Siegel modular forms associated to Weil representations
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913750949298176 |
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| author | Wang, Chun-Hui |
| author_facet | Wang, Chun-Hui |
| contents | We study some explicit Siegel modular forms from Weil representations. For the classical theta group $Γ_m(1,2)$ with $m > 1$, there are some eighth roots of unity associated with these modular forms, as noted in the works of Andrianov, Friedberg, Maloletkin, Stark, Styer, Richter, and others. We apply $2$-cocycles introduced by Rao, Kudla, Perrin, Lion-Vergne, Satake-Takase to investigate these unities. We extend our study to the full Siegel group $\operatorname{Sp}_{2m}(\mathbb{Z})$ and obtain two matrix-valued Siegel modular forms from Weil representations; these forms arise from a finite-dimensional representation $\operatorname{Ind}_{\widetildeΓ'_m(1,2)}^{\widetilde{\operatorname{Sp}}'_{2m}(\mathbb{Z})} (1_{Γ_m(1,2)} \cdot \operatorname{Id}_{μ_8})^{-1}$, which is related to Igusa's quotient group $\tfrac{\operatorname{Sp}_{2m}(\mathbb{Z})}{Γ_m(4,8)}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_12140 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Siegel modular forms associated to Weil representations Wang, Chun-Hui Number Theory Representation Theory We study some explicit Siegel modular forms from Weil representations. For the classical theta group $Γ_m(1,2)$ with $m > 1$, there are some eighth roots of unity associated with these modular forms, as noted in the works of Andrianov, Friedberg, Maloletkin, Stark, Styer, Richter, and others. We apply $2$-cocycles introduced by Rao, Kudla, Perrin, Lion-Vergne, Satake-Takase to investigate these unities. We extend our study to the full Siegel group $\operatorname{Sp}_{2m}(\mathbb{Z})$ and obtain two matrix-valued Siegel modular forms from Weil representations; these forms arise from a finite-dimensional representation $\operatorname{Ind}_{\widetildeΓ'_m(1,2)}^{\widetilde{\operatorname{Sp}}'_{2m}(\mathbb{Z})} (1_{Γ_m(1,2)} \cdot \operatorname{Id}_{μ_8})^{-1}$, which is related to Igusa's quotient group $\tfrac{\operatorname{Sp}_{2m}(\mathbb{Z})}{Γ_m(4,8)}$. |
| title | Siegel modular forms associated to Weil representations |
| topic | Number Theory Representation Theory |
| url | https://arxiv.org/abs/2501.12140 |