Action of Atkin--Lehner involution on CM points

Question

In their first paper on Heegner points and derivatives of $L$-series, Gross and Zagier describe the action of Atkin--Lehner involutions on certain CM $\mathbf{C}$-points of the modular curve $X_0(N)$. The following are equivalent descriptions of the CM points considered by Gross--Zagier.

1. The isomorphism class of a cyclic $N$-isogeny $\phi : E_1 \to E_2$ of elliptic curves over $\mathbf{C}$ with $E_1$ and $E_2$ both having CM by a maximal order (hence the same maximal order since they are isogenous). Two cyclic $N$-isogenies $\phi_i : E_i \to E_i'$ for $i \in \{1, 2\}$ are isomorphic if there are $\sigma : E_1 \to E_2$ and $\sigma' : E_1' \to E_2'$ isomorphisms with $\sigma' \circ \phi_1 = \phi_2 \circ \sigma$.

2. A pair $(\mathscr{A}, \mathfrak{n})$ where $\mathscr{A}$ is an ideal class of a maximal imaginary quadratic order $\mathcal{O}$ and $\mathfrak{n}$ is an integral $\mathcal{O}$-ideal with $\mathcal{O}/\mathfrak{n} \cong \mathbf{Z}/N\mathbf{Z}$. To obtain an isogeny from a pair $(\mathscr{A}, \mathfrak{n})$, let $\mathscr{A} = [\mathfrak{a}]$ for $\mathfrak{a}$ a fractional $\mathcal{O}$-ideal and consider the isogeny $\mathbf{C}/\mathfrak{a} \to \mathbf{C}/\mathfrak{a}\mathfrak{n}^{-1}$ that lifts to the identity on the universal cover. Clearly $\mathcal{O} \subseteq \textrm{End}(\mathbf{C}/\mathfrak{a})$ and since $\mathcal{O}$ is maximal and the endomorphism ring is an order, we have equality.

3. An imaginary quadratic $\tau$ in the upper half plane $\mathbf{H}$ generating the maximal order $\mathcal{O}$, up to the action of $\Gamma_0(N)$. Gross--Zagier give formulas that allow one to obtain the corresponding pair $(\mathscr{A}, \mathfrak{n})$, but allow me to not reproduce them here.

I am interested in the action of the Atkin--Lehner involutions $w_d$ on these points, where $d$ is a Hall divisor of $N$, i.e., $d$ divides $N$ and $\gcd(d, N/d) = 1$. Gross--Zagier give the following descriptions of this action corresponding to the descriptions of CM points on $X_0(N)$.

1. Let $\phi : E_1 \to E_2$ be a cyclic $N$-isogeny. The image of $\phi$ under $w_d$ is \begin{align*} E_1/C \to E_1/\ker{\phi} \to E_2 \to E_2/\widehat{C} \end{align*} where $C \subseteq \ker{\phi}$ and $\widehat{C} \subseteq \ker{\widehat{\phi}}$ are the unique subgroups of order $d$, $\widehat{\phi}$ being the dual isogeny.

2. Given a pair $(\mathscr{A}, \mathfrak{n})$, let $\mathfrak{n}'$ by the ideal obtained by changing each prime $\mathfrak{p} \mid (\mathfrak{n}, d)$ to $\mathfrak{p}^h$. Then $w_d$ sends the pair to $(\mathscr{A}[(d, \mathfrak{n})], \mathfrak{n}')$.

3. Let $x, y \in \mathbf{Z}$ with $dx - Ny/d = 1$ and let $w_d$ act on $\tau \in \mathbf{H}$ by the fractional linear transformation $\tau \mapsto (dx\tau + y)/(N\tau + d)$.

In particular, I am considering 3. to be a definition for the action of $w_d$ on the CM points, and I want to know how to prove 1. and 2..

Is there a reference where these things are proved rigorously? I am familiar with the Heegner points on $X_0(N)$ article, but it seems as though proofs are not given.