Siegel modular forms associated to Weil representations

2025/01/21 by Chunhui Wang, Wang, Chun-Hui
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Number Theory (math.NT) #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2501.12140

openalex publication_date 2025/01/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 Sp_2m(ℤ) and obtain two matrix-valued Siegel modular forms from Weil representations; these forms arise from a finite-dimensional representation Ind_{\widetildeΓ'm(1,2)}^\widetildeSp'_2m(ℤ) (1_Γm(1,2) ⋅ Id_μ8)^-1, which is related to Igusa's quotient group \tfracSp_2m(ℤ)Γ_m(4,8).

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