Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2505.09163 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
Table of Contents:
- 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.