Inverse limits of CM points on certain Shimura varieties

TL;DR

The paper constructs inverse systems of CM points on Shimura varieties $Y_1(N)^\pm$ and $Y(N)^\pm$ by restricting to CM points primitive modulo $N$ and analyzes their inverse limits. It shows that $\mathcal{CM}(D,\widetilde{\mathbb{H}}_1^\pm) \cong \mathrm{Gal}(K^{\mathrm{ab}}/\mathbb{Q})$ and $\mathcal{CM}(D,\widetilde{\mathbb{H}}^\pm) \cong \mathrm{Gal}(K^{\mathrm{ab}}(t^{1/\infty})/\mathbb{Q}(t))$, where $K=\mathbb{Q}(\sqrt{D})$ and $t$ is transcendental. This is achieved by connecting CM-point data to definite form class groups $\mathcal{Q}(D,N)^\pm$, establishing bijections with CM points via $\rho$, and identifying the inverse limits with ray class fields $K_{\mathcal{O},N}$ and their transcendental extensions. The results provide an explicit geometric interpretation of class field theory, with an explicit p-adic description for primes $p$ and a decomposition into contributions from all imaginary quadratic fields. Overall, the work links CM theory, Shimura varieties, and explicit Galois actions through inverse limits.

Abstract

Let $N$ be a positive integer, and let $D\equiv0$ 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.

Theorems & Definitions (10)

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