Inverse limits of CM points on certain Shimura varieties

Let $N$ be a positive integer, and let $D\equiv 0$ or $1\text{Mod}4$ be a negative integer. We define the sets $\mathcal{C}\mathcal{M}(D,Y_1(N{)}^{\pm })$ and $\mathcal{C}\mathcal{M}(D,Y(N{)}^{\pm })$ as subsets of the Shimura varieties $Y_1(N{)}^{\pm }$ and $Y(N{)}^{\pm }$, respectively, consisting of CM points of discriminant $D$ that are primitive modulo $N$. By using the theory of definite form class groups, we show that the inverse limits \begin{equation*} \varprojlim_N,$\mathcal{C}\mathcal{M}$(D,,Y_1(N)^\pm)\quad$\text{and}$\quad \varprojlim_N,$\mathcal{C}\mathcal{M}$(D,,Y(N)^\pm) \end{equation*} naturally inherit group structures isomorphic to $\mathrm{G}\mathrm{a}\mathrm{l}(K^{\mathrm{a}\mathrm{b}}\mathrm{/}\mathbb{Q})$ and $\mathrm{G}\mathrm{a}\mathrm{l}(K^{\mathrm{a}\mathrm{b}}(t^{1\mathrm{/}\mathrm{\infty }})\mathrm{/}\mathbb{Q}(t))$, respectively, where $K=\mathbb{Q}(\sqrt{D})$ and $t$ is a transcendental number. These results provide an explicit and geometric interpretation of class field theory in terms of inverse limits of CM points on the associated Shimura varieties.