Counting intersection numbers of closed geodesics on Shimura curves
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2021
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| _version_ | 1866909973958623232 |
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| author | Rickards, James |
| author_facet | Rickards, James |
| contents | Let $Γ\subseteq\text{PSL}(2, \mathbb{R})$ correspond to the group of units of norm $1$ in an Eichler order $\mathrm{O}$ of an indefinite quaternion algebra over $\mathbb{Q}$. Closed geodesics on $Γ\backslash\mathbb{H}$ correspond to optimal embeddings of real quadratic orders into $\mathrm{O}$. The weighted intersection numbers of pairs of these closed geodesics conjecturally relates to the work of Darmon-Vonk on a real quadratic analogue to the difference of singular moduli. In this paper, we study the total intersection number over all embeddings of a given pair of discriminants. We precisely describe the arithmetic of each intersection, and produce a formula for the total intersection. This formula is a real quadratic analogue of the work of Gross and Zagier on factorizing the difference of singular moduli. The results are fairly general, allowing for a large class of non-maximal Eichler orders, and non-fundamental/non-coprime discriminants. The paper ends with some explicit examples illustrating the results of the paper. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2104_01968 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Counting intersection numbers of closed geodesics on Shimura curves Rickards, James Number Theory 11R52 (Primary) 11Y40, 16H05 (Secondary) Let $Γ\subseteq\text{PSL}(2, \mathbb{R})$ correspond to the group of units of norm $1$ in an Eichler order $\mathrm{O}$ of an indefinite quaternion algebra over $\mathbb{Q}$. Closed geodesics on $Γ\backslash\mathbb{H}$ correspond to optimal embeddings of real quadratic orders into $\mathrm{O}$. The weighted intersection numbers of pairs of these closed geodesics conjecturally relates to the work of Darmon-Vonk on a real quadratic analogue to the difference of singular moduli. In this paper, we study the total intersection number over all embeddings of a given pair of discriminants. We precisely describe the arithmetic of each intersection, and produce a formula for the total intersection. This formula is a real quadratic analogue of the work of Gross and Zagier on factorizing the difference of singular moduli. The results are fairly general, allowing for a large class of non-maximal Eichler orders, and non-fundamental/non-coprime discriminants. The paper ends with some explicit examples illustrating the results of the paper. |
| title | Counting intersection numbers of closed geodesics on Shimura curves |
| topic | Number Theory 11R52 (Primary) 11Y40, 16H05 (Secondary) |
| url | https://arxiv.org/abs/2104.01968 |