Siegel modular forms associated to Weil representations

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 ${\mathrm{Sp}}_{2m}(\mathbb{Z})$ and obtain two matrix-valued Siegel modular forms from Weil representations; these forms arise from a finite-dimensional representation ${\mathrm{Ind}}_{{\tilde{\Gamma }}_m^{\prime }(1,2)}^{{\widetilde{\mathrm{Sp}}}_{2m}^{\prime }(\mathbb{Z})}(1_{{\Gamma }_m(1,2)}\cdot {\mathrm{Id}}_{{\mu }_8}{)}^{-1}$, which is related to Igusa's quotient group $\frac{{\mathrm{Sp}}_{2m}(\mathbb{Z})}{{\Gamma }_m(4,8)}$.

57 pages, correct some mistakes, comments welcome