Inverse limits of CM points on certain Shimura varieties

Abstract

Let $N$ be a positive integer, and let $D\equiv 0$ or $1\Mod{4}$ be a negative integer. We define the sets $\mathcal{CM}(D,Y_1(N{)}^{\pm })$ and $\mathcal{CM}(D,Y(N{)}^{\pm })$ 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{CM}(D,\,Y_1(N)^\pm)\quad\textrm{and}\quad \varprojlim_N\,\mathcal{CM}(D,\,Y(N)^\pm) \end{equation*} naturally inherit group structures isomorphic to $\mathrm{Gal}(K^{\mathrm{ab}}/\mathbb{Q})$ and $\mathrm{Gal}(K^{\mathrm{ab}}(t^{1/\infty })/\mathbb{Q}(t))$, 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.