Siegel modular forms associated to Weil representations
Chun-Hui Wang
TL;DR
The work addresses constructing and understanding Siegel modular forms of half-integral weight arising from Weil representations via metaplectic central extensions. It develops explicit transformation laws for theta functions $\theta_{1/2}$ and $\theta_{3/2}$ under the theta-group $\Gamma_m(1,2)$ using Perrin–Rao–Kudla–Lion–Vergne cocycles and extends these to $\operatorname{Sp}_{2m}(\mathbb{Z})$ through induced finite-dimensional representations. A key contribution is the explicit coset parametrization of $\Gamma_m(1,2)$ in $\operatorname{Sp}_{2m}(\mathbb{Z})$ and the cocycle modification that yields a representation factoring through the finite Igusa quotient $\operatorname{Sp}_{2m}(\mathbb{Z})/\Gamma_m(4,8)$. The results illuminate the interaction between Weil representations, metaplectic covers, and Siegel modular forms, offering concrete matrix-valued instances and paving the way for automorphic applications and arithmetic interpretations of these central extensions.
Abstract
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 $\tfrac{\operatorname{Sp}_{2m}(\mathbb{Z})}{Γ_m(4,8)}$.