An archimedean approach to singular moduli on Shimura curves
Fuente:
arXiv
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| Autor principal: | |
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| Formato: | Preprint |
| Publicado: |
2026
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| Acceso en línea: | |
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| _version_ | 1866908874018127872 |
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| author | Nicolau, Mateo Crabit |
| author_facet | Nicolau, Mateo Crabit |
| contents | We give a new proof of a recent generalization to Shimura curve of genus 0 of the work of Gross and Zagier in `On singular moduli'. This generalization was conjectured by Giampietro and Darmon and proved by Daas by using $p$-adic $Θ$-functions as an analogue of the $j$-invariant. Instead of working $p$-adically, we prove this result by evaluating Green's function at CM points on the Shimura curve. Our strategy is inspired by the analytic proof of Gross--Zagier. We put a special emphasis on both the similarities and the differences with the $p$-adic proof. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_08569 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | An archimedean approach to singular moduli on Shimura curves Nicolau, Mateo Crabit Number Theory We give a new proof of a recent generalization to Shimura curve of genus 0 of the work of Gross and Zagier in `On singular moduli'. This generalization was conjectured by Giampietro and Darmon and proved by Daas by using $p$-adic $Θ$-functions as an analogue of the $j$-invariant. Instead of working $p$-adically, we prove this result by evaluating Green's function at CM points on the Shimura curve. Our strategy is inspired by the analytic proof of Gross--Zagier. We put a special emphasis on both the similarities and the differences with the $p$-adic proof. |
| title | An archimedean approach to singular moduli on Shimura curves |
| topic | Number Theory |
| url | https://arxiv.org/abs/2603.08569 |