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| Main Author: | |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2502.08760 |
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| _version_ | 1866909490457083904 |
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| author | Rosen, Esme |
| author_facet | Rosen, Esme |
| contents | Using the framework relating hypergeometric motives to modular forms, we define an explicit family of weight 2 Hecke eigenforms with complex multiplication. We use the theory of ${}_2F_1(1)$ hypergeometric series and Ramanujan's theory of alternative bases to compute the exact central $L$-value of these Hecke eigenforms in terms of special beta values. We also show the integral Fourier coefficients can be written in terms of Jacobi sums, reflecting a motivic relation between the hypergeometric series and the modular forms. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_08760 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Modular Forms and Certain ${}_2F_1(1)$ Hypergeometric Series Rosen, Esme Number Theory 11F67 Using the framework relating hypergeometric motives to modular forms, we define an explicit family of weight 2 Hecke eigenforms with complex multiplication. We use the theory of ${}_2F_1(1)$ hypergeometric series and Ramanujan's theory of alternative bases to compute the exact central $L$-value of these Hecke eigenforms in terms of special beta values. We also show the integral Fourier coefficients can be written in terms of Jacobi sums, reflecting a motivic relation between the hypergeometric series and the modular forms. |
| title | Modular Forms and Certain ${}_2F_1(1)$ Hypergeometric Series |
| topic | Number Theory 11F67 |
| url | https://arxiv.org/abs/2502.08760 |