Siegel modular forms associated to Weil representations

TL;DR

This paper explores explicit Siegel modular forms derived from Weil representations, extending previous work to the full Siegel group and constructing new matrix-valued forms related to Igusa's quotient group.

Contribution

It introduces two new matrix-valued Siegel modular forms from Weil representations associated with the full Siegel group, expanding the understanding of roots of unity in modular forms.

Findings

Identified eighth roots of unity linked to classical theta groups.

Extended analysis to the full Siegel group $ ext{Sp}_{2m}( ext{Z})$.

Constructed two new matrix-valued Siegel modular forms from 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 $\operatorname{Sp}_{2m}(\mathbb{Z})$ 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 $\tfrac{\operatorname{Sp}_{2m}(\mathbb{Z})}{\Gamma_m(4,8)}$.