Traces of Hecke Operators via Hypergeometric Character Sums
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909523252346880 |
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| author | Hoffman, Jerome W. Li, Wen-Ching Winnie Long, Ling Tu, Fang-Ting |
| author_facet | Hoffman, Jerome W. Li, Wen-Ching Winnie Long, Ling Tu, Fang-Ting |
| contents | In this paper we obtain explicit formulas for the traces of Hecke operators on spaces of cusp forms in certain instances related to arithmetic triangle groups. These expressions are in terms of hypergeometric character sums over finite fields, a theory developed largely by Greene, Katz, Beukers-Cohen-Mellit, and Fuselier-Long-Ramakrishna-Swisher-Tu. Our approach, in contrast to the previous works, is uniform and more geometric, and it works equally well for forms on elliptic modular curves and Shimura curves. The same method can be applied to obtain eigenvalues of Hecke operators as well. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2408_02918 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Traces of Hecke Operators via Hypergeometric Character Sums Hoffman, Jerome W. Li, Wen-Ching Winnie Long, Ling Tu, Fang-Ting Number Theory 11F11, 11F80, 11L05, 33C20 In this paper we obtain explicit formulas for the traces of Hecke operators on spaces of cusp forms in certain instances related to arithmetic triangle groups. These expressions are in terms of hypergeometric character sums over finite fields, a theory developed largely by Greene, Katz, Beukers-Cohen-Mellit, and Fuselier-Long-Ramakrishna-Swisher-Tu. Our approach, in contrast to the previous works, is uniform and more geometric, and it works equally well for forms on elliptic modular curves and Shimura curves. The same method can be applied to obtain eigenvalues of Hecke operators as well. |
| title | Traces of Hecke Operators via Hypergeometric Character Sums |
| topic | Number Theory 11F11, 11F80, 11L05, 33C20 |
| url | https://arxiv.org/abs/2408.02918 |