$, but only about Galois action on modular functions.