Siegel modular forms associated to Weil representations (2501.12140v2)

Abstract: We study some explicit Siegel modular forms from Weil representations. For the classical theta group $\Gamma_m(1,2)$ 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 $\operatorname{Ind}{\widetilde{\Gamma}'m(1,2)}^{\widetilde{\operatorname{Sp}}'{2m}(\mathbb{Z})} (1_{\Gamma_m(1,2)} \cdot \operatorname{Id}{\mu_8})^{-1}$, which is related to Igusa's quotient group $\tfrac{\operatorname{Sp}{2m}(\mathbb{Z})}{\Gamma_m(4,8)}$.