Integrality of Hecke eigenvalues and the growth of Hecke fields
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866918531508994048 |
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| author | Sakugawa, Kenji Sugiyama, Shingo |
| author_facet | Sakugawa, Kenji Sugiyama, Shingo |
| contents | We prove that Hecke eigenvalues for any Hilbert and Siegel modular forms are algebraic integers. Our method does not rely on cohomologicality nor Galois representations. We apply the integrality of Hecke eigenvalues for Hilbert modular forms of non-parallel weight to the estimation of the growth of Hecke fields of Hilbert cusp forms with non-vanishing central $L$-values. As a further application, we give the growth of the fields of rationality of cuspidal automorphic representations of ${\rm GL}_{2d}(\mathbb{A}_\mathbb{Q})$ for a prime number $d$ with non-vanishing central $L$-values. We also apply the integrality of Hecke eigenvalues for holomorphic Siegel cusp forms of general degree in order to give the growth of the Hecke fields of those forms. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_11716 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Integrality of Hecke eigenvalues and the growth of Hecke fields Sakugawa, Kenji Sugiyama, Shingo Number Theory Primary 11F60, Secondary 11F41, 11F46, 11F75, 11R04 We prove that Hecke eigenvalues for any Hilbert and Siegel modular forms are algebraic integers. Our method does not rely on cohomologicality nor Galois representations. We apply the integrality of Hecke eigenvalues for Hilbert modular forms of non-parallel weight to the estimation of the growth of Hecke fields of Hilbert cusp forms with non-vanishing central $L$-values. As a further application, we give the growth of the fields of rationality of cuspidal automorphic representations of ${\rm GL}_{2d}(\mathbb{A}_\mathbb{Q})$ for a prime number $d$ with non-vanishing central $L$-values. We also apply the integrality of Hecke eigenvalues for holomorphic Siegel cusp forms of general degree in order to give the growth of the Hecke fields of those forms. |
| title | Integrality of Hecke eigenvalues and the growth of Hecke fields |
| topic | Number Theory Primary 11F60, Secondary 11F41, 11F46, 11F75, 11R04 |
| url | https://arxiv.org/abs/2401.11716 |