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| Main Author: | |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2304.10696 |
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| _version_ | 1866915404837814272 |
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| author | Zhu, Baiqing |
| author_facet | Zhu, Baiqing |
| contents | We establish the arithmetic Siegel-Weil formula on the modular curve $\mathcal{X}_{0}(N)$ for arbitrary level $N$, i.e., we relate the arithmetic degrees of special cycles on $\mathcal{X}_{0}(N)$ to the derivatives of Fourier coefficients of a genus 2 Eisenstein series. We prove this formula by a precise identity between the local arithmetic intersection numbers on the Rapoport-Zink space associated to $\mathcal{X}_{0}(N)$ and the derivatives of local representation densities of quadratic forms. When $N$ is odd and square-free, this gives a different proof of the main results in [SSY22]. This local identity is proved by relating it to an identity in one dimension higher, but at hyperspecial level. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_10696 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Arithmetic Siegel-Weil formula on $\mathcal{X}_{0}(N)$ Zhu, Baiqing Number Theory We establish the arithmetic Siegel-Weil formula on the modular curve $\mathcal{X}_{0}(N)$ for arbitrary level $N$, i.e., we relate the arithmetic degrees of special cycles on $\mathcal{X}_{0}(N)$ to the derivatives of Fourier coefficients of a genus 2 Eisenstein series. We prove this formula by a precise identity between the local arithmetic intersection numbers on the Rapoport-Zink space associated to $\mathcal{X}_{0}(N)$ and the derivatives of local representation densities of quadratic forms. When $N$ is odd and square-free, this gives a different proof of the main results in [SSY22]. This local identity is proved by relating it to an identity in one dimension higher, but at hyperspecial level. |
| title | Arithmetic Siegel-Weil formula on $\mathcal{X}_{0}(N)$ |
| topic | Number Theory |
| url | https://arxiv.org/abs/2304.10696 |