Inverse limits of CM points on certain Shimura varieties
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915287134109696 |
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| author | Jung, Ho Yun Koo, Ja Kyung Shin, Dong Hwa |
| author_facet | Jung, Ho Yun Koo, Ja Kyung Shin, Dong Hwa |
| 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_09163 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Inverse limits of CM points on certain Shimura varieties Jung, Ho Yun Koo, Ja Kyung Shin, Dong Hwa Number Theory 11R37, 11E57, 11G18 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. |
| title | Inverse limits of CM points on certain Shimura varieties |
| topic | Number Theory 11R37, 11E57, 11G18 |
| url | https://arxiv.org/abs/2505.09163 |