CM points on Shimura curves via QM-equivariant isogeny volcanoes
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866909422495727616 |
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| author | Saia, Frederick |
| author_facet | Saia, Frederick |
| contents | We study CM points on the Shimura curves $X_0^D(N)_{/\mathbb{Q}}$ and $X_1^D(N)_{/\mathbb{Q}}$, parametrizing abelian surfaces with quaternionic multiplication and extra level structure. A description of the locus of points with CM by a specified order is obtained for general level, via an isogeny-volcano approach in analogy to work of Clark and Clark--Saia in the $D=1$ case of modular curves. This allows for a count of all points with CM by a specified order on such a curve, and a determination of all primitive residue fields and primitive degrees of such points on $X_0^D(N)_{/\mathbb{Q}}$. We leverage computations of least degrees towards the existence of sporadic CM points on $X_0^D(N)_{/\mathbb{Q}}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2212_12635 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | CM points on Shimura curves via QM-equivariant isogeny volcanoes Saia, Frederick Number Theory We study CM points on the Shimura curves $X_0^D(N)_{/\mathbb{Q}}$ and $X_1^D(N)_{/\mathbb{Q}}$, parametrizing abelian surfaces with quaternionic multiplication and extra level structure. A description of the locus of points with CM by a specified order is obtained for general level, via an isogeny-volcano approach in analogy to work of Clark and Clark--Saia in the $D=1$ case of modular curves. This allows for a count of all points with CM by a specified order on such a curve, and a determination of all primitive residue fields and primitive degrees of such points on $X_0^D(N)_{/\mathbb{Q}}$. We leverage computations of least degrees towards the existence of sporadic CM points on $X_0^D(N)_{/\mathbb{Q}}$. |
| title | CM points on Shimura curves via QM-equivariant isogeny volcanoes |
| topic | Number Theory |
| url | https://arxiv.org/abs/2212.12635 |