Algebraicity of Spin $L$-functions for $\mathrm{GSp}_6$
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866916521405579264 |
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| author | Eischen, Ellen Rosso, Giovanni Shah, Shrenik |
| author_facet | Eischen, Ellen Rosso, Giovanni Shah, Shrenik |
| contents | We prove algebraicity of critical values of certain Spin $L$-functions. More precisely, our results concern $L(s, π\otimes χ, Spin)$ for cuspidal automorphic representations $π$ associated to a holomorphic Siegel eigenform on $GSp_6$, real Dirichlet characters $χ$, and critical points $s$ to the right of the center of symmetry. We use the strategy of relating the $L$-values to properties of Eisenstein series, and a significant portion of the paper concerns the Fourier coefficients of these Eisenstein series. Unlike in prior algebraicity results following this strategy, our Eisenstein series are on a group $G$ that has no known moduli problem, and the $L$-functions are related to the Eisenstein series through a non-unique model. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2408_03442 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Algebraicity of Spin $L$-functions for $\mathrm{GSp}_6$ Eischen, Ellen Rosso, Giovanni Shah, Shrenik Number Theory We prove algebraicity of critical values of certain Spin $L$-functions. More precisely, our results concern $L(s, π\otimes χ, Spin)$ for cuspidal automorphic representations $π$ associated to a holomorphic Siegel eigenform on $GSp_6$, real Dirichlet characters $χ$, and critical points $s$ to the right of the center of symmetry. We use the strategy of relating the $L$-values to properties of Eisenstein series, and a significant portion of the paper concerns the Fourier coefficients of these Eisenstein series. Unlike in prior algebraicity results following this strategy, our Eisenstein series are on a group $G$ that has no known moduli problem, and the $L$-functions are related to the Eisenstein series through a non-unique model. |
| title | Algebraicity of Spin $L$-functions for $\mathrm{GSp}_6$ |
| topic | Number Theory |
| url | https://arxiv.org/abs/2408.03442 |