A note on the spaces of Eisenstein series on general congruence subgroups
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866914017508851712 |
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| author | Sahu, Soumyadip |
| author_facet | Sahu, Soumyadip |
| contents | This article proposes a new approach to studying the spectral Eisenstein series of weight $k$ on a congruence subgroup of $\text{SL}_2(\mathbb{Z})$ using Hecke's theory of Eisenstein series for the principal congruence subgroups. Our method provides a gateway to analytic and arithmetic properties of the spectral Eisenstein series using corresponding results for the principal congruence subgroup. We show that the specializations of the weight $k$ spectral Eisenstein series at $s = 0$ give rise to a basis for the space of Eisenstein series on a general congruence subgroup, and the Fourier coefficients of the basis elements lie in a cyclotomic number field. Our philosophy also yields an explicit basis parameterized by cusps for the space of Eisenstein series with a nebentypus character. We utilize the spectral basis for the space of Eisenstein series to provide a simple proof of the Eichler-Shimura isomorphism theorem for the entire space of modular forms. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_10627 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A note on the spaces of Eisenstein series on general congruence subgroups Sahu, Soumyadip Number Theory This article proposes a new approach to studying the spectral Eisenstein series of weight $k$ on a congruence subgroup of $\text{SL}_2(\mathbb{Z})$ using Hecke's theory of Eisenstein series for the principal congruence subgroups. Our method provides a gateway to analytic and arithmetic properties of the spectral Eisenstein series using corresponding results for the principal congruence subgroup. We show that the specializations of the weight $k$ spectral Eisenstein series at $s = 0$ give rise to a basis for the space of Eisenstein series on a general congruence subgroup, and the Fourier coefficients of the basis elements lie in a cyclotomic number field. Our philosophy also yields an explicit basis parameterized by cusps for the space of Eisenstein series with a nebentypus character. We utilize the spectral basis for the space of Eisenstein series to provide a simple proof of the Eichler-Shimura isomorphism theorem for the entire space of modular forms. |
| title | A note on the spaces of Eisenstein series on general congruence subgroups |
| topic | Number Theory |
| url | https://arxiv.org/abs/2312.10627 |